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CHAPTER 5

Lagrange and Bolza Problems of Optimal Control and Other Problems

As we have shown in Section 1.9, Lagrange, Bolza, and Mayer problems can be essentially transformed one into the other by simple changes of space variables. Thus, the necessary condition and the other statements of Chapter 4 for Mayer problems have their counterpart for Lagrange and Bolza problems. We present here briefly the main statements for the Bolza problems for easy reference, those for Lagrange problems being the same (with 9 = 0).

5.1 The Necessary Condition for Bolza and Lagrange Problems of Optimal Control

We consider here a Bolza problem concerning the minimum of the functional

(5.1.1) g,/o scalars,

with x(t) = (Xl, ... ,x"), u(t) = (ul , ... ,u"'), satisfying the differential system

(5.1.2) dx/dt = f(t, x(t), u(t)), f = (flo ... ,/.),

the boundary conditions (tl' X(tl), t2 , x(t2)) e B C R 2n + 2, and the constraints (t, x(t))e A, u(t) e U.

Here all n + 1 scalar functions Ji(t, X, u), i = 0, 1, ... , n, are assumed to be defined and continuous in the set M of all (t, X, u) with (t, x) e A, u e U(t) (or M = A x U if U is a fixed set), together with their partial derivatives fit, Jix j , i = 0, 1, ... , n. Also, 9 is a real valued function of class C l , and 9 == 0 for Lagrange problems. The class Q of the admissible pairs x(t), u(t), tl ~ t ~ t2 , is defined as in Section 4.1, though we must now require explicitly that fo(t, x(t), u(t)) be integrable in [tl' t2J. As in Section 4.1, we assume that x(t), u(t), tl ~ t ~ t 2 , is an optimal pair and we need not repeat the assumptions (a)-(d) of Section 4.1 for the situation under consideration. In particular, the graph

196

L. Cesari, Optimization—Theory and Applications© Springer-Verlag New York Inc. 1983

5.1 The Necessary Condition for Bolza and Lagrange Problems 197

of the optimal trajectory x is in the interior of A, U is a fixed set (not necessarily bounded), and the optimal strategy u is bounded. (Alternatively, assumptions for the case of u unbounded and for other variants can be made.)

We shall need now an (n + 1)-vector A = (Ao,A1> ... ,An), and the Hamiltonian function

(5.1.3) H(t,x, u, A) = Ao!o{t, x, u) + Ad1(t, x, u) + ... + A.fn(t, x, u).

Also, we shall need the related function

(5.1.4) M(t, x, A) = inf H(t, x, u, I). NEV

The necessary condition for Lagrange problems now takes the form below, which we shall deduce explicitly from (4.2.i) in Section 5.2. A direct proof is indicated in Section 7.31.

S.Li (THEOREM). Under the hypotheses listed above, let x(t) = (xl, ... , xn), u(t) =

(ut, ... ,u"'), t1 ~ t ~ t2, be an optimal pair, that is, an admissible pair x, u such that I[ x, u J ~ I[ X, uJ for all pairs X, u of the class Q of all admissible pairs. Then the optimal pair x, u has the following properties:

(P1') There is an absolutely continuous vector function I(t) = (Ao, A1, ... , An), t1 ~ t ~ t2 (multipliers), which is never zero in [t1> t2J, with Ao a constant in [t1> t2J, Ao ~ 0, such that

dAJdt = -HAt,x(t),u(t),I(t)), i = 1, ... , n, t E [t1, t 2J (a.e.).

(P2') For every fixed t in [t1, t 2J (a.e.), the Hamiltonian H(t,x(t), U, A(t)) as a function of u only (with u in U) takes its minimum value in U at u = u(t):

M(t, x(t), A(t)) = H(t, x(t), u(t), A(t)), t E [t 1, t2J (a.e.).

(P3') The function M(t) = M(t, x(t), i(t)) is absolutely continuous in [t 1, tlJ (more specifically, M(t) coincides a.e. in [t1> t2] with an AC function), and

dM/dt = (d/dt)M(t, x(t), A(t), u(t))

= H,(t, x(t), u(t), A(t)), t E [t1> t 2J (a. e.).

(P4') Transversality relation: n n

Aodg - M(t1)dt1 + I Aj(t1)dx{ + M(tl)dtl - I Aitl)dx~ = 0 j=l j=l

for every (2n + 2)-vector h = (dtbdX1>dtl,dxl) E B', that is,

(5.1.5) Aodg + [ M(t)dt - J1 Aj(t)dxI = O.

The transversality relation is identically satisfied if t1, Xl, tl , X2 are fixed, that is, for the boundary conditions which correspond to the case that both end points and times are fixed (dt1 = dXi1 = dt2 = dx~ = 0, i = 1, ... , n). For Lagrange problems of course g == O,dg == O.

Here x, u is an admissible pair itself, so that the differential equations

(5.1.6) dxijdt = j;(t, x(t), u(t)), i = 1, ... , n,

198 Chapter 5 Lagrange and Bolza Problems of Optimal Control

hold, and these equations, together with (PI), yield the canonical equations

(5.1.7) dxi oH dAi vH dt = VA;' dt - oxi' i = 1, ... , n.

These same equations as well as (P3) can also be written in the explicit forms

dxi/dt = Ji(t, x(t), u(t», i = 1, ... , n,

• dAddt = - L Ait)./jAt, x(t), u(t», i = 1, ... , n,

j=O

If we denote by XO an auxiliary variable satisfying the differential equation dxo/dt = fo(t, x, u) and initial condition XO(t1) = 0, then

I[x,u] = f,/l fodt = XO(t2) - x°(td = XO(t2), I,

and the two equations

can be added to the canonical equations (5.1.7), since VH/OAo = fo and H"o = 0, and hence Ao is a constant, as stated.

Finally, let us note that for autonomous problems, that is, for problems where fo, flo ... ,f. do not depend on t, but only on x and u, we have Jil = oJi/ot = 0. Hence, (P3') yields dM/dt = 0, and the function M(t) is a constant in [t1' t2]. In particular, for autonomous problems between two fixed points Xl = (xL i = 1, ... ,n) and X2 =

(x~, i = 1, ... , n), in an undetermined time, then we can take tl = 0, dxi = dx~ = 0, i = 1, ... , n, dt2 arbitrary. Hence, (P4') yields M(t2)dt2 = 0, or M(t2) = 0, and M(t) is the constant zero in [t1' t2]. This occurs, for instance, in a problem of minimum time (between two fixed points), where fo = 1, and then I = t2 (see Section 5.3A).

Remark 1. There are cases where we can guarantee that Ao oF 0, and hence Ao > ° and we can take AO = 1. For instance, if U = R"', m ~ n, and the m x n matrix [Jiu.(t, x(t), u(t», i = 1, ... , n, s = 1, ... , m] has rank n, then AO oF 0. Indeed, for the property minimum of H we need that, along x(t), u(t), we have AoftlUo + D= 1 AiJiuo = 0, s = 1, ... , m, and Ao = ° would imply either that the n vectors Jius' s = 1, ... , m, are linearly dependent, or that Ao = 0, Al = ... = A. = 0, in both cases a contradiction. Forinstance, ifU = R"', m ~ n, and allJi, i = 1, ... , n, are linear in u, then/; = Ai(t,x) + L~=1 B;.(t,x)u', and we require that the m x n matrix [Bi.(t,x(t))] have rank n along x(t). This is exactly what· occurs for Lagrange problems of the calculus of variations treated as problems of optimal control, that is, m = n, U = R', f = u, i.e., Ji = ui,

i = 1, ... , n, Ai = 0, B;. = !5;., det[ Bi.] = 1. On the same question of AO > 0, cf. Remark in Section 7.4E for problems written in the equivalent Mayer form.

Remark 2. As for Mayer problems in Section 4.3, we briefly state here a particular existence theorem for Lagrange and Mayer problems of optimal control. This theorem will be contained in the more general statements we shall proved in Chapter 9 and in

5.2 Derivation of Properties (Pl')-(P4') from (Pl)-(P4) 199

the even more general statements in Chapters 11-16. Thus, we are concerned here with a functional I[ x, u] of the form (5.1.1) and we state a theorem guaranteeing the existence of an absolute minimum of I[ x, u] in the class Q, we assume not empty, of all pairs x(t), u(t), tl ~ t ~ t2, x AC, u measurable, satisfying (5.1.2) and the constraints (t,x(t)) E A c R n+ 1, u(t) E U c Rm, and (t I, x(t l ), t2, x(t2)) E B C R 2n+ 2. We shall need the sets Q(t, x) = [(ZO, z)1 ZO :? fo(t, x, u), z = f(t, x, u), U E U] c Rn+ I.

S.l.ii (FILIPPOY'S EXISTENCE THEOREM). If A and U are compact, B is closed, fo, fare continuous on M = A x U, 9 is continuous on B, Q is not empty, and for every (t, x) E A the set Q(t, x) is convex, then I[ x, u] has an absolute minimum in Q.

If A is not compact, but closed and contained in a slab to ~ t ~ T, x ERn, to, T finite, then the statement still holds if the conditions (b), (c) of Section 4.3 hold.

If A is contained in no slab as above, but A is closed, then the statement still holds if (b), (c) hold, and in addition if (d') g(t l , XI' t2, x2) -> + 00 as t2 - tl -> + 00 uniformly with respect to XI and x2, and for (tl, XI' t2, x2) E B; and (dO) there are numbers J.1 > 0, M 0 :? 0 such that fo(t, x, u) :? J.1 for all It I :? M 0 (t, x) E A, u E U. Actually, if x, Ii is a known admissible pair for which I has a value I[x, Ii] = m, then all we need to know instead of (d) is that there is some M such that for all admissible pairs x, u with t2 - tl > M we have I[x,u] > m.

Example

Take n = 1, m = 1, fo = 1 + X4 + u4, differential equation x' = f = tx + t2 + tu + u, U = [ -1 ~ u ~ 1], tl = 0, x(O) = 1, x(t2) = 0, t2 undetermined, t2 :? 0, 9 = O. For every (t,x) the set Q(t,x)=[(zO,z)lz°:?I+x4+u4,z=tx+t2+tu+u,-I~u~l] is certainly convex since f is linear in u and fo is convex in u. We note that u = -1, x = 1 - t, 0 ~ t ~ 1, is an admissible pair, with I = Il, and thus Q is certainly not empty. Moreover, since j~ :? 1, any admissible pair with t2 > Il gives to I a value larger than V and can be disregarded. Therefore, we can limit ourselves to admissible pairs with 0 ~ t2 ~ V. With this remark, if Ao = [0 ~ t ~ ¥], then A = Ao x R, B = (0) x (1) x Ao x (0), and thus A and B are closed, while U is compact. Condition (b) is trivially satisfied since the initial point (0, 1) is fixed. Condition (c) is satisfied since f is linear in x, and both t and t2 + tu + u are continuous and bounded in A. All requirements of Theorem (5.l.ii) are satisfied. The integral has an absolute minimum under the constraints.

5.2 Derivation of Properties (PI')-(P4') from (PI)-(P4)

We shall now deduce (P1')-(P4') from the analogous relations (P1)-(P4) of Section 4.2. To this purpose let us first transform the Bolza problem above into a Mayer problem. As mentioned in Section 1.9, we introduce an auxiliary variable xn+ I, the extra differential equation dxn+ I/dt = j~(t, x(t), u(t)), and the extra boundary conditionxn+ I(td = O.


Thus

We denote by x the (n + I)-vector x = (Xl, ... ,xn,xn+1), and by j(t,x,u) the (n + 1)vector function l(t, x, u) = (f1' ... ,f..!o). We have now the Mayer problem of the minimum of the functional

with differential equations

dxi/dt = f;(t, x, u), i = I, ... , n,

boundary conditions (t1' x(t1), t2 ,X(t2)) E B, Xn+1(t1) = 0, and the same constraints as before: (t,x(t)) E A, u(t) E U, where A = A x R. The boundary conditions can be expressed in the equivalent form (t1' x(t1), t2 , x(t2)) E ii, where ii is the closed subset of the space R2n +4 defined by ii = B x (x~+ 1 = 0) X (X2+ 1 E R), since we have assigned the fixed value X~+l = 0 for xn+1(t1), and we have left undetermined the value X2+ 1 = xft + l(t2). Thus, in any case, we have dx~+ 1 = 0, and dX2+ 1 is arbitrary. In other words, the new set ii' is the set ii' = B' x (0) x R, since dX2+ 1 is arbitrary. According to Section 4.2, we need now an (n + I)-vector A = (AI' ... ,An, Aft + I)' the Hamiltonian function

H(t, x, u, A) = Ad1 + ... + Aftfn + An+ do,

and the related function

M(t,x,1) = inf H(t,x,u,X), ueU

where we have written x instead of x, since these functions do not depend on xn+1. Also, we shall need a nonnegative constant A.o ~ O.

We are now in a position to write explicitly statements (Pl)-(P4) of (4.2.i), for the so obtained Mayer problem, where n + 1 replaces n of (4.2.i). The following remarks are relevant. First, from (PI) of (4.2.i), we obtain for the multiplier An+ 1 the equation dAn+1/dt = -Hxn+l' where now Hxn+1 = 0; hence dAn + l/dt = O,andAft + l(t) is a constant in [tl' t2]. Secondly,let us write (P4) for the new problem, namely,

[ n+ 1 J2

Ao[dg + dX2+1] + M(t)dt - i~l Ai(t)dxi 1 = 0,

or

where X~+l = 0, dx~+1 = 0, dX2+1 arbitrary. Hence Aft + 1 = Ao ~ 0, a constant, so that H as defined above reduces to H as defined by (5.1.3), and M as defined above reduces to M as defined by (5.1.4). Then, relation dM/dt = H, reduces to dM/dt = H" that is, (P3'). The transversality relation becomes now

Aodg + [ M(t)dt - itl Ai(t)dxT = O.

We have derived relations (Pl')-(P4') from (Pl)-(P4).

5.3 Applications of the Necessary Conditions for Lagrange Problems

5.3 Examples of Applications of the Necessary Conditions for Lagrange Problems of Optimal Control

A. The Example of Section 4.2B

201

The example discussed in Section 4.2B can be written as the Lagrange problem of the minimum of the integral

J.'2 I[x,y,u] = dt=t2 -t1 "

(where we can take t1 = 0), with differential equations, constraints, and boundary conditions

dx/dt=y, dy/dt=u, uEU=[-lsusl],

x(t1) = a, y(t1) = b, x(t2) = 0, y(t2) = 0.

Then Jo = 1, J1 = y, J2 = u, and the Hamiltonian is H = Ao + A1Y + A2U, with Ao a constant. The analysis is now the same as for the example in Section 4.2B.

Note that here the Hamiltonian is H = Ao + A1y + A2U, with Ao ~ ° constant, and A1, A2 satisfying dAddt = 0, dA 2/dt = -A1; hence A1 = c1, A2 = -c1t + C2, C1, C2

constant, as for the example in Section 4.2B. Here again H, = 0, dM/dt = 0, and M is a constant. On the other hand, 9 = 0, dt1 = 0, dX 1 = dX 2 = dY1 = dY2 = 0, dt2 arbitrary, and (P4') yields M(t2) dt2 = 0, or M(t2) = 0.

B. Exercises

1. Write the problem of the curve C:x = x(t), t1 S t S t2, of minimum length between two points 1 = (t1' Xl), 2 = (t2, x 2), t1 < t2 (or between a point and a curve) as a Lagrange problem of optimal control, and use the necessary conditions of this chapter to obtain the results in Section 3.1.

2. In analogy with Section 4.4A, write the transversality relations in finite form for the Bolzaproblem whenBisdefined by k - 1 equations1>/t lo x 1,t2 ,X2) = O,j = 2, ... ,k, and I[ X, u] = 1>1(t1, Xl' t2, x 2) + S:: Jo dt.

3. In analogy with 4.4C(c), derive the transversality relations in finite form for the Lagrange problem with first end point 1 = (t 1,X1) fixed, second end point X 2 on a given set B in Rn (target), when B has to be reached at some undetermined time t2 > t1. Case (el): B = Rn is the whole x-space. Case (c2): B is a given curve in R", B: X = b(r), r' s r s r", r a parameter, b(r) = (b 1, ... , bn), and X hits B at a time t2 > t1, X 2 = x(t2) = b(ro), r' < ro < r". Case (c3): B is a given k-dimensional manifold in R", B:x = b(r), b(r) = (b 1, ... ,bn), r = (r1,'" ,rk), X = x(t2) = b(ro), t2 > tlo ro = (r 10, ... , rkO) interior to the region of definition of b(r).

Ans.: (cl) M(t2) = 0, A;(t2) = 0, j = 1, ... , n; (c2) M(t2) = 0, D~ 1 A)t2)bj(ro) = 0; (c3) M(t 2 ) = 0, D= 1 AAt2 )bjr.(ro) = 0, s = 1, ... , k.


4. The same as Exercise 3, case (c3), where B is given by n - k equations flo(x) = 0, II = 1, ... , n - k. (Note that M(t2) = 0, and that LJ= 1 Ait2) dx~ = 0 for all dX 2 = (dx1, ... , dx~) satisfying D= 1 flaxAx 2) dx~ = 0, II = 1, ... , n - k.)

5. In analogy with Section 4.4D derive the transversality relations in finite form for the Lagrange problem, with first end point 1 = (tbX1) fixed, terminal time t2 > t1 fixed, and second end point X 2 on a given set B in R2 (target). Same cases as (c 1-3) above.

Ans.: Same as in Exercises 3 and 4 with no conclusion on M(t2).

5.4 The Value Function

We consider here the Bolza problem in Section 5.1 for the case in which any given fixed point (tbX1) in a set V has to be transferred to a target set Be V by means of an admissible trajectory x(t), t1 ~ t ~ t2, in V, and corresponding control function u(t), for which the functional

l[x,u] = g(t2,X(t2)) + it2 fo(t,x(t),u(t))dt Jt,

has its minimum value under the same requirements as in Section 5.1. In other words, 9 is here a given function of (t, x) on B, and for the AC trajectory x and the measurable control function u we require as usual u(t) E U, x(t 1) = Xl, (t2, x(t2)) E B, and (t, x(t)) E V.

For any (tb Xl) E V we denote by Qt,x, the family of all admissible pairs x(t), u(t), t1 ~ t ~ t2, transferring (t1,X1) to B in V; we shall always assume below that (t2,X(t2)) is the only point on B of the trajectory (t,x(t)); and we take

W(t b X1)= inf l[x,u] = inf [g(t2,X(t2)) + tfo(t,X(t),U(t))dt] . .ottXl OttXl

The function W(tb Xl) is thus defined in V and may have the value - 00. Also w(t1, Xl) = + 00 whenever Qt,x, is empty.

First we shall consider all points (tb Xl) for which the problem above has a minimum.

Let 3 = (I, x) be any point of an optimal pair xo(t), uo(t), t1 ~ t ~ t2, say x = xo(I), t1 ~ I ~ t2, with x(t1) = Xl' Clearly the point 3 = (I, x) can be transferred to B in V, since the pair xo(t), uo(t), I ~ t ~ t2-that is, the restriction of xo, Uo to the subinterval [I, t2]-evidently performs the task. Thus, the class Q of all (admissible) pairs x, u

transferring 3 = (I, x) to B is not empty.

S.4.i (PROPERTY OF OPTIMALITY). Every point (I, x) of the trajectory xo, say x = xo(I), t1 ~ I ~ t2, is transferred to B in V optimally by the pair xo(t), uo(t), I ~ t ~ t2·

Proof. Let 1 = (t1,X1), 2 = (t2,Xz(t2)). If the statement were not true, then there would be another admissible pair x1(t), U1(t), I ~ t ~ t~, transferring 3 = (I, x) to some point, say 4 = (t~,X1(t~)), on B, with (briefly) 134 < 132, or (briefly) J 34 + g(4) < J32 + g(2), where J 34, J 32 are the values of the integrals on 34 and 32, and g(2), g(4) are the values of 9 at 2 and 4. Now let us consider the admissible pair X(t), v(t), t1 ~ t ~ t2, defined by X = X o, v = Uo on [t1,7], X = Xl' V = U1 on [I,(2]. Then X, v transfers 1 = (tb Xl)

5.4 The Value Function 203

to 4 in V with J 13 + J 34 + g(4) < J 13 + J 32 + g(2), that is, 114 < 112 , a contradiction, since xo, Uo transfers 1 to 2 in V optimally. This proves (5.4.i). 0

We now state a number of theorems concerning the value function for the Bolza problem. Clearly, they can all be derived from those in Section 4.5 for Mayer problems. The proofs, therefore, are omitted, and left as an exercise for the reader.

S.4.ii. For every admissible trajectory x(t), t l :5; t:5; t2, transferring any point (t1,X(tI)) to B in V, the function

it2 S(t) = w(t, x(t)) - Jt fo(s, x(s), u(s)) ds,

is not decreasing in [t l ,t2] and is equal to g(t2,X(t2)) for t = t2. If x, u is an optimal pair for (t1,Xtl in V, then S(t) is constant in [t l ,t2].

From now on in this section we assume that (a) any point (tl, XI) E V can be transferred to B in V by some admissible pair x(t), u(t), tl :5; t:5; t2 , u(t) sectionally continuous, u(t) E U; and (b) U is a fixed subset of Rm.

S.4.iii. Under hypotheses (a), (b), if (t, x) is any interior point of V where the value function w(t, x) is differentiable, then

n

wt(t, x) + fo(t, x, v) + I W;,;«t, x)!;(t, x, v) ~ 0 i= 1

for all v E U. Moreover, if there is an optimal pair x*(s), u*(s), t:5; s :5; t~, transferring (t, x) to B in V, u* also sectionally continuous, then

r::i~ [ wt(t, x) + fo(t, x, v) + itl wAt, x)!;(t, x, V)] = 0

and the minimum is attained by v = u*(t + 0).

S.4.iv. Under hypotheses (a) and (b), if all points (t,x) E V can be optimally transferred to B in V by an admissible optimal pair x(s), u(s), t :5; s :5; t2, if there is a function p(t, x) in V - B, continuous in V - B, such that u(s) = p(s, x(s)) for any such pair, and if the value function w(t, x) is continuous in V and of class C1 in V - B (w = g on B), then w(t, x) satisfies in V - B the partial differential equation

n

wt(t, x) + fo(t,x,u) + I wAt, x)!;(t, x,p(t, x)) = 0, i= 1

or

wt(t,x) + H(t, X,p(t, X), wAt, X)) = o. Here H is the Hamiltonian defined by (5.1.3) (with ).0 = 1; cf. Section 5.1).

S.4.v. Under hypotheses (a), (b), let (t1,XI) be an interior point of V, let x*(t), u*(t), tl :5;

t:5; t2, be an optimal pair transferring (tlo XI) to B in V, and such that the entire trajectory (t, x*(t)) is made up of points interior to V, except perhaps the terminal point (t2,X*(t2)) E B. Suppose that the value function w(t, x) is continuous on V and of class c2 on V - B. Then


the functions Ai(t) = W,;«t,x(t», t1 :5; t:5; t2 , i = 1, ... , n, satisfy the relations dA;/dt = -H,At,x*(t), u*(t), A(t» and H(t,x*(t), u*(t), A(t»:5; H(t,x*(t),U,A(t» for all u E U and t E [t1> t2] (a.e.). In other words, A(t) = (AI' ... ,An) are the multipliers.

5.5 Sufficient Conditions for the Bolza Problem

As in Section 4.6, we begin with the converse of (5.4.iii).

S.S.i (A SUFF1cmNT CONDITION FOR OPTIMALITY FOR A SINGLE TRAJECTORY). Let w (t, x) be a function on V such that w(t,x) = g(t, x) for (t,x) E Be V, and let (t1>xd be a point of V such that, for every pair x(t), u(t), t 1 ::;; t ::;; t2 , in QtlXl' w(t, x(t» is finite and S(t) =

w(t, x(t» - J:2 fo(s, x(s), u(s»ds is nondecreasing, and for some pair x*(t), u*(t), t1 :5; t:5; t2 ,

in Q'lXl' S(t) = w(t,x*(t» - J:2 fo(s, x*(s), u*(s»ds is constant on [t1>t2]. Then, the pair x*, u* is optimal for (t1,xd in V.

S.S.ii (A SUFF1cmNT CoNDITION FOR OPTIMALITY FOR A SINGLE TRAJECTORY). Let w(t, x) be a given function in V with w(t, x) = g(t, x) on B, w(t, x) continuous in V and continuously differentiable in V - B, satisfying

n

w,(t,x) + fo(t,x,u) + L wAt,x)!;(t,x,u) ~ 0 i=l

for all (t, x) E V - Band u E U. If x*(t), u*(t), t1 ::;; t::;; t2 , is an admissible pair transferring (t1,X1) to B in V, and

n

w,(t,x*(t» + fo(t,x*(t),u*(t» + L wAt,x*(t»!;(t,x*(t),u*(t» = 0, i=l

(in other words, w(t,x*(t» - J:2 fo(s,x*(s),u*(s»ds, t1 :5; t::;; t2 , is constant), then (x*,u*) is optimal.

S.S.iii. If w(t, x) is a continuous function in V, of class C1 in V - B, with w(t, x) = g(t, x) on B, if x(t), u(t), t1 :5; t:5; t2 , is an admissible pair in V transferring (t1,XI) to B in V, and if S(t) = w(t, x(t» - J:2 fo(s, x(t), u(s» ds is constant on [t1' t2], then

w,(t, x(t» + H(t, x(t), u(t), wx(t, x(t» = 0, t E [t 1, t2] (a.e.).

S.S.iv (BOLTYANSKll'S SUFF1cmNT CONDITION FOR A SINGLE TRAJECTORY). Let V, B, and S be subsets of R1 +n, B eSc V, S of measure zero. Let w(t, x) be a given function in V, with w(t, x) = g(t, x) on B, w(t, x) continuous in V and continuously differentiable in V - S, such that

n

w,(t,x) + fo(t,x,u) + L wAt,x)!;(t,x,u) ~ 0 i= 1

for all (t, x) E V - S and all u E U, and such that S(t) = w(t, x*(t» - J:2 fo(s, x*(s), u*(s» ds, t1 ::;; t:5; t2 , is constant on a given admissible pair x*(t), u*(t), t1 ::;; t ::;; t2 , transferring (t1, Xl) to Bin V. Then x*, u* is optimal for (t1,X1) in V.

Bibliographical Notes 205

Remark. The concepts of synthesis, feedback control, marked trajectories, and regular synthesis are completely analogous to those we have discussed in Section 4.6 for Mayer problems. We leave their formulation as an exercise for the reader, together with the analogous statement of Boltyanskii, (4.6.v).

As a further exercise the reader may derive the statements of Sections 5.4-5 from those of Sections 4.5 -6. Alternatively, the reader may prove the statements of Sections 5.4-5 directly. The proofs are analogous to the ones for Sections 4.5-6.

Bibliographical Notes

The statements of this chapter are only technical variants of those of Chapter 4, to which we refer.

optimization—theory and applications || lagrange and bolza problems of optimal control and other...

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