asymptotic behavior and stability problems in ordinary differential equations || the concept of...

Chapter I

The concept of stability and systems with constant coefficients

§ 1. Some remarks on the concept of stability

1.1. Existence, uniqueness, continuity. We shall often consider systems of n first order normal differential equations

X~ = I.(t, Xl' ... , X,,), i = 1, 2, ... , n, (1.1.1)

where x~=dx.ldt and where t is real and I., Xi real or complex. We shall often denote X = (Xl' ... , X,,) as a "point" or a "vector" and t as the "time", and E" shall denote the space of the points x. Often (1.1.1) is obtained by a transformation of the m second order Lagrange equations relative to a mechanical system with m degrees of freedom, and thus n = 2m.

Sometimes we shall denote also the (n + 1)-tuple (t, Xl' ... , X,,) as a "point", and E,,+! shall denote the space of the points (t, Xl' ... , X,,), or (t, X). We shall be concemed with the behavior of the solutions of (1.1.1) for t ~ to and t-+ 00 (at the right of to), or for t:::;; to and t-+ - 00

(at the left of to) for some given to. The functions I, are supposed to be defined in convenient sets 5 of "points" (t, Xl' ... , X,,). Expressions like open, or closed sets 5, are considered as self-explanatory. We shall say often that a set 5 is open at the right [left] of to if 5 is open when we rf'strict ourselves to points with t:;;;;: to [t::;;; to]' Finally expressions like continuity of the functions I. at a point (t, Xl ... , X,,), or of the functions Xi (t) at a time t do not need explanations. By using vector notations, the system (1.1.1) can be written in the form

X' = I(t, xL (1.1.2)

where x' =dxfdt, and where x, I are the vectors

X=(Xl, .. ·,X,,), x,=x.(t), 1=(11, ... ,/,,), 1.=I,(t,x).

If I does not depend on t, then system (1.1.2) is called autonomous. If I is periodic in t, of some period T, i.e., 1 (t + T, x) = t (t, x) for all t and x, then (1.1.2) is called periodic.

We shall denote by 1/ ul/ = I Ut I + ... + I u" I the norm of any vector u. If to, xo= (xlO ' ... , x"o), b> 0, a> 0, are given, we shall often consider sets 5 (tubes) defined by 5= [to::;; t~ to+a, I/x-xol/:::;;b], or 5= [to:::;; t< + 00, I/x - xol/ :::;;b]. It is only for the sake of simplicity that

Ergebn. d. Mathem. N. F. Bd.16, Cesari, 2. Aufl.

L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations© Springer-Verlag Berlin Heidelberg 1963

2 I. The concept of stability and systems with constant coefficients ( 1 .1)

we have used the norm lIull mentioned above in defining a tube S instead of

lIull = max [Iuli, ... , lu"I], or lIull = (lul I2 + ... + lun I2)l,

for instance. The last one, or Euclidean norm, will be denoted by lIulle •

WeH known theorems of existence and uniqueness hold for system (1.1.2). Wementionhere only a few (see E. KAMKE [1]; S. LEFSCHETZ[2]).

(1.1. i) If 1 (t, x) is continuousin aset S = [to ~t ~to + a, IIx - xoll ~b], then there is a vector function

x = x(t) = x(t; to, xo), to ~ t ~ to +~, IIx(t) - xoll ~ b, (1.1.3)

which is a solution of (1.1.2) in [to, to + al] and satisfies the initial condition x(to; to, xo) = xo, where ~ is some number 0 <al ~a (it may weH occur that the solution x (t; to, xo) has actuaHy a maximal interval of cxistence [to, to+cx] with 0 <cx <al.

The same condition assures also the existence of x (t; to, Xl) in to~t~tO+al' for aH Xl with IIxl-xolI~bl for some ~,bl,c<~~a, 0< bl ~b. It may be pointed out that, if ~ is the maximum number o < ~ ~a, for which IIx (t) - xoll ~b for aH to ~t ~to + ~, then either IIx(aJ - xolI=b, oral =a. If/(t, x) iscontinuousforaHt>to, IIx-xolI ~b, then x(t; to, xo) may exist in [to, + 00), or in some maximal interval [to, to + ~], al < + 00, and then IIx(to + aJ - xoll = b.

A Lipschitz condition of the type II/(t, x2) -/(t, Xl) 11 ~KIlX2 - xIII for aH (t, xJ, (t, xJ E Sand some constant K> 0 assures the uniqueness of the solution x(t; to, xo) and also its continuity with respect to the initial vector xo; that is, given e>O there exists 15 = l5(e; I, xo, aJ >0, O<I5~bl' such that IIxl-xolI~15 implies IIx(t;to,xJ-x(t;to,xo}\l~e for aH to ~ t ~ to + ~ . Thus X (t; to, xo) can be said to be uniformly continuous in [to, to +~] with respect to the initial vector xo.

The same condition assures also the continuity of x(t; to, xo) as a function of t, Xo (and even of to) with analogous restrictions as to t, to, Xo (see E. KAMKE [1]; S. LEFSCHETZ [2]).

More precisely let us mention the following statements.

(1.1. ii) If I(t, x) is continuous in aset S = [to ~t ~to + a, IIx - xoll ~b], and II/(t, Xl) -/(t, xJII ~Kllxl - x2 11 for all (t, Xl), (t, x2) E S, and some constant K, then x(t; to, xo) is uniquely determined by the initial condition x(to)=xo. In addition, if x(t;to,xo) exists in [to,to+~], O<~~a, ~<+oo, and IIx(t)-xolI<b in [to,tO+al], then there is abi, 0 <bI ~b, such that x(t; to, u), u = (uv ... , u,,), exists in the same interval [to, to +~] for all lIu - xoll ~bl' and x (t; to, u) is a continuous function of t and u for to ~t ~to + al and lIu - xoll ~bl'

(1.1. iii) Under the conditions of (1.1. ii), if the components li of 1 have continuous partial derivatives lii=olilox; in S,i,i=1, ... , n, then

t. Some remarks on the concept of stability 3

the components %i of %(t; ' 0 , u) have continuous partial derivatives %ij=8%i/8uj with respect to uj.i.i=1 •...• n. for to~I~Io+lZt. Hu-%oll~ht. If let. %.",) is a continuous function of t. %. and a parameter ",./11.:;;'", :;;''''1' then %(t; tOt u. "') is a continuous function of t. u. ",. Analogous statements hold for the continuity of the partial derivatives of higher order of % provided the components /i of / have corresponding continuous partial derivatives (cf. E. KAMKE [1]. E. A. CODDINGTON and N. LEVINSON [3]).

The Lipschitz condition in (1.1. ii) may be replaced by II/(t. xJlet, %')U~LOIXt-%~1) for all (t. %v. (t. %.)ES. where L(,).0:;;,,.:;;'2b. denotes a continuous function with L(O)=O. L(,»O for 0<,:;;'2b.

111 f d,/L(,) = + 00 (W. F. OSGOOD [1]).

0+ It may be interesting to notice that any condiuon of uniqueness is

also a condition of continuity with respect to the initial values. Also, it is noteworthy that a lower bound for the number lZt in statement (1.1. i) can be given: if II/(t, %)II<M for all O:;;.t:;;'a.II%-%oll:;;.b, then, one may choose lZt;;:;: min [a, Mb-I] (E. KAMKE [1]; S. LEFSCHETZ [2]). If / (t. %) is continuous for all t;;:: to and real vectors %. and a solution %(t; tOt %.) exists only in a maximal interval [10. t). to<I<+ 00. then 1I%(t. to; %.)II~+ 00 as t~l-o. This fact is a consequence of (Li. i) and of the evaluation of lZt given above (E. KAMKE [1]; or A. WINTNER [4]). An example of a system presenting this behavior is given below in (1.3). example 1.

We shall be interested in the existence of solutions %(t; tOt %0) in the whole interval [10. + 00). Linear systems. i.e .• systems of the form

• %; = L aij(t) %j + Mt). i = 1. 2 •...• n. i-I

(1.1.4)

where ai j (t). Mt) are continuous functions of t for t;;:: 10 , ha ve such a property. i.e .• everysolution %(t; tOt %0) existsin [10. + 00) (E. KAMKE [1]) and. by (1.1. ii). is unique.

For system" (1.1.2) conditions for the existence of ~(I; 10, ~o) in the whole interval [10, + 00) have been given by A. WINTNER as follows:

(1. 1. iv) If 1 (~) is a continuous function of ~ for all vectors ~, and 111 (~lIl ~ Kli ~II for some constant K and all ~, then ~(I; 10, ~o) exists in [10, + 00) for all Xo (A. WINTNER [4]).

(1.1. v) If 1 (I, ~) is a continuous function of I and ~ for all I;;:: 10 and ~, if there +00

is a function L(,.) > 0, ,.;;:: 0, with f d,.JL(,.) = + 00, and 11/(1, ~lIL ~Lal~IL) for all I;;:: 10 and ~, then ~(I; 10. ~o) exists in [10, + 00) for all ~o (A. WINTNER [4]).

P"oolol (1.I.v). Let ,.=(~+ ... +~:)l. Then for any solution ~=~(I), '0 ~ 1 < T, we have, by differentiation and manipulation, (,. ,.')1 ~ (~~ + ... + x:) (X;I + ... + X~I), and hence, by (1.1. i), also (,. ,.')I~"'(f~ + ... + I:). Thus (,. ,.')1::;; "IIIC, and tU;;:: Id"IIL(,.). If ~(I; '0 , ~o) exists only in [10, T). k + 00, then 11 X (llIL

1*

4 I. The concept of stability and systems with constant coefficients (1.2)

+00 _ + 00. hence y = y (t) _ + 00. and. by integration. + 00 > t - to ? J drjL(r) =

o + 00. a contradiction. (1.1. v) is thereby proved. hence. also (1.1. iv) is proved.

Condition (1.1. v) can be replaced by the more general one 111 (t. x)IL:::;; m (t) L(lIxIL), where m(t). t:::: to• is any continuous function and L(r) is as in (1.1. v) (R. CONTI [7]). An analogous condition holds for I complex.

In this book we will encounter a number of conditions assuring the existence of x(t; to• xo) in [to• + (0). Let us mention that N. P. ERUGIN [7. 8). M. A. KRASNOSELSKII and M. G. KREIN [2). T. YOSHIZAWA [1). have given other conclitions to the same purpose. Obviously the considerations above concerning t 2 to can be repeated for t ~ 10 and are omitted. This will be the rule in the present book.

According to C. CARATHEODORY [I. pp. 665-688) the concept of a solution of a system of ordinary differential equations can be generalized by postulating that (1.1.1). or (1.1.2). or (1.1.4) by satisfied only for almost all t. provided we restrict ourselves to solutions x (t) which are absolutely continuous (AC). This is essentially due to thc fact that x(t; ' 0 , xo) is actually a solution of the integral equation

t x(/) = x(to) + J I[u. x(u)] du.

t.

The existence. uniquepess. and continuity theorems given above hold also in this new setting. which may allow so me relaxation in the conditions required on I. A condition of uniqueness replacing condition (1.1. ii) as weil as the Osgood condition mentioned abovc is the following one due to L. TONELLI [5]; !I(f (t. xl )-

1 (t. x2)11 :;;; fP (t) L (11 Xl - x2 11l. where fP (t) is an L-integrable function in [to. to+a] and L(r). 0:;;'Y:::;;2b. is a continuous function with L(O) =0. L(r»O forO<r:::;;2b,

2b and J drjL(r) = + 00.

0+ Remark. If 1 (t. x) is a continuous function of t and x for a:::;; t:::;; band all x.

if xn(t). a:::;;t:;;'b. n = 1.2 •.... are solutions of (1.1.1) in [a. b] with 11 xn (t)1I :::;;M. and xn(t) _x(t) as n_oo for all t. a~t~b. then x(t). a:::;;t:::;;b. is a solution of (1.1.1). Indeed I(t • . -r) is continuous in a:;;' t:;;. b. IIxll S; M. and hence 111 (t. x) 11 :::;; N for so me N. Since x~ = I(t. XII)' we conclude that 11 x~ (t) 11 :;;. N for all t and n. and this implies that the functions XII (t). a S; t:;;. b. n = 1. 2. ...• are equicontinuous.

t Thus the convergence xn (t)....,. X (t) is uniform in [a. b]. By xn (t) = xn (a) + J 1 Eu.

. t a x(u)] du. asn_oo. wededucex(/)=x(a)+JI[u.x(u)]du. and finally x'(t) = 1 [t. X (t)). a :::;; t :::;; b. a

1.2. Stability in the sense of LVAPUNOV. We shall now be concerned with the existence of a solution x (t; to, xo) of (1.1.2) for all t;::. to and its behavior as t _ + 00 (asymptotic behavior). There are many properties which may be used in the characterization of such a behavior. We shall mention the Lyapunov stability in the present artic1e, the boundedness in (1.4), and a score of other properties in (1.5).

If I(t, x) is continuous in a set 5 of points (t, x) and 5 is open at the right of to, if a solution x Ct) = x (t; to, xo) of (1.1.2) exists in the infinite interval to:;;;: t< + 00, and [t, x (t)] E 5 for all t 2 to, then x (t; to, xo) is said to be stable (at the right) in the sense 01 Ly APUNOV if (IX) there exists a bl > 0, such that every solution x (t; to, Xl) exists in to:;;;: t< + 00 and

1. So me remarks on the concept of stability 5

[t, X (t) JE 5 for all t ~ to whenever the initial vector Xl satisfies 11 Xl - xoll ;S; b1 ;

(ß) given e> 0, there is a 15 = 15 (e; I, xo), 0< i5 .:s;;:b1 , such that IIxl - xolI;S; 15 implies IIx(t; to, Xl) - x(t; to' xo)l!::;;;e for all fo:s:t< + 00. Thus stability in the sense of Lyapunov turns out to be the same as uniform continuity of x(t; to, xo) in [to, + (0) with respect to the initial vector XO' The same solution X (t; to, xo) is said to be asymptotically stable (at the right) if (IX), (ß) hold, and (ß') there exists a 15 = 15 (j, xo), ° < 15;S; b1 , such that IIxl - xolI.:s;;: 15 implies IIx (t; to, Xl) - X (t; to, xo)ll-+o as t-+ + 00.

If 1 satisfies a Lipschitz condition as in (1.1) in each set 5 = [to;S; t ~ t1 ,

IIx - xoll ::;;;;: b, t1 ;;:;: toJ, with b sufficiently large, then the solution X (t; to, xo) is uniquely determined not only by the point (to, xo) but also by any point (t2 , xJ with X 2 =X(t2), tO;;;;.t2 :;;;.t1 • The continuity theorem we have mentioned in (1.1) assures then that the Lyapunov (asymptotic) stability (or instability) of the solution x(t; to; xo) is not affected by replacing the couple (to, xo) by any other couple (t2 , x2), x2 = X (t2),

to ;;;;. t2 :;;;. t1 •

If we replace the interval [to, + (0) by the interval (- 00, toJ in an previous considerations, then we will define Lyapunov and asymptotic stability at the left. Lyapunov and asymptotic stability at both sides are then self-explanatory. To simplify the notations in the present book, by stability and asymptotic stability we will always mean stability and asymptotic stability in the sense of Ly APUNOV. Also, unless indicated otherwise, we will always understand stability at the right.

If the conditions (IX), (ß), or (ß') above are satisfied only by the solutions X (t; to, xo) of a given manifold M of solutions of (1 1.2), then we have Lyapunov or asymptotic conditional stabilities.

For linear systems .,

x~ = L ail. (t) Xh + I. (t), i = 1, 2, ... , n, (1.2.1) n~l

to .:s;;:t<+ 00, where aih(t),li(t) are continuous functions in [to, + (0), condition (IX) is always satisfied. In addition if one solution X (t) is stable, then an solutions are stable, and we may say that the system (1.2.1) is stable. If one solution is asymptotically stable, then an solutions are asymptotically stable, and we may say that the system (1.2.1) is asymptotically stable.

Indeed, let xo, X be two solutions of (1.2.1) with initial values uo, u at t = to, and Xo + Ax, X + Ax the solutions with initial values Uo + A u, u + A u. Since X o is stable, given e> 0, there is 15 > ° such that 11 A xII < e for allIlA ull < 15, t;;;:. to, and this in turns implies the stability of x. The same holds for asymptotic stability.

According to the definitions above, asymptotic stability implies Lyapunov stability (also, see remark at the end of 6.2).

6 I. The eoncept of stability and systems witb constant eoefficients (1.3)

A nonnal differential equation of the n-Ih order

x<") = / (I, x', ... , X("-l», (1.2.2)

where x(,,) =tlx"/tlr, h=O, 1, ... , n, can be reduced to a system (1.1.1), or (1.1.2), by the standard transfonnation x=x1 , x' = x., ... , X("-l) = x". Hence a solution x(t)=x(I;lo,110'''',11,,-v,10::;;;1<+oo, of (1.2.2), satisfying the initial condition x(,,) (10) = 11", h = 0, 1, ... , n -1, will be said to be stable at lhe right in lhe sense 0/ LVAPUNOV if (IX) there is a b1> 0 such that every solution %(1) = x(t; '0 , iio, ... , ii,,-J exists in [10 , + 00) for every (iio, ... ,ii,,-J with Ilii"-11,,I<b1 , where Eis extended over all h = 0, 1, ... , n -1; (fJ) given 8 >0, there exists a 6, 0< 6 ::;;;b1 , such that Ilii,,-11,,1::;;;6 implies II%(")(I)-x(")(I)I::;;;8 for all 1,10::;;;1<+00. The same solution x (I) shall be said to be asymptotically stable at the right if (IX), (fJ) hold, and (fJ') there exists a 6.,0:S;;60 ::;;;b1 , such that Elii,,-l1,,1 ::;;;60 implies EI%(")(I) - x(")(t)I-O as t_+ 00. Analogous definitions hold for stability at the left, or at both sides.

Finally, any nonnal differential system of order n = ~ + ... + nk,

(1.2·3)

i = 1, 2, ... , k, can be reduced to a system (1.1.1) by a transfonnation analogous to the one used for (1.2.2), and the concept of stability in the sense of LVAPUNOV may be given in a similar way. For reference see A. LVAPUNOV [3].

For nonnonnal systems of differential equations, the questions of existence, uniqueness, and continuity are much more difficult to answer. Nevertheless the d,efinitions of stability given above for systems (1.1.1), (1.2.2), (1.2.3) apply fonnally also to nonnonnal systems, as

~(X1' ... , x"' x~, ... , x~) = 0, i = 1,2, ... , n;

F( ' ("» - O· %,%, ••• ,% - ,

(1.2.4)

(1.2.5)

~(X1' ... , X~"l), x2 , ... , x~"'), ... , x/" ... , x~"tl) = 0, i = 1, 2, ... , k. (1.2.6)

1.3. Examples. t. The solution x = 0 of the equation x' = Xl, X real, is stable neither at the right nor at the left since every real solution x = x. (t - 1 X.)-l with x. > 0 (x.< 0) eeases to exist at 1 = xöl • This shows that eondition (Ct) does not hold in the present case (see illustration).

2. The solution x = - t of the equation !Ir' = 1 - Xl is not stable at the right sinee all (real) solutions x=tanh(I-I.+ k), k=aretanhx •• -1 <xo< + 1. approach +1 as ''''+00. Every solution x=tanh(I-Io+k). as weH as the solution x = + 1. is asymptotieally stable at the right (see illustration).

3. Every solution of the equation !Ir' = 0 is stable at both sides. but asymptotieally stable at neither side. In<{eed x (I; I •• x.) = x. = constant for every I. and 1 x(t;,.. Xl) - x(l; I •• x.lI = 1 Xl - x.1 for every t.

4. Every solution of the equation x' + x = 0 is asymptotically stable at the right and unstable at the left. sinee x (I) = Ce-I. C eonstant. and thus 1 x(t; I •• Xl) -

x(I;I •• x.)I ... o as ''''+00 and "'+00 as , ... -00.

1. Some remarks on tbe concept of stability 7

S. Tbe solution x = 0 of tbe equation x" - x = 0 is conditionally stable at tbe rigbt witb respect to tbe manifold M of tbe solutions of tbe form x = Ce-I.

6. Tbe solution x = 0 of tbe equation x' -lxi = ° is (conditionally) asymptotically stable witb respect to tbe manifold M of tbe solutions x (I) :::;; O.

x 2 x

------~~~~-----t

7. Every nonzero solution of tbe equation x"=-2-l {x2+ (x4+4x'2)i)x is stable neitber at tbe rigbt nor at tbe left. Indeed, every solution bas tbe form x = C sin (cl + d), c, d constants, and tbus for every two solutions xl = cl sin (Cl 1+ dl), xt=c.sin(c.l+dt), witb Cl*O, C2*O, Ca/Cl irrational, we bave limlxl-x.1 = I Cl I + I CI I as 1-+ + 00, as weil as t -+ - 00 . Tbe zero solution is stable at both sides.

1.4. Boundedness. Given a system (1.1.2) of n first order differential equations, a solution x=x(t; to, xo), lo:;;:t< + 00, is said to be bounded at the right if IIxll:;;:M for all t;;;;;, to and some finite M> 0; that is, if I' Ix; (t)1 :;;:M for all t;;;;;,to, i = 1,2, ... , n.

Given a differential equation (1.2.2) of order n, a solution x(t) = x (t; to, 1/0' ... ,1/"-1) will be called bounded at the right up to the derivatives 01 order kif Ix(h)(t)I:;;:M for all t;;;;;,to and h=O,1, ... ,k. The same solution will be called bounded at the right if it is bounded at the right up to the derivatives of order n -1 . Analogous definitions hold for boundedness at the left.

Boundedness and stability are independent concepts as the following examples show.

Example 8. Every solution of tbe equation x' = 1 is of tbe form x = C + t and thus is unbounded at both sides tbougb stable at both sides.

Example 9. Every nonzero solution x(t) of the equation considered in example 7 is bounded at botb sides, tbougb unstable at botb sides.

Nevertheless, boundedness and stability are strictly connected for linear systems, as the following remarks show.

For homogeneous systems

" x: = ~ a; h (t) x/t, i = 1, 2, ... , n, (1.4.1 ) 11=1

where a;/t (t) are continuous functions in [to' + 00), the following elementary theorem holds: The solutions x(t) of (1.4.1) are allstable at the right if and only if they are all bounded at the right (3.9. i).

8 1. The concept of stability and systems with constant coefficients ( 1.5)

For nonhomogeneous linear systems

" x~ = Lai/I (t) Xh + ti (t), i=1,2, ... ,n, (1.4.2) h=l

where ai/I (t), ti (t) are continuous functions in [to, + 00), the fonowing theorem holds: If the solutions x (t) of (1.4.2) are an bounded at the right, they are an stable at the right; if they are an stable at the right and one is known to be bounded at the right, then they are an bounded at the right (3.9. i).

Analogous theorems hold for boundedness and stability at the left. Also analogous theorems hold for n-th order linear equations

x(n) + ~ (t) x(1O-1) + ... + an (t) x = 0, to ;:;:;;: t < + 00,

x(1o) + ~(t) x(1O-1) + ... + an(t) X = t(t), to ;:;:;;: t < + 00,

(1.4·3)

( 1.4.4)

where t(t), ai(t), i=1, 2, ... , n, are continuous functions in [to, + 00)

and where boundedness has to be understood as boundedness up to and including the derivatives of order n - 1.

For equations (1.4.3), where the coefficients ai/.(t) are continuous and bounded in [to, + 00), a nonelementarytheorem of E. ESCLANGON [1J (see also E. LANDAU [1J) states that if Ix(t)I;:;:;;:M for an to:;;;'t< + 00 and some M, then there exists another constant Mi such that IX(h) (t)1 ;;;;'M1 for an to ;;;;;:t<+ 00 and h=O, 1, 2, ... , n-1.

1.5. Other types of requirements and comments. Stability in the sense of LVAPUNOV and boundedness are only two of the possible requirements which can be demanded of a solution x(t) of a system (1.1.1) existing in [to, +00); that is, two of the relevant properties whose presence may or may not be used to characterize the behavior of a solution x (t) of (1.1.1) as t ...... + 00.

First of an we have already mentioned various modifications of the LVA

PUNOV stability; asymptotic stability, conditional stability. If we suppose that an the functions li(t; Xl' ... , xn) in (1.1.1) are defined for t '2;to and an (Xl' ... , Xn )

of a given region R of points (vectors .1'= (Xl' ..• , Xn ) containing .1'0' we may ask whether

as t ...... + 00 for every initial vector xlER and then we will say that .1'(1; 10 , .1'0) is asymptotically stable in the large relalively to R. Of course R may be the whole xl' ... , xn-space (real or complex) and in any case we have to assure that x(t; to, Xl)

exists in [to, + 00) for every xlER. Stability in the large has been recently investigated by A. 1. LURE [7], N. P. ERUGIN [6], M. A. AIZERMAN [1, 2, 3,4], E. A. BAR

BASIN and N. N. KRASOVSKII [1, 2J. We may require stability only for some components xi of x, or for an. For

instance a solution .1'0 (t) of an n-th order differential equation, n :2: 2, may be caned stable, if given e > 0 and to, there is a <5 > 0 such that 1 x (to) - .1'0 (toll:::; <5 implif s 1 x (I) - .1'0 (t) 1 :::; e for an t > 10 , no matter whether an analogous fact holds when the derivatives up to the order n - 1 are taken into consideration as usual.

The same modifications hold for boundedness. For instance we may ask whether the solutions of a differential equation (1.2.1) are bounded, no matter if so me of the derivatives are not (see e.g., A. LVAPUNOV [3]).

1. Some remarks on the concept of stability 9

It may be mentioned here that E. ROUTH in a particular question proposed to denote as stable a solution x (t) of a second order equation x" = F(x, x' t), x real, which (a) exists for all t;;;: to' (b) x(t)~O as t~+oo, and (c) x(t) =0 at infinitely many points t = tn with tn ...... + 00 as n ~ 00 (oscillatory solutions approaching zero as t~+oo).

Given a solution x (t; to' xo) of system (1.1.1), to:;;;' t< + 00, we may require that given e > 0 there exists a IJ > 0 such that for all t ~ to and x with IIx - x(t; to' xo)1I < IJ, the solution x(t; t, x) exists in Ci, + 00) and IIx (t; to, xo)x(t; l,x)lI<e for all ,;;;: t. Then x(t; to' xo) is said to be unilormly stahle in the sense 01 LVAPUNOV at tbe right. Analogous definitions can be given at the left, or for uniform asymptotic stability. The concept of uniform stability was already considered by A. LVAPUNOV [3], and then discussed by K. PERSIDSKII [6], L. ERMOLAEV [1], and others [see (3.9) for uniform stabilityand (6.8), (9.3) for other concepts].

Let us consider the functions D (t) = IIx (t; to, Xl) - x (t; to' Xo) 11, to :::;; t < + 00,

and e (IJ) = SupD (t) where the supremum is taken for all to:;;;, t< + 00 and 11 Xl -xolI:;;;,lJ. In case of Lyapunov stability we have e(IJ) ~o as IJ~O. We may ask wh ether e(IJ) =MIJ, or e(IJ):;;;'M6IX for some M;;;: 1 and O<at:;;;' 1, and all IJ sufficiently small.

In the case of asymptotic stability we have D(t) ~O as t~ + 00 for IIxl-xoli sufficiently small, and we may ask whether D (t) is an infinitesimal of some prescribed type, e.g., D(t) :;;;,Me-IXt, or D(t) <Mt-IX for some M, at > 0 and all to:;;;,t< +00 (exponential, harmonic stability, etc.). We may inquire whether D(t) is integrable in [to' + 00), or more generally whether D (t) ELIX in [to' + 00) for some at >0; i.e., DIX(t) is integrable in [to' +00). Even in the case where D(t) does not approach zero as t ~ + 00, it may occur that D (t) F.LIX for some at > O. In addition an evaluation of the magnitude or growth of D (t) in [to' t] as t ~ + 00,

or of its integral in [to' t], may be of interest. When only real solutions are considered, we may ask whether the components

x. (t) of x (t) remain of constant sign for t large, or if they are oscillatory in caracter, and, if they approach zero, whether they approach zero monot-:mically or not. In this sense the classical and recent oscillation theorems for linear and nonlinear differential equations fall in the frame of the present discussion. Since the word "stability" is often misused, the expression "qualitative theory of differential equations" may be preferred.

The considerations above refer mainly to t ~ + 00, or t -+ - 00 on the real axis, and we will suppose that this is the case most of the times. Nevertheless, t could approach 00 in the complex field, either in any neighborhood of 00, or in some sector, or some other set of the complex plane, and then tbe question of the behavior of the solutions as t~ 00 could be discussed analogously.

Finally it should be pointed out that the transformation t = 1/Z (or others) transforms a neighborhood (real, or complex) of t = 00 into a neighborhood (real, or complex) of Z = 0 for the transformed equation. Should Z = 0 be an "ordinary" or a "regular singular" point for the transformed equation, then Cauchy or Fuchs theories would yield complete information on the behavior of the solutions (cf. § 3 and § 10).

1.6. Stability of equilibrium. The concept of LYAPUNOV'S stability as given in (1.2) was considered long be fore Ly APUNOV in connection with the question of the "mechanicalstability" of a position of equilibrium of a conservative system E with constraints independent of t. If E has m degrees of freedom, if ql, q2' ... , qm is any system of Lagrangian coordinates and V = V(ql' ... , qm) the potential energy of E, then the


behavior of X is described by the system of m second order Lagrangian equations

aL/aq. - (d/dt) (aL/aq:) = o. i = 1. 2 •...• m. (1.6.1)

where L = T - V. and T = T(q1' ...• q".. q~ • ...• q:") is the kinetic energy of X. If (1.6.1) has a constant solution q.=qiO' q~=O. i=1. 2 ..... m. then the solution represents a possible stationary state E of X. or a position of equilibrium. The mechanical stability of E is generally considered as expressed by the Lyapunov stability of the above constant solution of system (1.6.1). By a displacement it is always possible to transfer the equilibrium point to the origin.

It may be mentioned here that by introducing the generalized momenta Pi=aT/aq~ and the Hamiltonian function H=H(q1 ..... q", • ... P1' ...• Pm) defined by H = T + V = 2 T - L = ~ Pi qj - L. equations

.=1 (1.6.1) are reduced to the 2m Hamilton equations

dq./dt = aH/aPi' dP./dt = - aH/aqi. i = 1 ..... m. (1.6.2)

A thwrem of J. L. LAGRANGE [1] assures that a position of equilibrium of a conservative system is stable if V has a minimum there. Conversely. A. Ly APUNOV [3] has proved under restrietions that the same position is unstable if V has no minimum there. All this is connected with the "second method" of A. LYAPUNOV and we will refer briefly to it in § 7.

1.7. Variational systems. Given a system (1.1.1) and a solution x(t; to• xo). to s;.t< + 00. contained in a region S of points (t. x) open at the right of to• then the question of the stability of x(t; to• xo) at the right can always be reduced. at least formally. to the question of the stability at the right of the solution u = 0 of some new system (1.1.1). Indeed. by putting

(1.7.1 )

we transform the solution x = x (t; to• xo) into a solution u = 0 and the system (1.1.1) into the new system

u' = /[t, x(t) + u] -/[t. x(t)] = F(t, u).

where F(t. 0) =0 for every t:2!to. If we suppose that the components li of 1 have partial derivatives lij = Ofi/ax, continuous in Sand we put

a'j(t)=I.j[t.x(t)]. I 1';(t. u) = f.[t. x(t) + u] - f.[t. x (t)] ( )

1.7.2 .: a.dt) "l + ai2~ U 2 + ... + ai,,(t) u,. + Xdt. u). J-1 ..... n. u-("t ..... u,,). tos;.t<+oo.

then the vector X(t. u) of components Xi(t. u) is a continuous function of t and u in a set S of the (t. U 1 • .... u,,)-space. open at the right of

1. Some remarks on the concept of stability 11

t=to and containing the half straight-li ne [t:;:::to• u=OJ. In addition. from (1.7.2) we have X(t. 0) =0 for every t. and. from TAYLOR'S formula. we deduce X(t. u)-+O as I/ul/-+O for every t:;:::to [uniformly in every to ::s;; t ;:;;: It. 11 < + 00 J. System (1.1.1) is transformed in to the system

.. u;=1:a;;(t)u;+X;(t.u). i=1 •...• n.

;=1 (1.7·3)

which admits of the trivial solution u = O. If we suppose that the components I; of 1 have both first and second partial derivatives continuous in S then. by TAYLOR'S formula. we have

X(t.u)/I/ul/-+O as I/ul/-+O.

for every t :;::: 0 [uniformly in every to::S;; t ::s;; t1 • It< + 00 J. If the functions I. are analytic in S with respecf to Xl' .•.• x,. for

every t:;:::to• then we have 00

Xi (t; u1 • ••.• u .. ) = 1: 1: Ps. i, ... I .. (t) u~· u;· ... u! ... h=2 '.+," +i .. =h

where i1 • i2 • •••• i .. ;;::; 0 are all integers. the coefficients Pi. I •... i .. (t) are continuous functions of t. and the series above converge absolutely and uniformly in a neighborhood U of (0. O ..... 0) for every t:;::: to.

In any case the stability of the solution x(t; to• xo) with respect to system (1.1.1) is reduced to the stability of u = 0 with respect to system (1.7.3). The system (1.7.3) is called the variational system relative to the solution x(t; to• xo) of system (1.1.1). Obviously. if system (1.1.1) is linear. then also (1.7.3) is linear [X=:;oJ and the transformation is trivial. If (1.1.1) is nonlinear. then X is not zero. and the system (1.7.3). is nonlinear.

It is natural to associate the linear system ..

u~ = 1: a;;(t) Ui ;=1

(1.7.4)

with system (1.7.3) and this one is said to be the linear variational system relative to x(t; to• xo) and system (1.1.1). It is natural also to ask whether the stability or instability of the solution u = 0 for system (1.7.4) implies stability or instability of the solution u = 0 for system (1.7,3). In general there is no such dose connection between systems (1.7.3) and (1.7.4) as the following examples show.

Exampll!s: 1. The solution x = 0 is stable at either side for the linear equation x' = O. but is unstable at either side for the nonlinear equation x' = x2 (cf. 1.3. no. 1).

2. The solution x = 0 is unstable at the right for the linear equation x'= x since the solutions have all the form x = C eI. On the other hand. the solution x = 0 is asymptotically stable at the right for the nonlinear system x' = x-ei Xl

since all solutions of this equation have the form x = C eI[ 1 + (t) CI (e31 _ 1)] -l. ;;:: O. and hence x -+ 0 as t -+ + 00.


Nevertheless, under mild conditions, the stability and instability of the solution u=O with respect to the linear system (1.7.4) imply stability and instability respectively of the solution u = 0 with respect to the nonlinear system (1.7.3), as A. LVAPUNOV essentially discovered [3 J. We shall give precise theorems along this line in § 6 and § 8.

The generality of the just mentioned statement has led to a definition of stability which is widely used in applied mathematics. A solution x = X o (t), for which the solution u = 0 of the corresponding linear variational system (1.7.4) is stable, is said to be infinitesimally stable.

1.8. Orbital stability. For the sake of simplicity we shall suppose now that the system (1.1.1) is autonomous, i.e., of the form x'=f(x), x=(x1 , ••• , xn). and that a solution x=X(t), - oo<t<+ 00, of it is known which is periodic of some period T, i.e., X(t+T) =X(t) for all t. Since the system is autonomous also x = X(t -y) is a solution for every constant y (phase). In this condition x = X(t) defines a closed path C (orbit) in the x-space En . We shall denote by {x, C} the usual distance of the point x from C (i.e., the infimum of all the distances from x to any point p of Cl. The orbit C is said to be stable provided, given any E> 0, there is a r5 > 0 such that every solution x (t) which passes at some time t = to at a point Xo E En with {xo, C} < r5 has the property {x (t), C} <: 13 for all t ~ to. The same orbit C is said to be asymptotically stable provided it is stable and there is an 130> 0 such that every solution x (t) which passes at some time t = to at a point xoE E with {xo, C} < 130

has the property {x(t), C}~O as t~+ 00. A further more particular case is the asymptotic stability with asymptotic phase for which the additional requirement is made that x(t) - X(t - y) ~ 0 as t~+ 00 for a convenient constant y (phase). For instance, the system of the example 1 of 1.9 presents orbits wh ich are all stable, but not asymptotically stable, according to the above definition. For these definitions see, for instance, E. ROUTH [3J, A. LVAPUNOV [3J, E. CODDINGTON and N. LEVIN

SON [H 1.9. Stability and change of coordinates. Neither Lyapunov stability, nor

boundedness have invariant characters with respect to a general change of coordinates (unknown functions) Xl' x 2 ' ••. , xn ' or of independent variable t. Thus for a mechanical system it may happen that a solution is stable with respect to a given system of Lagrangian coordinates and is unstable with respect to another system.

Examples. 1. The system

x' = - Y (x2 + y2)~ y' = X (x2 + y2)~ has the 2-parameter family of solutions

x = ccos(et + d), y = esin (et + d),

(c, d arbitrary constants). The solution x = 0, y = 0 is stable, all other solutions are unstable. Nevertheless, if new unknowns T, d are introduced by means of the re-lations

x = r cos () , y = r sin () , () = r t + d,

1. Some remarks on the concept of stability

then the given system is transformed into the system

1"= O. d'= o.

whose solutions I' = Ii. d = CI' (Cl' CI constants) are all stable.

13

2. The pendulum equation x" + sin x = 0 has a family of solutions of the form x=csin[tp(c)l+d]. (c, d constants). where tp(e) denotes a function of C which can be expressed (though not easily) in terms of elliptic functions. The solution x = 0 is stable while all other solutions of this family are unstable. The transformation

x = I'sin [tp(l') t + d]. Y = !'Cos [tp(l') t + d]

transforms the given differential equation into the same system ,'= O. d' = 0 above whose solutions are all stable.

3. If a material point P in a given plane IX is attracted by a fixed center Q in IX proportionally to the inverse of the square of its distance from this center. then we may consider the elliptic trajectories described by P which are all unstable (except the equilibrium solution P = Q which is stable). Such instability is due to the fact that the times of revolutions are functions of the axes of the ellipses and thus points p. P' very elose to each other at the time t = 0 but traveling on slightly different ellipses. may find themselves at opposition and thus at great distance from each other in due course of time. Such instability occurs both in cartesian and polar coordinates. Nevertheless the movement of P is stable with respect to the variable

U = T - P (1 + e cos 0)

where p and e are the parameter and the eccentricity of the ellipse described by P (A. LVAPUNov [3]).

4. Any system (1.1.2) which has a system of n first integrals <Pj{x. t) =c •• i = 1 •...• n. that can be solved for Xl' .... xn' X. = tp.(t. c) for tos:. t < + 00. where C = (Cl' .... cn). can be reduced to equilibrium Y' = O. i = 1 ..... n. by the coordinate transformation Yi=<P.(x.t). All the solutions Yj=c •• i=1 ..... n. of the trans-formed system are stable.

1.10. Stability of the m-th order in the sense of G. D. BIRKHOFF. Motivated by astronomical research of the nineteenth century. and following N. POINCARE [6. 7]. G. D. BIRKHOFF has deeply investigated a concept of stability [20. 23] which has some interest in general dynamics and astronomy (e.g .• the three body problem). We shall give here only the formal definition and mention a few properties (see G. D. BIRKHOFF [20]). Let x' = I(x). x = (Xl' .... • "rn). be an autonomous differential system and let (~l' .... x,,) be a point of equilibrium. i.e .• I(~) = O. Such a point is said to be stable 01 the m-th ol'del' if it possesses the following properties:

(A) There exist two constants K. L such that every solution x = x (t; xo. to) with IIxo-xlls:.s satisfies the relation IIx(t;xo.to)-xlls:.Ks for all t with I t - tal s:. Ls-"'.

(B) For all solutions x(t; xo. to) as in (A) and all fixed T and all polynomials P{x) in xl ..... xn • whose terms have degrees :':5. the function P[x(t);to.xo]. [in particular. the components x.(t; to• xo) of x] can be represented in [to - T. to + T] by means of trigonometrical sums

Ao + E(A cos Ä.t + B.sin Äit). IÄ. - Äil :.: Ä > O.

of not more than N + 1 terms with an error s:. Qs"'+s in [to - T. to + T]. (Q. Ä. N constants. s = 1. 2 .... ). Stability of order m implies stability of every ordert ..... m - 1. Property (A) alone is denoted by BIRKHOFF as .. perturbative stability" (of order m); property (B) alone is denoted as .. trigonometric stability" (of order m).

14 I. Tbe concept of stability and systems with constant coefficients (1.11)

Stability of all orders is denoted as complete stability. Perturbative stability (of first. or higher order) is similar. but independent from LVAPUNOV stability at both sides. Nevertheless. perturbative stability of a linear system with constant eoeffi-

" cients. x, = ~ aijx;. is equivalent to the boundness in (- 00. + (0). of all solu-;

tions x. and. henee (1.4). to LVAPUNOV stability at both sides.

If we suppose that the components of I(x). say 1,(x1 • •••• XII)' i = 1 •.... n. are analytic funetions of Xl' ...• X". around (SI' ...• K,,). and we perform an analytic transformation X= tP(y). where tP is also analytic around the transformed point of equilibrium y. then system x' = 1 (x) is transformed into an analogous system y'=g(y). Both perturbative and trigonometrical stabilities are invariant with respect to analytic transformations tP. On these questions see G. D. BIRKHOFF [20. 23. 25].

1.11. A general remark and bibliographica1 notes. There are many different definitions of "stability" which have little to do with one another. though there are intricate connections between them. Eaeh represents a desirable property of a given solution. On the other hand. the property indieated by LVAPUNOV'S definition whieh seems the most natural of all. is not present in many important cases. No known definition is aeceptable in all cases. For an analysis of the eoncept see A. WINTNER [35]. J. J. STOKER [1.4]. G. D. BIRKHOFF [20]. and other expositions.

For eollateral reading on the general questions dealt with in this book the reader is referred to the following weil known text books and reports: A. ANDRONOV and C. E. CHAIKIN [1]. R. BELLMAN [12]. N. BOGOLIUBOV and N. M. KRvLov (S. LEFSCHETZ. ed.) [34], B. V. BULGAKOV [10]; E. A. CODDINGTON and N. LEVINSON [3]. D. GRAFFI [16]. A. L. LURE [8]. E. KAMKE [1]. S. LEFSCHETZ [2. 3]. I. G. MALKIN [22]. L. A. MACCOLL [1]. N. MINORSKV [6]. V. V. NEMVCKII [5]. V. V. NEMVCKIl and V. STEPANOV [1]. G. SANSONE [16]. G. SANSONE and R. CoNTI [2]. V. M. STARZINSKII [5]. J. J. STOKER [1], S. P. STRELKOV [1], A. WINTNER [35]. A. M. LETOV [7]. Yu. A. MITROPOLSKII [8]. I. M. RApOPORT [4]. N. BOGoLIUBOV and Yu. A. MITROPOLSKII [1]. M. A. AIZERMAN [7]. G. N. DUBOSIN [2]. N. G. CETAEV [5]. I. G. MALKIN [9]. W. KAPLAN [3]. N. W. McLACHLAN [5J.

§ 2. Linear systems with constant coefficients 1.1. Matrix notations. By A. B • ... we shall denote mX n matrices [as;J.

[bjj] • ...• i.e. matriees with m rows and n eolumns. whose elements as;' bs;. i = 1 •...• m. i = 1 •...• n. are real or eomplex numbers. By A = B we mean that the matrices are equal. or identieal. i.e. they both have m rows and n eolumns and aij = bij. i = 1, ...• m. i = 1 •... , n. By the zero matrix. or O. we denote the matrix whose elements are all zero. If m = 1. n = 1, the matrix is a number; if m = 1. n > 1. or m> 1. n = 1. the matrix is called a prime. or a vector. If not indieated otherwise we think of a veetor as an n X 1 matrix and denote it by a smal11etter. If m = n the matrix is said to be a square matrix of order n and the terms aH' i = 1 •...• n. are said to form its main diagonal; their sumo t,. A. is ealled the trace of A. If m = n and a, ; = 0 for i =F i. then A is said to be a diagonal matrix; if m = n, ai; = 0 for i =F i. a,; = 1 for i = i. than the matrix is said to be the identy matrix or unit matrix and denoted by A = I = ["0]' By C = A + B. D = a.A = A a. (a. real or eomplex. both A and B mx n matriees) we denote the matriees [Ci ;]. [d.;] defined by Cij=aij+bij. dij=a.a,;=aija.. Finally if Ais an mxp matrix. and B a p X n matrix. by E = AB we denote the mx n matTix [eij] defined by eij = ail bJ ; + ... + aif>bf>;' Equality. addition. and multiplieation. as defined

2. Linear systems with constant coefficients 1S

above, have the weil known formal properties sketched below:

(2.1. i) (a) A = A; (b) A = B implies B = A; (c) A = B, B = C implies A =C.

(2.1.ii) (a) A +B=B+A; (b) (A +B)+C=A +(B+C);

(2.1.iii) (a) a.(A+B)=a.A+a.B; (b) (a.+{J)A=a.A+{JA;

(2.1.iv) (a) (AB)C=A(BC); (b) (A+B)C=AC+BC, C(A+B)= CA +CB.

Given any mx n matrix A = raH]' then the complex conjugate matrix Ä = [äij] of A is the mxn matrix whose elements äij are the complex conjugates of a." and the transpose A_1 = [b.,] of A is the n X m matrix defined by b;; = afj. If m = n, and A = A -1' then A is said to be symmetrie, if A = Ä -1 then A is said to be Hermitian. From these definitions it follows that AB=AB, and (AB)_I= B_1 A_1 •

Any m' X n' matrix B = [airi.]' 1:::;; m'::;;; m, 1:::;;: n':::;;: n, obtained by extracting from an mx n matrix A = [ai i] the m' rows of indices ;1' ... , i",., and the n' columns of indices ;1' .... ; .. " is said to be a minor of the matrix A. If m' = n'. m = n. ir =;r' ,. = 1 •...• n'. then Bis said to be a principal minor of A. If m' = n' = n - 1. m = n. then B can be thought of as obtained from A by suppressing the i-th row and the ;-th column of A and is then denoted by Ai!' or minor of the element aii of A.

By det A = det raH] we shall denote as usual the determinant of the n X n matrix A = [a.,]. If A. B, C. are n X n matrices and C = AB. then detC =detA . detB; and thus detC=o if and only if detA =0. or detB=o. The number IXii = (-1)S+idetA., is said to be the cofactor of aij in A. An nxn matrix B= [bij] = A-l is said to be the inverse of the n X n matrix A = raH] if BA = I. It is weil known that a matrix A has an inverse A-l if and only if detA =t= 0 and then A-l = [bil] is uniquely defined by biS = IXu/detA and A-l A = A A-l = I. Also detA-l = (detA)-I; more generally detA'" = (detA)'" for all integers m:<;: 0 and even for all integers m ~ 0 if detA =t= o. Finally (A B)-1 = B-l A-l for all n X n matrices A. B with detA. detB=t= O. and (A-l)_1 = (A_1)-I. Thus the symbol A:~ is unambiguous.

If A is an n X n matrix. by the rank h of A. 0:::;;: h:::;;: n. we denote the maximum order of its square minors whose determinant is =t= o. By the nullity 11 of A we mean 11 = n - h, O:::;;:,,:::;;:n. Thus" > 0 if and only if detA = o.

The polynomial of degree n,t(e)=det(eI-A)=det[~Ue-aii] is said to be the charactcristic polynomial of A. and

t(e) = e" - Sle,,-l + S.(!"-I1_ ••• ± S".

where Sr denotes the sum of all principal minors of A of order,.. and thus S,. = detA. The equation t(e) = 0 is then the characteristic equation of A and its distinct roots (h, ...• e",. 1 :::;;:m:::;;:n. the characteristic roots of A. For each root er we shall consider the multiplicity Pr. 1:::;;: pr:::;;:n. of er for the equation t(e) = O. and the nullitY"r. 1:::;;:"r:::;;:n. of the matrix erI-A. ,.=1.2 •... , ,n. Obviously Pt + ... +p",=n.

If A = [a.,] is an n X n matrix and P = [P.,] any n X n matrix with det P=t= O. then the n X n matrix B = p-l A P is said to be obtained from A by a similarity transformation and this relation is denoted by B,..., A . Any matrix B obtained from a matrix A by performing a given permutation of its n rows and the same permutation of its columns is certainly similar to A. The following statements are immediately proved:

(2.t.v) (a) A -A; (b) A,...,B implies B,...,A; (c) A ,...,B. B,...,C implies A,...,C;


(2.1. vi) (a) A - B implies det A = det Band A and B have the same characteristic polynomial and the same characteristic roots with the same multiplicities and nullities.

If A=[a.i] is an llXn matrix and Bs=[bl:L], s=1, ... ,N, are nsxn" matrices with n l + ... + nN = n, and a, i = buv whenever i = n l + ... + n S - 1 + U, j =

1l1 +···+ns_ l +v, U, v=1, ... ,ns ' s=1, ... ,N, and aij=ootherwise, then we say that A is the direct sum (cf. S. PERLIS [1]) of the matrices B s and we write A =diag(B1 , B z' ... , B N ). In other words, Ais made up by the "boxes" BI' ... , B N , adjacent to one another along the main diagonal, while all other elements of A are zero. If Ais the direct sum of the matrices BI' ... , B N , then we have detA =

detBl . detBz ..... det B N . Since eI - A is also the direct sum of the matrices eI - B s ' s= 1, ... , N, we also have

det (e I - A) = det (e I - BI) ..... det (e I - B N) ;

in other words, the characteristic polynomial of A is the product of the characteristic polynomials of the matrices B s '

In matrix theory the following theorem is proved concerning the reduction of a matrix to one of its canonical forms (J ORDAN'S) in the complex field by means of similarity transformations.

(2.1. viii) Given an n X n matrix A of characteristic roots er' r = 1, ... , m, with multiplicities P.r and nullities vr' there are complex matrices P with detP=!= 0 such that ] = ~l A P is the direct sum of N matrices Cs = [c~s/J of orders n$ with cii=e;,i=1, ... ,ns' ci,i+l=1,i=1,2, ... ,ns -1, Cij=O otherwise, where e; = er for some r = 1, ... , m. The N matrices Cs are uniquely determined up to an arbitrary permutation.

There are in the theory of matrices various proofs of (2.1. viii). Most of them construct the matrix P in steps, as a product of matrices p,., Pz' ... , so chosen that the matrices Pl-l Ap,., Pz- 1 p t 1 AP1 Pz' ... , assurne more and more the diagonal aspect in the sense of the theorem (processes of diagonalization by similarity transformation). It is possible to proceed in such a way that at each step tbe elements of the matrices PI' Pz' ... , are obtained by solving algebraic linear systems whose determinants are always =!= O.

The matrices Cs are diagonal only if n s = 1. Also, we have in any case det (e 1-Cs) = (e - es)"-, and since the characteristic polynomial of A is the product of the characteristic polynomials of the Cs (2.1. vi), we conclude that for every characteristic root er' of A there must be a number of matrices Cs baving e~ = er in the main diagonal with 1:(r) ns= P.r where 1:(r) ranges over all s with e~ = er' These matrices Cs are called the companion matrices of er (S. PERLIS [1]). On the other hand, for each Cs the matrix e~I - Cs has determinant zero but by suppressing the first row and the last column of e~I - Cs ' (whose elements are all zeros) wc obtain a diagonal matrix whose diagonal elements are all - 1. Thus e~I - Cs has rank ns- 1 and nullity 1. Analogously the matrix erI - ] has dcterminant zero, but by suppressing the rows and columns mentioned above corresponding to the companion matrices of er we get a minor of maximal order n - 1:(') (1) and determinant =!= O. Thus thc nullitY"r of e,I -A is given byv,=1:(r)(1). From thc two equalities

wherc we always have ns :2 1, we conclude as folIows:

(2.1. ix) For each characteristic root er of an n X n matrix A we have Vr ::::;: Pr' i.e., the nullity Vr of the matrix erI - A is always ::;;; the multiplicity Pr of er m thc characteristic polynomial f(e) = det (eI - A).

2. Linear systems with constant coefficients 17

(2.1. x) The matrix A has a diagonal canonical form if and only if '11,= p,. ,,= 1 •...• m; i.e .• if and only if for each characteristic root e, the nullity v, is equaI to the multiplicity Pr'

(2.1. xi) For each characteristic root e, the companion matrices Cs (with e; = e,) have a11 orders n.= 1 Ü and only if v,=p,.

Given a matrix A (I) = [ai; (I)] whose elements are a1l differentiable functions of I. by derivative A'(I) =DA(I) =dA/dt is meant the matrix [ai;(t)].

b Similarly. we may define f A (I) dt. The formal theorems on derivatives and

• integrals hold as usual. say. e.g .• (A +B)'=A'+B'. (AB)'=A'B+AB'. (aA)'= a' A +aA'. If A(I) is an nxn matrix as above with detA * O. then from AA-l=I. by differentiation and manipulation. we may deduce (A-l)'= - A-IA' A-l.

The formula for the derivative of a determinant is better deduced directly .. from the definition of determinant as usual. and reads (detX(I»)'= ~ detXi(f).

t=l where Xi(l) is the matrix obtained from X by replacing the elements of its i-th row (column) by their derivatives.

We shall denote by the norm IIAII of a nxn matrix A = [aO] the sum IIAII = 1:laol of the absolute values of all its elements. Thus. if C = AB. it follows immediately IICII:s;;IIAIIIIBIl. In particular. if ,,=Ax. where x." are n-vectors (nX1 matrices) and A an nxn matrix. we have 1I,,1I:s;;IIAllllxll where IIxll.II,,1I are the norms of the vectors x. ,,(1.1). Also IIcAIl = I ciIiAIl. IIA + BII:s;; IIAII +IIBII. and.

if A(t) is function of I. a:S;; I:S;; b. then II/A(I) dlll:s;;fIIA(t)1I dt.

If A 1. h = O. 1. 2 •...• denote real or complex mx n matrices the concepts of limit and series

00

A = lim All. S =~A1. 1-+00 1=0

are selfexplanatory since they reduce to the mx n limits or series of corresponding elements.

Given areal or complex nXn matrix A. by the matrix eA (exponential matrix of A) is denoted the sum of the series

eA= 1+ (1/t!)A + (t/2!)AB+ (t/3!)A3+ ...•

and this series is certainly convergent since IIA "li ;;;; IIA 11" for all h;;;; 0 and. therefore. each of the n l series components is minorant of the convergent series whose elements are (1/h I) IIA Ir. It is immediately proved that eA eB = eA +B for any two n X n matrices provided A B = BA. Note that. if A,..... B then eA ,.....eB • Indeed. if B = p-l A p. then we have B"= p-l A 1 P for all h ~ 0 and finally eB = P-1eA P. If A = diag (4t ••..• a .. ). then eA = diag ('.' ..... e .... ). If A is the direct sum of matrices BI' ...• B .... then ,A is the direct sum of the matrices eB, • ...• ,B",. Thus if A = ] is the Jordan canonical form of (2.1. viü) then ] is the direct sum of the matrices C. of orders n. discussed there. n1+ ... +n ... =n. and ,J is the direct sum of the matrices eC'. Suppose C.= [Cik] is one of these matrices. say cii=". Ci.i+l= 1. cik=O otherwise. and put C.="l +Z. where Z= [Zik]. zii=O. zi.i+l = 1. zi k = 0 otherwise. By direct computation we see that ZIo= [zJf] is the

matrix with Z~~J+k= 1. z!f=o otherwise. if 1 :S;;h:S;;n.-- 1. For h~ns we lIave Z1= O. i.e .• alllarge powers of Z are zero. Thus the series for eZ reduces to a polynomial expression in Z and we have eZ= [~ik] with ~ii= 1. ~j.i+l = 1/1!. ~j.i+I= 1/2! •...• and ~;k= 0 ü;> 11. Thus eZ is completely defined and we have ,C.= ereZ.

ErgebD. d. Mathem. N. F. Bel. 16. Cesari. 2. Auß. 2

t8 I. The concept of stability and systems with constant coefficients (2.2)

Finally BI is the direct sum of the matrices eC'. Note that. if I is any real or complex number. then CI = 1'tI + ZI. eC'= e"eZ'. and eZ'= [Ci"] with C;;= 1. Cj.Hl = I/t I. Ci.iH = 1'/21 •...• and Cj"= 0 .for;> k. It is important for our purpose to observe inally that if A is any n X n matrix and I real or complex. we have

(d/dl),A' = A ,A'.

This identity can be proved by direct differentiation of the series for BA '.

eA' = 1 + (1/1 I) A + (1/2!) AI + (1/3!) AI + ...•

or by reduction of A to canonical form and the consideration of the matrices eC,1 above. Finally we will need the following theorem:

(2.1. xü) For any real or complex nxn matrix A with detA * 0 there are (infinetely many) complex matrices B with BB=A. and we will denote them as B =lnA.

P1'ool. Suppose first A =J have the canonic form of (2.1. viii). If all ns= t. then J is diagonal. A = diag(At •...• ).,,). Äj* o. ; = t •...• n. and we have B = diag(ln At •...• ln i.,,). Otherwise we may consider Jas the direct sum of the matrices C. of orders ns• and determine B as the direct sum of matrices B. of the same orders n s• For each matrix Cs put C.= l' 1 +Z. Let us observe that for any complex number ".1,,1< 1. we have 1 +" = exp In(1 + ,,). and hence

1 +" = f (1/h!) (f (-1)"k-1yk)". "=0 11=0

and this identity could be verified by actual computations. Thus the same identity holds when " is replaced by the matrix 1'-1 Z. that is. we have

1 + 1'-1Z = f (1/hl) ( f (-1)"k-11'-"Z")". 11=0 11=0

where Z"=O for all k:i::ns. Consequentely. we may assume

and finally

00

In (1 + 1'-1Z) = ~ (-t)"k-1 ,-1IZ". "=0

.. -1 InCs=ln(rI+Z) = (In1')I+~(-1)"k-l1'-"Zk.

10=0

Thus 1nJ is defined as the direct sum of the matrices InC •• s = 1 •...• m. For any n X n matrix A we have A = P J p-l for some matrix P with det P * O. and we may assume B = InA = PlnJ P-l.

2.2. First applications to differential systems. If A = [a, i (t)] denotes an n X n continuous matrix function of t. t ';i:: to• and x (t) = [Xi (t)] a vector function of t. we shall consider the homogeneous linear system

.. x; = L aöj(t) xi'

;=1 i = 1 •...• n. or x' = A x. (2.2.1)

By a fundamental system of solutions X(t) = [Xij(t)]. of (2.2.1) we shall denote an n X n matrix X(t} whose n columns are independent solutions of (2.2.1). Sometimes we may suppose that these n solutions


are determined by the initial conditions

x.;(to) =6.;, i=1, ... ,n, {j=1, ... ,n),

where 6.;=1, or 0, according as i=j, or i=t:=j; i.e., X(to) =1 where I is the unit matrix. Then we have x (t) = X(t) x(to) for every solution x(t) of (2.2.1). Indeed X(t) x(to), as a linear combination of solutions of (2.2.1), is a solution of (2.2.1), and since X(to) x(to) = x (to) , the product X(t) x(to), by the uniqueness theorem (1.1. ii), coincides with x(t). From the formula for the derivative of a determinant we obtain also, as usual, that det X(t) satisfies the first order equation

(d/dt) (detX) = (trA)detX,

and hence we have the jacobi-Liouville formula I

det X(t) = det X(to) exp f (tr A) dt. I.

(2.2.2)

If f(t) = [I. (t), i = 1, ... , nJ denotes any n-vector, we shall consider also the nonhomogeneous linear system ..

x~=~a.;(t)xi+I.(t), i=1, ... ,n, or x'=Ax+l. (2.2.3) ;=1

Then if x(t) is any solution of (2.2.3). y(t) the solution of the homogeneous system y' = A Y determined by the same initial conditions y (to) = x (to). if Y (t) is the fundamental system of solutions of y' = A Y with Y(to) =1, then the following relation holds:

I

x(t) = y(t) + f Y(t) Y-1(ot) I(ot) dot. t.

(2.2.4)

Indeed, the second member verifies (2.2.3), satisfies the same initial conditions as x (t), and thus coincides with x(t) by the uniqueness theorem (1.1. ii).

If A is a constant matrix, and we assume to = O. then Y(t) Y-1 (ot) is the fundamental system of solutions of (2.2.1) determined by the initial conditions Y(t) Y-1 (ot) = I at t = ot. and the same for Y (t - ot) ; hence Y(t) Y-l (ot) = Y(t - ot) and finally (2.2.4) becomes

I

x(t) = y(t) + f Y(t - ot) I(ot) dot, t.

for A a constant matrix and Y(O) = I. 2.3. Systems with constant coefficients. The system of first order

homogeneous linear differential equations ..

x~ = ~ a.; x;, i = 1, ...• n, ;-1

(2.3.1)

can be written in the form x'=A x where A = [al;] is a constant nxn matrix. x = (Xl' ...• x .. ) is an n X 1 matrix, or n-vector, function of t.

2·


and x'=dxldt. If Pis any constant nxn matrix with det P=!=O and Y=(Yl' .•. , Y,.) is an nX1 matrix, or n-vector, function of t, related to xby the formula x=Py, or Y= P-1 X, then (2.3.1) is transformed into the system y'=By, where B=P-IAP. Thus, by (2.1), there are matrices P which transform A into its canonical form J discussed in (2.1), J=diag [Cl' CI' ... , CN ], where each matrix C. of order n. is defined as in (2.1) and~+ ... +nN=n. Forevery s let h=~+ ... +n.-1• Then, if n. = 1, the (h + 1 )-th equation of the system y' = J Y has the form

yi-h = e~YHl' (2·3·2)

If n. > 1, then the equations of indices h + 1, ... , h + n. of the same system have the form

Yi+1 = e~ YH1 + YHI, } I I " YII+2 = e. YII+! + YII+3, .•. , YII+,., = es YIl+""

(2·3·3)

where s = 1,2, ... , N. Each system {2.3.2} has the solution

YHl = ~;'.

Thus a corresponding solution of the system y' = J Y is obtained by putting Yi=O for a1l1$.j$.h, h+2$.j$.n. Each system (2.3.3) has n. independent solutions of the form

Y1I+1 = (P.-l/(rx - 1)!) ~;', YHlI = (F- 8/(rx - 2)!) ~;t, ... , }

YII+« = ~;', Y"+«+1 = ... = Y'" = 0, (2·3.4)

where 17. is one of the integers 17. = 1, 2, ... , n.. The corresponding ns solutions of the system y' = B Y are then obtained by putting Yj = 0 for all 1:;;' i :;;'h' h + n. + 1 $. j $. n. If we denote by Y(t) the matrix of all n solutions of system y' = J Y defined above, we have Y(O) =1, and thus Y{t) is certainly a fundamental system. According to (2.1) system x' = A x has the system of solutions X = eA', and since X(O) = 1 obviously X(t) is a fundamental system of solutions of (2. t). For A = J, Y = el ' is a fundamental system of solutions of y' = J Y, where el ' is the direct sum of the matrices eC". By comparison with (2.1) it is easy to recognize that (2.3.4) is exactly eC". Since the x. [Y.] are linear combinations of the Ys [x,] with constant coefficients, we conclude that system (2.3.1) has all solutions x. bounded in [0, + 00) if and only if the same occurs for the solutions Y of y' = J y, and this occurs if and only if R(e1'):;;'O, "=1, ... , m, and if, for those roots (if anyJ with R(e1') =0 all companion matrices Cs have orders n. = 1. By (2.1) we know that this occurs if and only if Pr = '11,. Thus we conclude as folIows:

(2.3.i) The system x'=Ax has solutions all bounded in [0, +00) if and only if R (e,) ~ 0, r = 1, ... , m, and if, for those roots (if any) with R(er)=O, we have 1-'1'=11,. Also, system x'=Ax has solutions all approaching zero as t_+ 00 if and only if R(e,) <0, ,.=1, "', m.

2. Linear systems with constant coefficients 2t

An n-th order linear homogeneous differential equation with constant coefficients

yC .. ) + ~ yC"-l) + ... + a .. y = 0 (2·3·5)

can be written in the form (2.3.1) by the substitution y = Xl' y' = X 2 , ••• ,

yC .. -l) = X .. and then it yields the linear system , , , ,

Xl = x2 , Xs = x3 • ••• , X .. _l = x .. , X .. = - a .. Xl - •.• - ~ X"'

whose matrix A = [a,;J has a quite typical form. It is an elementary exercise to prove that det (eI -A) =e"+~e .. -l+ ... +a .. and that for every characteristic root e, we have 'P, = 1. A consequence of (2.3. i) is then

(2.3. ii) Equation (2.3.5) has an solutions bounded in [0, + 00) together with an their derivatives if and only if R (e,) ::::;; 0, r = 1, 2, ... , m, and if, for an roots e, (if any) with R (e,) = 0, we have ft, = 1. Also, equation (2.3.5) has an solutions approaching zero (with an their derivatives) if and only if R (e,) < 0, r = 1, 2, ... , m.

We add here the following simple remark conceming system x' = A x. (2.3. iii) If a is any real number a > R (e,). r = 1 •...• m. then there is a constant

C > O. (C = C[A. a]). such that IIx(t)lI::;; Cllx(O)lIexp(at) for every solution Xl') of x'=Ax.

Proof. Let IX = max R (es)' ,,= max ns• and let c > O. be a constant such that t. t • ••.• t"::;;cexp(a-lX)t for all t;;:: o. Such a constant certainly exists since tS/exp(a -IX)I-+O as t-+ + 00. S = O. 1 ....• ". Then. by I exp (estl!::;; exp(1X I). we deduce I y,(t) I< cexp (at) for each element of the matrix Y(t) above. Hence. IIY(t)lI::;;n2cexp(at) for all t;;:: o. Now for every solution x of x'=Ax we have X= PY. y= p-1x. and y(t) = Y(t) y(O) since Y(O) = I. Thus x(t) = Py(t) = PY(t) y(O) = PY(t) P-1x(0) and 11 x (t)1I ::;; IIPII'II Y(t)II'IIp-11lIlx(0)1I::;; Cllx(O)1I exp (at) for some constant C.

2.4. The ROUTH-HuRWITZ and other criteria. The considerations above show that the question of the boundedness in [0. + 00] of an solutions of a system (2.3.1) or an equation (2.3.5) is reduced to a quest ion of algebra. Thus. any condition assuring that the characteristic roots e. have the properties above may be of interest for the problem under discussion. One of the best known conditions is due to E. J. ROUTH [1] and A. HURWITZ [1J.

(2.4. i) If F(z) = z" + ~ Z .. -l + ... + a.. is a polynomial with real coefficients, let D1 = al ,

a l aa a5 aU-l 1 a s a, au-s

D,,= det 0 al aa aU-3 k=2,), ... ,n, 0 1 as au-,

o 0 0 ... a"


with ai = 0 for i> n. If all determinants D" are positive, k = 1, 2, ... , n, then aIl zeros of F(z) have negative real parts.

For instance, if. F(z) = z3+ 11 Zl+ 6z + 6, we have DI = 11, DI = 60, Da= 360, and F(z) has a11 roots with negative real parts.

See for references M. MARDEN [1, p.141]. The same book refers also to analogous conditions for polynomials F(z) with complex coefficients. A more involved condition assuring that the roots of F(z) either have real parts negative, or have real parts zero and are simple (as required by 2.3. ii), has been given by T. VIOLA [1]. The Hurwitz criterion is also a particular case of more comprehensive statements concerning the number of zeros of F(z) whose real parts are above, or below a given number, or between two given numbers. Either theory of residues, or Sturm sequences, are used in the proofs of these statements (M. MAR

DEN [1]).

By the remarks of (1.4) conditions (2.3 i) or (2.3. ii) are also sufficient conditions for the stability in the sense of LYAPUNOV (resp. asymptotic stability) at the right of all solutions of system (2.3.1) [or differential equation (2.3.5)].

For n large the use of the Routh-Hurwitz criterion is impractical, and other equivalent processes replace it quite weIl, namely the very same processes by means of which that criterion is usually proved. We mention here briefly some pertinent statements.

(2.4. ii) A necessary condition in order that the real polynomial F(z)=z"+alz"-l+.··+a .. have all its roots with negative real part~, is that all (real) coefficients ~, ... , a .. are positive.

P1'oof. Indeed. the roots Zl' ...• z .. (each repeated as many times as its multiplicity) are real. or in complex conjugate pair.;. Hence the polynomial F(z) is the product of factors either of the form (z - at - iß) (z - at + iß) = Z2 - 2u + (atZ + pa) = Z2 + az+ b with a> O. b> o. or of the form Z-at = z +a. with a > o. By successive multiplications we nece&sarily obtain a polynomial F(z) which has its coefficients all =t= 0 and positive.

Now let us consider together with F(z), the polynomial G(z) = ~Z"-l + aaz .. -a .. , whose last terms is a .. _Iz, or alS according a.'5 n is even or odd. We may weIl suppose now al , al •...• alS aIl real and positive. Let us perform on F(z), G (z) the usual finite process for the determination of their highest common factor. i.e., determine the Sturmian finite sequence

F=Gdl +/Z ' G=/zdz+/a, /"=/ada+/'.···

where d1 = bl Z + 1, da = baz, da = baz. . . .• are aIl polynomials of the first degree.

(2.4. iii) A necessary and sufficient condition in order that all roots of the real polynomial F(z) have negative real parts is that the numbers b1 , bl' ... a11 be positive.


For instance, if F(z) = z8 + t t Zl + 6z + 6, we have G (z) = t t Zl + 6, and, by successive divisions, we have ~=(t/tt)z+t, d.=(t2t/60)z, da =(tO/11)z, and hence lIt, ba, ba are positive, and F(z) has all roots with negative real parts.

Prool 01 (2.4. iii). Indeed, the numbers b. and HURWITZ' determinants D. are related by formulas usually proved in algebra, namely

b1 = DI l , b. = D~ Dl l , b. = D: D i l Däl , ba = D: D.l D;l, ....

In general b.=D:Di!.lDi~l (see e.g. D. F. LAwDEN [t]; M. MARDEN [1]). Thus the numbers b. are a11 positive if and only if the numbers D. are all positive, and thus (2.4. iii) follows from (2.4. i).

Let us observe finally that the study of F(z) on the imaginary axis of the complex z-plane can be done easily by putting z=iy. Then

F(iy) = A(y) + i B(y) = (a .. - a .. _ II yll + ... ) + i (a .. _1 y - a .. - 3 y3 + ... ) and the zeros of F on the imaginary axis are the common real roots of the two real polypomials A and B. An important theorem of the

w

theory of complex functions states that the number of the zeros of an analytic regular function F(z) within a closed path C(F =t= 0 on C) is given by D/2n where D is the variation of the argument of F(z) along C. In other words, as z describes C, the complex variable w =F(z} describes a closed path r in the w-plane and ID/2nl is the number of times by which r encircles the origin w = O. (In Topology tbis number is ealled the topological index of r with respect to the origin w = 0 (see P. ALExANDRovand H. HOPF [1], p. 462). If C is the path which is the eom-

posite of the half cireumferenee c [c=Rexp (iO), - ~ 5;.05;. -~l, and

the segment s between the points Ri and -Ri then C for large R will eontain all roots with positive real parts. If F(z) has no imaginary root, then F(z} =t=0 on C for large R. On c tbe term z" of F(z) is predominant and henee the variation of the argument along c is nn (1 +0 (R-l). Along s we have arg F(z) = are tan [B(y)/A(y)], and thus a detailed study of this real funetion of y in (- 00, + oo) yields the variation of the argument of w along s, and finally D. Thus we have the following:

(2.4. iv) A neeessary and sufficient condition for asymptotie stability is that F(z) have no pure imaginary roots and that D = o.

For instance if F(z) = z8 + 11 Zl + 6z + 6, we have n = 3, A (y) = 6 - 11 y., B(y)-=6y-yl, and the graphs of A(y), B(y) show that (1) A and B have no


common root, hence F(z) has no purely imaginary root; (2) if co (y) = arc tan (BI A), n Sn

and we assume co (0) =n, then co(-oo) = --, co(+oo} =-. Thus argF(z) 2 2

has variations - 3 n on s, and + 3 n on C; i.e., Q = 0, and F(z} has aJl its roots with negative real part.

For a great number of applications of the methods discussed above (2.4. i, H, iii, iv) see the recent book by D . F. LAWDEN [1] . The method which has lead to (2.4. iv) is closely related to the Nyquist diagram (2.8. vii) .

2.S. Systems of order 2. The considerations of (2.3), (2.4) may be usefully exemplified by the following examples.

a, b, c, d real constants. (2.5.1)

The characteristic equation is e2-(a+d}e+(ad-bc}=0 and (2.5.1), for t -+ + 00 presents the folJowing cases: 1. a + d < 0, a d - bc > 0; roots with nega-

X,

- __ --=~E_----x,

- _--?lIrE---...... - x,

tive real parts (or negative and real), alJ solutions x(t} -+0 as t-+ + 00. 2. a + d < 0, a d - bc = 0; one root zero, one negative and real, all solutions are bounded. 3. a + d< 0, a d - bc < 0 ; two real roots one of wh ich positive, infinitely many solutions unbounded. 4. a + d = 0, ad - bc > 0; both roots purely imaginary, all solutions bounded. 5. a+d=O, ad-bc=O, a, b, c, d not all zero; one double root zero with p, = 2, 17 = 1; infinitely many solutions unbounded. 6. a = b = c = d = 0; one double root zero with p, = 2, 17 = 2, all solutions bounded (and constant) . 7. a + d = 0, ad - bc > 0; two real roots one of which is positive, infinitely many solutions unbounded. 8. a + d > 0; at least one root with real positive part (or real and positive) , infinitely many solutions unbounded.

Another viewpoint in the analysis of the solutions of system {2.5.1} is the following one which has far reaching consequenccs in the discussion of nonlinear systems. Wf'; shall consider the solntions of {2.5.1} as trajectories in the Xl Xz plane, and study their behavior as t -+ + 00. We shall suppose ad - bc '* 0 which excludes zero roots for the characteristic equation. The following cases shall be taken into consideration.

(a) Two real distinct roots of the same sign, say ez< el < 0, or 0 < ~ < es; i.e. {a+d}2-4{ad-bc»o, and a+d<O, or a+d>O. Then system (2.5.1)


is transformable by means of a linear real transformation to the canonical form u'=~u. v' = e,v. whose solutions are u=Aexp(~/). v=Bexp(e,tl. A. B arbitrary constants. These solutions represent the u-axis (A =1= O. B = 0). the v-axis (A = O. B=I= 0) and the curves vJB = (uJA)/hII/. (A =1= O. B=I= 0). If el < ~ < O. then u-+O.v -+0. vJu=(BJA)exp(el-~)t-+O as t-++oo; if O<~<el the same occurs as t -+ - 00. The trajectories are represented in the illustrations where the arrows correspond to increasing values of t. The point (0. 0) is said to be a stable. or unstable node. according as ~. e. < O. or el' e, > o.

(b) One double root e (necessarily real) with I' = 2. v = t; i.e. (a + d)'- 4 (adbe) = O. (and thus e = (a + d)J2 and a - d. b. e not a11 zero. Then system (2.5.1) is transformable by means of a real linear transformation to the canonical system u' = e u. v' = ev + u whose solutions are u = A exp (et). v = (A t + B) exp (et). Thus

Xz

v

v

-1---I--tlo!E'-+-+- - .XI

for A = O. the corresponding solutions form the v-axis. If I! < O. i.e. a + d < O. then u-+O. v .... o vJu .... oo as t-+ + 00; if e > O. i.e .• a + d > O. the same occurs as t -+ - 00 The trajectories (for A =1= o. B =1= 0) cross the u-axis at t = - BJA . The trajectones are represented in the illustrations. The point (0. 0) is said to be a stable. or unstable node according as e < O. or e > o.

(c) One double root e (necessarily real) with I' = 2. v = 2; i.e .• a = d = e. b = e = O.

Then system (2.5.1) has the form Xl = e Xl' X~ = e x, and its solutions are Xl = A exp(e t). x, = Bexp(e t). which form the xeaxis. the xs-axis. and the straight lines XIJXl = BJA . (A =1= O. B =1= 0). If e = a = d < O. then Xl -+0. x,-+O as t -+ + 00.

If e = a = d> O. the same occurs as t -+ - 00 . The trajectorips are represented in the illustrations. and the point (0.0) is said to be a stable. or unstable node according as e < O. or e > O.

(d) Two real distinct roots of different signs. say ~ < 0 < e,; i.e .• (a + d)2_ 4 (ad - be) > O. (a + d) (ad - be) < O. Then. the discussion proceeds as in (a) only now for A =1= o. B=I= O. the curves have the equations (uJA)I1.(vJB)I1' = 1 and u-+O. v -+ 00 as t -+ + 00. The trajectories are represented in the illustrations. and the point (0. 0) is said to be a saddle point.

(e) Two complex conjugate roots e . e = rx. ± iß. rx. =1= 0; i.e .• (a + d)' - 4 (adbe) < O. a + d=l= O. and rx. = (a + d)J2. Then. as we have noticed in (2.1) . system (2.5.1) is transformable. by means of a linear transformation. to the system u' = eu. v' = ev. and each real solution (Xl' XI) of (2.5.1) is a linear combination with conjugate coefficients of complex conjugate solutions u of u' = eu. and v = Ü of v' = ev. Now in the complex u-plane the first equation has solutions of the form fl=Ael1'=me'''e«'e'P'=me«'e'(n+p'J . Thus if u=1'e'8. 1'=1= O. we have 1'=mea.,. 8=n+ßt+2kn. If rx.<0. then 1'-+0 as t-++oo; if rx.>0. the same OCCUlS as


I ~ - 00. In any case (J ~ 00 as I ~ 00. The trajectories are represented in the illustrations. The point (0. 0) is said to be a slahle or unslahle spit'al poinl according as Ot < O. or Ot > o.

(f) Two purely imaginary complex roots e.e=±iP. P*o; i.e .• a+d=o, {JI = ad - bc > o. The discussion proceeds as in (e) only that here t' = constant. i.e .• the trajectories are circles in the complex u-plane. and are ellipses in the Xl x.plane (see illustration). The point (0.0) is said to be a ceniM.

11. x" + 2g x' + I x = o. I. g real constants, (2.5.2)

This equation. by putting Xl = X. XI = x', is reduced to the system x~ = Xl'

X; = -IXl -2gx,; hence a=O. b= 1. c= -I. d= -2g. a+d= -2g. ad-bc=l. The discussion is analogous to the one above. only the cases (b) and (c) are

excluded. The Xl x.-plane. now that Xl = X. X. = X'. is said to be the phase plane. The point (0. 0) is a center if g = 0, I> 0, is a stable (unstable) spiral point if f > gl. g > 0 [g < 0], is a saddle point if I< ga. I< 0, is a stable (unstable) node if l::'g8• g>O[g<O]. Thesolutionsof (2.5.2) are of the form

x=Ae«tsin(yl-Ä) if I>gl.

X = A e'" + B ,.'

X = (A I + B) e«'

if I<gl.

if I=gl.

where A, B. Ä are arbitrary constants. If I>gl then Ot= -go y= (/-g1)i>0; and. as usual. the following terms are used: A amplitude, 1 phase. d= -2nOt/" logarithmic decrement. T=2n/y period, ,,= T-l=y/2n frequency. the nonzero solutions being all oscillatory (5.1). If I<gl. then t'l' t'a= -g± (gl-f)I; if 1= gl. Ot = - g. and the solutions are not oscillatory in either case. All solutions X (t) together with x' (t) approach zero as t ~ + 00 if and only if g > 0, / > 0; are all bounded if and only if g';;! 0, I > O. or g > o. I';;! o.

If X (I) is the displacement of a physical system at the time I. and g';;! O. I> 0, thcn its motion is said to be aperiodic and overcritically damped if g > o. I< gl; aperiodie and critically damped if g > o. 1= gl ; oscillatory and undercritically damped if g > o. I > gl; simply harmonie if g = o. I > o.

It is typical of all linear oscillations that the frequency (" = y/2n above) is a constant. independent of the amplitude A. This independence ceases in general with nonlinear systems.

2.6. Nonhomogeneous systems. If A is a constant matrix and I (t) a vector function. then we shall consider the nonhomogeneous system

x'=Ax+l· (2.6.1 )

(2.6. i) If 1I/(t)II::.ceb'. R(ei) <b, i= 1 •...• n. then also 11 x (t)1I < Neb,. t';;!to' for every solution x(t) of (2.6.1) and some constants N and to.

In particular if I (t) is bounded. and R (ei) < o. i = 1, ...• n. then all X (I) are bounded (t';;! to). Indeed. if y (t) is the solution of the system y' = A Y having the same initial conditions at t = to= O. if Y(t) is the fundamental solution of y' = A Y with Y(oj = I. then ,

X (t) = Y (t) + f Y(t - Ot) I (Ot) dOt. o ,

IIx(tlll::.lly(t)1I + fllY(t - Ot)IIII/(Ot)lIdOt. o


From IIN)!I=:;; Cebt, R(ei) =:;; a <b, i = 1, •.. , n, it follows that lIy(I)II, 11 Y(I) 11 <Me4' for some constant M and 1:0:: 0, and hence

, 11 X (I)!I =:;; M e4' + M C f eA(t-llt) eblltda.

o = M eA' + MC(b - a)-l(ebt - eilt) =:;; M [1 + C(b - a)-l]eb'.

The conclusion of (2.6. i) does not hold necessarily if R (ei) = b for some i, even if for the corresponding roots ei we have Pi = "i' as the following example shows: x~ = XI' x; = - Xl + 2 cos 1 (i.e., y" + y = 2 cos I) whose solutions Xl = I sin 1 + C sin (I-y), xl=lcosl+sinl+Ccos(l-y) are all unbounded. For the case now excluded we mention the statement: (2.6. ii) If the system x' = A X has all its

+00 solutions bounded in [0, + (0), if f 11/(1)11 dl < + 00, then also the solutions of the system x'=Ax+1 are all bounded in [0, +(0). The proof is a modification of the previous one. (See § 3.) Analogous statements hold for a nonhomogeneous differential equation

x( .. ) + a1 x( .. -I) + ... + a .. X = I (t)

with constant coefficients.

2.7. Linear resonance. We have already mentioned that the equation

x"+ kx'+ w2 x = 0, k ~ 0, w> 0, (2.7.1)

has solutions all bounded and oscillatory if kl < 4w2, i.e., k < 2w. We shall nov. consider the nonhomogeneous equation

x"+kx'+w1x=Acosml, k:O::O, W>O, m;<:o, m=l=w, (2.7.2)

and its limiting case

x"+kx'+w2 x=Acoswl, k:O::O. W>O. (2.7·3)

As usual. k X' is said to be the damping force, w2 X the restoring spring force. A cos mt an extern al sinusoidal force (input). If Xl' XB' xa are the solutions of the equations (2.7.1), (2.7.2). (2.7.3) respectively satisfying the initial conditions x (0) ="10' x'(o) ="11' then

xII') = ["10 cos YI + y-1("I1- "Ioa.)sinyl]e«t,

x. (I) = [("10 - A LI cos Ä) cos y I + y-1 ("11 - A LI m si n Ä -

- a. "10 + a. A LI cos Ä) sin y t] elltt + A LI cos (m I - Ä) •

xa(l) = ["Iocosy 1+ y-1 ("11 - a."Io - A k-1) sin y I] elltt + A k-1 w-1 sinco I,

where a. = - r 1 k, y = (w2 - 4-1 k2)l > 0,

cos Ä = (w2 - m2) LI, sin Ä = k mLl, LI = [(w2 - m2)1 + k2 m2]-I.

For m = O. w> O. we have LI = w-I , cosÄ = 1, sinÄ = 0, and

For k = 0, w> O. we have a. = 0, y = w, and


Obviously y .... HU as k -+0 and w/2n is said to be the "natural frequency" for (2.7.2). Now, if k > 0, then 0( < 0, and the terms containing e"-t above approach zero as t -+ + 00 (transient), so that, for large t, we have

%a(t} "" A k-1 w-1 sinwt

(steady state). In other words, the "input" Acosmt generates through (2.7.2) an analogous "output" A .1 cos (mt - A.) of the same frequency, different phase, and amplitude A .1, while the constant input A generates a constant output A w-2•

The ratio

is called the (dimensionless) amplification factor. If ~ = m/w, 'YJ = k/w, we have p=p(~}=[(1_~2)2+'YJ2~2]-~ and p(O)=1, p(+oo)=O for every 'YJ. We have

now dp/d~ = - ~p3 (2~2 + 'YJ2 - 2); hence, if 11;; V2, i.e., k;;;:: V2w, then p(~) is a decreasing function of ~ and 1> P > 0 for all ~ > o. If 'YJ <V2, i.e., k <V2w, then p (~) has a maximum at ~ = (1 - 2-1 'YJ2)i given by p ~ 'YJ .... 1(1-'YJ2/4)-t. It is usually said that for k ;;;;: V2w there is no resonance while for k < V2w there is resonance around the resonance frequency m/2n = w (1 - k2/2w2)!/2n and that, at this frequency, the amplification factor, or resonance factor, is Pres = Wk-1 (1-k2/4w2}-l. Obviously mres-+w, Pros -+ + 00 as k -+ o. In other words, the resonance is most remarkable for small damp-ing coefficient k and m/2n elose to the reso

nance frequency mres!2n, where mres!2n -+w/2n (natural frequency) as k -+ o. The phenomenon considered above, though quite significant, is rarely completely

materialized in any application. In all practical cases the resonance phenomenon is less crude. The deep reason is that all mechanical, or physical system are not linear, but they are more and more similar to linear systems the smaller are their deplacements from their position of equilibrium (see pendulum, elastic spring, elastic string). As soon as the oscillations build up, the system becomes essentially nonlinear, and then the natural frequency is not a constant [end of (2.4)], but a function of the amplitude. Generally the natural frequency changes with the amplitude making the resonance less stringent. See this book in (§ 8) and E. W. BROWN [1].

2.8. Servomechanisms. (a) General considerations. Let u(t) be a (known) function and %(t) another function (unknown) related by the differential equation

(2.8.1)

where au' ... , a", bu' •.. , b,.. denote constants. By using as usual the operator D we have

P(D) % = Q(D) u, (2.8.2)

where P, Q denote polynomials in D. If f (t) = Q(D) u, then any solution % of (2.8.1) is related to the solution y of

P{D) y = 0,

2. Linear systems with constant coefficients

which has the same initial conditions as x at t = 0, by the formula

t x(t) = y(t) + f Y(t - er.) I (er.) der.,

o

29

where Y(t) is the solution of (2.8.3) with Y(O) == ••• = yC"-2)(0) = 0, YC"-l)(O) = 1 (2.2). An analogous formula in terms of u can be obtained as follows for 0:::;: m:::;: n - 1. Let Wo (t) be the particular solution of (2.8.3) which satisfies the conditions Wo (0) =710' ... , WJ,,-l)(o) =71"-1' defined by

ao71o = bm-"+l' a0711 + al 710 = bm_"+I' .", a071"-l + ... + a"-1710 = bm ,

where bi = 0 if i< O. Then we have

as is easy to verify.

t x(t) = y(t) + fWo(t - Ot) u(Ot) dOt

o (2.8.4)

A great number of physical systems are regulated by (2.8.1) and they are often called (linear) "servomechanisms" (in a general sense), where some "input" u (t) (signal, flow, power, etc.) generates, through a device V, an "output" x (t) of the same or of different kind, an:! u(t), x(t), functions of time,

d -"D-z are connecte by relation (2.8.1), though sometimes, in the appli-cations, the coefficients a., b. are never actually determined. The device V may be as simple as the one-mesh electric circuit mentioned below, or as vast and complicated as mentioned in the remark at the beginning of (2.9).

The statements below are well-known for linear problems. The first one is usually called the principle of superposition. The remaining ones are really corollaries of it.

(2.8. i) P(D) x. = Q(D) u.' i = 1, ... , N, implies P(D) (2' x.) = Q(D) (2'u,).

(2.8. ii) P(D) Xo = 0, P(D) Xl = Q(D) u implies P(D) (xo + Xl) = Q(D) u.

(2.8. iii) P(D) x. = Q(D) u, i = 1,2, implies P(D) (Xl - XI) = O.

The wide use of (2.8. i) in all kind of applications (since it allows the separation of various kinds of "inputs ''l, need not be mentioned here. The following statement is also a trivial consequence of (2.8. ii, iii) :

(2.8. iv) If u is.a given function and X a solution of (2.8.1), then xis stable (asymptotically stable) for equation (2.8.1) if and only if the solution y = 0 is stable (asymptotically stable) for equation (2.8.3).

Therefore, the stability of the solutions X of (2.8.1) is reduced to the question of the stability of the solution Y= 0 of (2.8.3) and this in turn to the questions of algebra discussed in (2.4). In a wide range of applications only asymptotic stability is of interest. Indeed plain stability is very labile and may change into instability by the slightest variation of the physical system of which the differential problem under discussion is, often, a very approximate representation. (Some purely imaginary roots may move to the right half of the complex plane and generate instability.) Thus asymptotic stability is essential. We have already seen in (2.4) some basic tools related to the Routh-Hurwitz criterion by means of which the question of stability can be answered. (For applications see, e.g., D. F. LAWDEN [1].) We shall supposP., in the following lines that all roots of the equation aoe" + ... + a" = 0 have negative real parts, and hence all coefficients a. are different from zero and have the same sign.

If u is a given function (input) then the output X depends upon the initial conditions. If Xl' X. are two different solutions of (2.8.1) corresponding to different and in reality, unpredictable initial conditions, then Xl - X. is a solution of (2.8.3)


(by 2.8. iii), and then xl - xl _ 0 as t _ + 00 because of the asymptotic stability (and even I Xl - xal < ce-'" for some constants a > 0, c > 0, because of 2.3. iii). Thus Xl and XI (if they do not approach zero themselves) have the "same behavior" as t_ + 00 (steady state). Thus any particular solution xl of (2.8.1) is valid for the description of the steady state as t _ + 00.

Now we have to describe this steady state (response) in terms of the input u(t) . A solution X (t) of (2.8.1) for m < n is given by

t X (t) = f Wo (t - IX) U (IX) dlX, (2.8.5)

-00

and this integral is absolutely convergent if u is bounded. If u is a constant A,* 0, then a partieular solution Xo of (2.8.1) is the constant B = A b"Ja,. (which may be zero) , and we may denote b"Ja" as an amplification factor (for a constant input). If u is a harmonie oscillation, say u = A sinwt, of frequency W/2:rt, then a solution X of (2.8.1) and (2.8.2) can be easily determined by the use of complex variable theory. Indeed, u=Aexp(iwt), x=Cexp(iwt) verify (2.8.2) provided CP(iw) =A Q(iw) . Hence, if p and (/I denote the modulus and the argument of Q(iw)/P(iw), we have

x=I(Ape'~e'''') =Apsin(wt+ (/I).

In other words, the response X (steady state) of (2.8.2) to the input A sinwt is a harmonie oscillation of the same frequency, amplitude A p, and phase (/I. The number p is then the amplification factor. Both p = P (w), (/I = (/I (w) depend upon w. By interpreting p and (/I as the polar coordinates of an auxiliary E7]-plane, wc have the harmonie response diagram (see the examples below). Frequencies wo' O<wo< +00, at wh ich p(w) has a maximum (if any) are called resonance frequencies.

The complex function Y(iw) = Q(iw)/P(iw) is called the frequency response function relative to equation (2.8.1) i.e., to the corresponding physical system E (a servomechanism) which, under this aspect is called, sometimes, a filter.

For instance, if u(t), x(t) are voltages between the :b(t) I l' 0: ~ Rd

C x(t) terminals ab, cd of the electric circuit of the illustration, ! 1:: j then LCx"+LR-lx'+x=u, or

." (2.8.6)

~ ...... - - - ... --- ---- --------., , I

I l ~x: : x I : I I , I L __ ...... _ _____ __ ______ __ J

where g, 1 are positive constants. Then the frequency response function is Y(iw) = 12 [f2-w2+2igw]- l . A harmonie oscillation u=Asinwt generates the output x = A P sin (w t + (/I) (steady state) with p = 12 [(j2 - ( 2)2 + 4 g2w2r~, tan (/I = _2gw(f2_ W 2)-l. The harmonie response diagram is given in the illustration.

As another example suppose that a quantity, say x, which is being controlled by some regulator is continuously confronted with the controlling quantity u and the difference u - x influences the output x through some device so that (aD2+ bD) x=c(u-x), a, b, c positive constants. Then, through simplifications, we again have the same equation (2.8.6).

For the equation (D + a)' x = a' u we have p = a' (a2 + ( 2)-2, (/I = - 4 are tan (w/a); hence, as w varies from 0 to + 00, the point of polar coordinates (p, (/I)

describes the curve shown in the illustration (harmonie response diagram) .


lb) The weighting function. It is convenient to define the function W(t). - 00 < t < + 00. by W(t) = w.,(t) if t~ 0, W(t) = 0 if t < o. Thus we have, from (2.8.5).

I +00

x (t) = J w., (t - Ot) U (Ot) dOt = J u (t - Ot) w., (Ot) dOt. -00 0

and finally +00

x(t) = J u(t - Ot) W(Ot) dOt. (2.8.7) -00

The function W(t) is called the weighting function of the filter. It is particularly important to consider the rectangular input u (t) = 0 for t < 0 and t > 1fm. u (t) = m

1/". for 0 S; t S; 1fm, m > O. Then we have x (t) = m J W(t - Ot) dOt. A graph of this

o function gives an idea how the filter behaves under a sudden impulse. shows the time lag (b) of the filter, and yields information on the transient.

Finally, we have x (t) ...... W(t) as m ...... + 00. It is usual to consider the Dirac delta function or impulse

+00

function, c5 (t) = 0 for t =t= 0, with J c5 (t) dt = 1, and to -00

conclude that the weighting function W(t) is the response of the filter to the impulse function. In the theory of distributions of L. SCHWARTZ this important point has received rigorous mathematical formulation.

Also, of particular interest, is the case where u (t) u (t) = 0 if t< 0, u (t) = 1 if t ~ O.

is the unit-step function

(c) The Fourier transform. Let us observe that if u(t), -oo<t<+oo, is any periodic function of period T, then we may write u(t) as a Fourier series, say

+00 n2 u (t),..., I: a"e'"wl, where a" = (1fT) J u (Ot) e-'"wOtdOt. We shall expect the output

,,~-oo -TI! to be given by

+00

x (t)....., I: a" Y(i n co) e'"wl. n=-oo

This is actually the case under the usual condition that u (t) is L-integrable in [0, T] and m S; n - 2. and then the series for x (t) above converges uniformly.

If u (t). - 00 < t < + 00, is any nonperiodic function, we may write u (t) as a Fourier integral

+00 +00

u(t),...,(1f2n)JU(ico)e· w1 dco. where U(ico) = Ju(t)r·w1dt. -00 -00

Then we may expect the output to be given by

+00

X(t) ....., (1f2n) J U(i co) Y(i co) (,wl dco -00

+00

and this is really the case under usual conditions on u (t), say J I u (t) I dt < + 00 -00

and m S; n - 2. Hence, if X(w) is the Fourier transform of x (t). we have

X(ico) = U(ico) Y(ico) , or:

(2.8. v) The Fourier transform of the filter output is equal to the Fourier transform of the input multiplied by the frequency response function.

32 I. The concept of stability and systems with constant coefficients

Finally by putting u (t) = e'IDI in (2.8.7), we have

+00 x (t) = e-iID1 f W(Ot) e- iIDoc dOt = eiID1 Y(i w) , or:

-00

(2.8)

(2.8. vi) The frequency response of a stable filter is the Fourier transform of the weighting function

+00 Y(iw) = f W(Ot) e-··'OCdOt.

-00

(d) The Nyquist stability criterion. It should be mentioned first that the frequency response function of a given filter, or servomechanism E, can be determined experimentally, and this is often done without any knowledge of the differential equa~ion (2.8.1) regulating E.

The combination of known systems EI' E. and a comparator LI according to the scheme of the illustration is called a feedback, or closed loop system of which

an example had been already given under (a). A controlled quantity x = x(t) is confronted with the controlling quantity u through a device EI which transforms x (t) into y (t), and the difference u - y produced by the comparator LI generates the output x through a device EI. Thus we will have

A(D) x = B(D) (u - y), P(D) y = Q(D) x,

where A, B, P, Q are polynomials in D, and, by elimination of y,

[A(D) P(D) + B(D) Q(D)]x = B(D) P(D) u.

The frequency response function of the closed loop system is

Z(iw) = B(iw) P(iw)/[A(iw) P(iw) + B(iw) Q(iw)]

and the stability of the closed loop system is assured if all the roots of the equation

A(Ä) P(Ä) + B(Ä) Q(Ä) = 0

have negative real parts, i.e., if the same can be said of the zeros of the function

S(z) = B(z) Q(z) + 1. A(z)P(z)

Now, by suppressing in E the comparator LI and the connection between LI and EI' we have an "open loop" system Eo regulated by A(D)x=B(D)u, and P(D) y = Q(D) x; hence by

A (D) P(D) Y = B (D) Q (D) u.


The response function of the open loop system 2'0 is Yo(iw) = B (iw) Q (iw)/A (iw) P (iw) = 5 (iw) - 1 and the stability of 2'0 depends on the equation A (,1.) P (,1.) = o.

If 2'0 is known to be stable, then A (,1.) P (,1.) = 0 has all its roots with negative real parts, hence 5 (,1.) has no poles on or at the right of the imaginary axis. If we consider again the c10sed path C of (2.4) in the complex z-plane (see illustration) we have another application of the theorem of the theory of complex functions mentioned there. Let r be the image of C in the complex w-plane, where w = Yo(z), i.e., the image of C under the open loop response function ZOo Then the number of roots of the equation 5 (z) = 0 within C is equal to the number of times r encirc1es the point w = - 1 [provided no root of 5 (z) is on CJ. For large R this number is equal to the number of roots of 5 (z) to the right of the imaginary axis. The following conc1usion can be drawn:

(2.8. vii) (NYQUIST'S stability criterion.) A sufficient condition in order that the c10sed loop 2' is stable is that the open loop 2'0 is stable and tha image r of the path C under the open loop response function does not encirc1e w = - 1 for large R.

2.9. Bibliographical notes. The tTemendous growth of the general theory of servomechanisms is typical of the last twenty years. The regulators of the movements of large telescopes controlled by the feeble light of a distant star, the regulators of an automatically run chemical plant, tbe feedback electric systems containing tubes, the nerve cell, are only a few examples of the applications. The considerations of (2.8) intend only to give a glimpse of the first elements of the theory of servomechanisms and to establish a bridge. For the theory we refer to the excellent books: L. A. MACCOLL [1J, H. M. JAMES, N. B. NICHOLS, and R. S. PHILLIPS [1J, and to G. S. BROWN and D. P. CAMPBELL [1], D. F. LAwDEN [1], R. OLDENBURG [1].

For extension of the theory of servomechanisms to the case of more than one controlling and controlled quantity, see the quoted books, and, e.g., the papers M. GOLOMB and E. USDIN [1], F. H. RAYMOND [1], A. M. POPOVSKII [1]. We should also mention, in this respect, the research of B. V. BULGAKOV [7, 9] concerning consistency and bounds for the solutions of linear differential systems of the form

.. .. L f;j(D) x; = L giP(D) up, i = 1, ... , n.

;=1 P=1

Differential-difference equations and systems (systems with time lags) have a very large literature and we mention here the c1assical papers of O. HILB [1], C. LOVE [1], and the more recent ones D. R. HARTREE [1]. A. D. MYSKIS [1-9], N. D. HAYES [1], L. A. PIPES [1], and H. SPÄTH [1], R. BELLMAN [13J, E. PINNEY [3J. The question of the stability of the solutions, in particular, is discussed in TA LI [1], A. ANDRONOV and A. MAYER [3]. Extensions to the non linear case will be mentioned in § 8.

From the very large amount of recent literature on linear servomechanisms we mention A. ANDRONOV and A. MAYER [1], B. FOGAGNOLO [1]. E. GRÜNWALD [1J, B. V. BULGAKOV [2, 3J, N. F. BARBER and F. URSELL [1]. Z. S. BLOH [1]. The last author has determined conditions in order that the response function to the unit step function be monotone.

In connection with stability of servomechanisms M. V. MEEROV [1] has discussed conditions under which the polynomial P (z) + m Q (z) = 0 has roots with negative real parts when both P(z), Q(z) are known to bave this property.

Ergebn. d. Mathem. N. F. Bd.16, Cesari, 2. Aufl. 3

asymptotic behavior and stability problems in ordinary differential equations || the concept of...

Documents