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Solution of some high-dimensional linear optimalcontrol problems by the method of region-analysisHO[Agrave]NG XUÂN PHÚ† aa Sektion Mathematik der KMU , Karl-Marx-Platz , Leipzig, 7010, DDR (GDR)Published online: 18 Jan 2007.

To cite this article: HO[Agrave]NG XUÂN PHÚ† (1988) Solution of some high-dimensional linear optimal control problems bythe method of region-analysis, International Journal of Control, 47:2, 493-518

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Solution of some high-dimensional linear optimal control problems by the method of region-analysis

A so-called method of region-analysis is given for solving a class of optimal control problems with phase restrictions in which the performance index is linear in the control variables and the state equation has the form i ( t ) = C(t) + Du(r). In par- ticular, an inventory problem is solved.

1. Introduction Optimal control problems in which the control variables appear only linearly are

interesting research subjects. There are many papers on the necessary and sufficient conditions for optimality (e.g. Spreyer and Jacobson 1971, Ioffe and Tichomirov 1979), o r on computational techniques for solving these problems (e.g. Soliman and Ray 1972, Aly and Chan 1973, Sirisena 1974). In particular, the determination of the properties of singular subarcs and of the necessary conditions for joining singular and non-singular subarcs is important. These problems were investigated by Bell and Boissard (1979), Bell and Turpin (1981), McDanell and Powers (1971) and Bortins et aL (1980).

Most of the above-mentioned papers deal with problems in which the control function is scalar and phase restrictions are absent. Masasuke and Yoshikazu (1971) researched into a linear problem with several control variables, but again without phase restrictions (moreover, the functions in the performance index and in the state equations were independent of tlfe time variable). Therefore if many other problems of great practical importance are to be solved, the development of additional methods is necessary.

In some previous papers (Phli 1984,1985,1986 a-c) it was shown how linear prob- lems with one state and one control variable may be solved by the method of region- analysis. In this paper, the method of region-analysis is developed for problems with several state and control variables and, of course, with phase restrictions. The results of this paper will be used in another paper to solve the control problem of systems with a circuit-free graph structure (e.g. a system of hydroelectric power plants). In 4 4 an inventory problem is considered.

The class of problems to which this analysis applies is the following. Determine the measurable right-hand continuous n,-vector function u* which

minimizes the functional

Received 5 January 1987. t Sektion Mathematik der KMU, Karl-Marx-Platz. Leipzig, 7010, DDR (GDR); from

September 1987: Institute of Mathematics, P.O. Box 631, Bo Ho, 10000 Hanoi, Vietnam.

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494 H. X. Phli

with L;f(t, <) = [L, ( t , t) L2(t, 5) ... Ln2(t, <)I, subject to the constraints

where x is an absolutely continuous n,-vector state function, to and t, are the fixed starting and final times. The function Lo( a , ) : [0, 77 x Rn' + R1 is continuous with respect to the first component (the lime variable) and continuously differentiable with respect to the second component (the state variables). The functions Li( -, - ) : [0, 7'l x R"'+ R' , i = 1,2, ..., n,, are continuously differentiable. Furthermore, D and Q are

constant n, x n, or n, x n, matrices and x,, x,, p- and 8' are constant n,- o r n2- vectors. Also, C is a given right-hand piecewise continuous n,-vector function, and q represents a continuously differentiable n,-vector function.

If Lo([, x(t)) = L,(t)x(t) and L, depends only on the time variable, then (1.1), (1.2) is an alternative formulation of a separated continuous linear program (see Anderson et al. 1983).

A pair (x, u) of an absolutely continuous n,-vector function x and a measurable right-hand continuous n2-vector function u is called an admissible process if it satisfies (1.2). If'an admissible process (x*, u*) minimizes the functional in (1.1), then it is denoted as an optimal process.

In the present paper we assume that all the problems considered possess at least one optimal process.

2. Necessary conditions for optimality Let

U = { ~ E R " ~ ~ ~ - < U < ~ + )

Moreover, let us denote the ith horizontal row of the matrix M by Mi. We can now formulate the following necessary condition for optimality.

Theorem 1 (Pontryagin's maximum principle-PMP) Let (x*, u*) be an optimal process of (1.1), (1.2). Then there exist a number I, 2 0,

two vectors p, and p, ( E R"'), a vector function p : [to, t,] + Rnl and n, regular non- negative measures pi, i = 1,2, ..., n,, concentrated on the set {t E [to, t,:l) Q:x*(~) =

qi(t)), which do not vanish a t the same time and which satisfy

This theorem follows directly from a theorem given by Ioffe and Tichomirov (1979, p. 208).

If an optimal process satisfies P M P for a certain I, > 0, we can set I, = I and the process is called normal. Note that a normal optimal process may also satisfy P M P for 2, = 0. Under certain conditions the control problem may have only normal

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High-dimensional linear optimal control problems 495

optimal processes. Since these conditions are often very strong in practice, rather than attempting to formulate them in this case, it is assumed that all optimal processes in the problems considered are normal. This assumption holds for many practical problems. If it does not hold, then only normal optimal processes are considered. (The .general term optimal processes is used throughout for brevity.) Thus we now set 1, = 1 a priori.

Definition A vector function @:[to, t,] -1 Rnl is called a switching function of the optimal

process (x*, u*) iff for every j = 1,2, ..., n, the following condition holds:

(where mj is the jth component of @)

Since the pontryaginian H is linear in the third component (i.e. with respect to the control variables), the last equation of Theorem 1 implies that Q, with

is a switching function of the optimal process (x*, u*), Let

and

Then we have the following necessary condition for optimality.

Theorem 2 Let (x*, u*) be an optimal process of ( l . l ) , (1.2). Then there exist pa E R"' and n,

measures pi, i = 1,2, ..., n,, with the properties represented in Theorem 1, such that @ defined by

@(t) = Dpo - L,,(to, xo) + 9( , x*, u*) dr + D @ dpi (2.1) i = " 1 S [ IO . I )

is a switching function of (x*, u*). Moreover, there exists a vector p, E R"' such that @(k) = DPr - Lu(t,, x0.

Proof We have seen that

@(t) = Dp(t) - L J t , x*(t)) (2.2)

represents a switching function of (x*, u*). Furthermore, Theorem 1 implies that there

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exist p,, pr and pi, i = l ,2 , ..., n,, with the necessary properties such that

In the following we write mi([ ) := dpi. Because pi is regular and non-negative, I l f 0 . t )

mi (for i = 1,2, ..., n,) and, with that, @ are of bounded variation and continuous on the left.

Corollary 1 Let (x*, u*) be an optimal process and Z be an open subinterval of ( to , t,) with

pJ: < u f ( t ) < p; for a certain j E {1,2, ..., n,} and Qx8(t) < q(t) for all t E Z, then B j ( t , x*, u*) = 0 for t E Z.

Proof The definition of the switching function, < uf ( t ) < /I; for all t E Z implies that

a j ( t ) = 0 for all t E Z. Since the measure pi is concentrated o n the set { t 1 Qix*(t) = qi( t )} , mi is constant in Z for all i. Therefore the assertion of this corollary follows from Theorem 2. 0

Corollary 2 Let (x*, u*) be an optimal process and Qix*(t) = qi(t) for a certain i and

Qi.x*(t) < qi.(t) for i' + i and for all t E Z t [to, t f ] Then, for every j = 1, 2, ..., n, and for almost all t E Z we have

uf ( t ) = /I; or u f ( t ) = /IJ: or (DjQ:)9j(t , x*, u*) < 0

Proof We have to show that ( D ~ Q : ) B ~ ( c , x*, u*) < 0 a.e. in Z' = { t E ZIPJ: < u f ( t ) <

/lj+ }. Clearly, Qj( t ) = 0 a.e. in Z', while mi, (i' # i) is constant in every open subinterval of Z'. Moreover, mi is a.e. differentiable because pi is regular and non-negative. Therefore Theorem 2 implies

g j ( t , x*, u*) + DjQ:mi(t) = 0 a.e. in Z'

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High-dimensional linear optimal control problems 497

The last corollary gives us a so-called lingering condition, i.e. it shows when the optimal process can 'linger' on the boundary

Remark 1 Corollaries 1 and 2 also hold if the integrand function L of the performance index

(1.1) is not linear with respect to the control variables and if the phase restriction Qx( t ) < q( t ) is replaced by g i ( t , x ( t ) ) < 0, i = 1,2 , ..., n,, since for this case we have

where

and

3. Some basic ideas of the method of region-analysis The main idea of the method of region-analysis is to find a function C;-: [ t o , t , ]

x Rnl-+iRL such that the ith component mj of the switching function may be rep- resented by 5. Then the so-called state-region

is analysed by means of 5, i.e. we investigate the structure of G where 5 is positive, negative or equal to zero. After that the sign of mj and the jth component of the optimal control may be determined. This will be developed further in Theorems 2 and 3 and in the application examples. Here, only cases where 5 is positive (or negative) in the whole region G are considered.

We now study the switching function of the optimal process ( x* , u*). In order to determine the dependence of the switching function on state and control variables, we calculate 9. Let ( x , u) be an admissible process of ( l . l ) , (1.2) , then

9 ( t , x , u) = DCLo<(t, ~ ( t ) ) + L;(t, x ( t ) )u( t ) l - CL,:(L x ( t ) ) + L.&L x ( t ) ) 4 t ) l

= DLoc(t , x ( t ) ) + DL:$, x ( t ) )u( t ) - L",(t, x ( t ) )

- L,(L 44) CC(4 + DTu(t)l

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Hence

g ( t , x , u) = DL,<([, x ( t ) ) - L,t(L 4 2 ) ) - L.<(L x( t ) )C(t) + A([, x(t))u(t) (3.1)

where

I L,,, ... LlL,

2 Lm<z ... LL, A!t> 5 ) := DL&(t, 5 ) - (DL;fS(t, 5)IT

Since all diagonal elements of the matrix A(t, 5 ) are equal to zero, the jth component Bj of the vector 9 ( t , x , u) is independent of the jth component u, of the control function u. Therefore, the jth component Dj of the switching function O, represented in (2.1) is independent of u f . This is an important property which is used now for determining uf by means of Dj.

Let Ajjt be the (j,j') element of the matrix A and

1: : = { i ~ { 1 , 2 , ..., n 3 } 1 ~ j ~ T > ~ )

1,; := { i E { I , 2, ..., n 3 } 1 DjQT c 0 )

Further on, define

It is obvious that (3.1) and (3.4) imply

' gj(t, x*, u*) = q ( t , x f ( t ) ) for t E [ to , t,], j = 1 , 2, ..., n2 (3.5)

We are now in a position to formulate a theorem for calculating u;.

Theorem 3 Let (x*, u*) be an optimal process of ( 1 . 1 ) , (1.2) and let j be a certain index from

{ I , 2, ..., n , } . Furthermore, aj is the jth component of the switching function of (x*, u*) represented in (2.1).

If Vj(t, 5 ) > 0 for all E G and I,: = 0, then there exists an s E [to, t,] [tl such that uJ(r) = p; and rnj(t) < 0 for t E ( to , s), u f ( t ) = P: and Q j ( t ) > 0 for t E (s , t o .

If V j ( r , 5 ) < 0 for all E G and If = 0, then there exists an s E [to, t,] [;I such that uf ( t ) = pf & d - ~ ~ ( t ) > 0 for t E ( t o , s), uf ( t ) = and @,(t) c 0 for t E (s, t f ) .

We have to prove only (a), since (b ) may be shown analogously. Equations (2.1) and (3.5) imply by I,: = @ that

=@,(to) + 1; v,(s x*(z)) dz + D j ~ : m i ( t ) i.1;

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High-dimensional linear optimal control problems 499

Since pi is non-negative, mi is monotone increasing. Therefore, because y ( t , <) > 0 for - - all 1 1 I E F and DjQ: > 0 for i E I,?, Q j is strictly monotone increasing. Hence, there is

L - A

an s E [ to , t f ] such that a j ( t ) < 0 for t E ( to , s) and @,(t) > 0 for t E (s, t,). Conse- quently, by the definition of the switching function, the assertion (a) holds. 0

Note that s may be equal to to or t,, i.e. ( t o , s) or (s , t,) may be empty and in this case the jth component uf of the optimal control u* has no switching.

Remark 2 If the u: are known for all k with Dki # O , i.e. x: may be determined completely,

then Y j can be replaced by with

By means of this the condition y(t, 5 ) > 0 (or < 0 ) for E G is weaker. Moreover, if L J - -

u: ( k Z j ) is still unknown, we use the following inequalities:

where

Example 1

i , = u , - 2u2, i2 = u, + 0.2t

O < u j ( t ) Q I , O<u2(t)<1

x , ( t ) < 1 , - x 2 ( t ) < -2, x , ( t ) -2x*(t) '< -4

I x,(O)=O, x 2 ( 0 ) = 2 , x , ( 4 ) = 1 , x 2 ( 4 ) = 4

l'(t2) 3 0 . for all 5 ,

First, we determine A. From (3.2) follows

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Therefore, (3.4) implies

and

= -25: + 25152 - ( r ( 5 2 ) + 35:)u:(t)

Furthermore, Dl Q: = 1, Dl Q: = 0, D, Qz = 1 , D,Q: = -2, D,Q: = - 1 and D,Q:= -4. Consequently, by (3.3), 1: = { I , 3),1; = @,I: = 0 and 1; = {1,2,3}.

Since 5 , 2 2, 1'(5,) 2 0 and u:(t) 2 0, we have V,(t, 5 ) > 0 for all [I] G.

Moreover, I; = 0, hence, (a) of Theorem 3 implies that there is an s , E [O,4] such that u:(t) = 0 for t E (0 , s l ) and u:(t) = 1 for t E ( s , , 4).

Because 5 , 6 1 < 2 6 r,, 1'(5,) 2 0 and u:(t) 2 0 for all

V2( t , t) = - 2 t 2 ( 5 , - 5 , ) - (r'(t2) + 35:)u:(t) > 0 for all E G [:I Thus, ( 6 ) of Theorem 3 shows from the result I: = 0 that there is an s, E [O,4] such that $ ( t ) = 1 for t E (0 , s,), and u:(t) = 0 for t E (s,, 4).

Now we can determine s , and s,, x: and xz by means of the state equations and boundary conditions.

Since i : ( t ) = u:(t) + 0,2t, x:(O) = 2 and xz(4) = 4, we have

i.e. s, = 0.4. Let y(t) = x:(t) + 2x:(t). Then j ( t ) = u:(t) + 0.4t and y(0) = 4 and y(4) = 9. Hence

i.e. s - 2.2. - \ Thus we have

0 for t E [O, 2.2) 1 for t E [O,0.4) u:(t) = ; u?(t) =

1 for t e [2.2,4] 0 for t E [0.4,4]

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High-dimensional linear optimal control problems

- 2t for t E [O,0.4]

-0.8 for t E p.4, 2.21

[ t - 3 for t E [2.2,4]

2 + t + 0.1 t z for t E [0,0.4] x?( t ) =

2.4 + 0.1 tZ for t E [O.4,4]

It may easily be shown that (x*, u*) is an admissible process of (3.6) (x* = [x: x?IT and u* = [u: $ I T ) . Moreover, (x* , u*) is the unique process which satisfies the necessary condition for optimality formulated in Theorem 3. Consequently, because the existence of an optimal process was assumed, (x* , u*) is the unique optimal process of (3.6).

It is interesting that the exact representation of the function 1 was not known, i.e. we can solve some problems even with insufficient information. This is an advantage of the method of region-analysis which is very useful for investigating proactical problems.

In the following, we consider the cases where Vj(t, () > O for all

I,: # 0 or Vj(t, 5) < 0 for all E G and I f # 0, i.e. the assumption of Theorem 3

does not hold. First, let us introduce two definitions.

The set &:= {[:I E GIQi i = q i ( t ) j is called the ith border of the state-region G,

We say that the jth component of the control is dominant on the ith border Qi if

Drvf E int {wl w = QiD1vj, oj E [ B y , bf 1) 11 1 (3.7 a)

for all vj. E [ b y , b;], j' # j and t E [ to , t , ] J This dominance condition guarantees that if an arbitrary admissible process (I, u) lingers on the ith border a t any t, i.e. Qix( t ) = qi( t ) and qi( t ) - Q i i ( t ) = qi( t ) - Qi(C(t) + DTu(t)) = 0, then fl,: < uj ( t ) < bjf. If uj ( t ) is equal to the maximal value Bj+ or the minimal value b,:, then (I, u) cannot linger on the ith border at t for any possible u f , j' # j.

For the determination of the optimal process (x*, u*), the dominance condition (3.7 a) may be weakened if some components of the optimal control u* are already calculated, namely

E int {wl w = QiDj'vj, vj E [PI:, B j f ] ) for all vY E [ & , p j f ] , jr E J' and t E [ to , t,] J

where all u;., j" E J", are the already known components of the optimal control u* and J' = { 1 , 2, ..., n,}\(J"u{j}). Clearly, (3.7 a) implies (3.7 b). Otherwise, if (3.7 b) holds

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502 H. X. Phu

for J" = 0 then (3.7 a) is satisfied. For the sake of brevity, (3.7) will be written for (3.7 a ) and (3.7 b). Furthermore, the jth component of the control is called dominant on the ith border Pi if (3.7 a) or (3.7 b) holds, according to the concrete situation.

Remark 3 If the jth component of the control vector is dominant on Qi then QiDT#O

(otherwise int { w l w = QiD:vj, vj E [b,:, P;]} = 0, which contradicts (3.7)). This also implies Dj # 0.

We now introduce the following theorem.

Theorem 4 \ ' Let (x*, u*) be an optimal process of ( l . l ) , (1.2) and let j be a certain index from

{ I , 2, ..., n,}. Suppose that q ( t , t;) > 0 for all E C and I j # (a. Moreover, let the

z' and z" with to Q z' Q z" < t , and

[I1 jth component of the control vector be dominant on Qi for all i E I j . Then there exist

mj(t) < 0 and uf (t) = for all t E (to, z')

Oj(t) > 0 and uf (t) = /I; for all t E (z", t,)

Oj(t) = 0 and E @ : = U Pi f o r a l l t ~ ( z ' , z " )

For every i E 1,: and almost everywhere in {t E (z', z") I Qix8(t) = qi(t)}, the relation

holds. Note that (to, 2') or (z', z") or (z", t,) may be empty. Before proving this theorem we

make some preparatory considerations. A pair (x, u) of an absolutely continuous n,-vector function x and a measurable

right-hand continuous n,-vector function u is named a process if i ( t ) = C(t) + DTu(t) and x(to) = xo and x(t,) = x,.

Consider now a process ( i , ti) defined by:

(a) 4 = uf for all j' # j

(b) there exist z' and z" with to 4 z' Q z" < t , , [sjt,] E Q for a" t E (z', z',

tij(t) = /3,: and [ifql $ P for all t E ( to, z'), tij(t) = and

all t E (z", t,) (where j is a certain index)

We shall show in Proposition 2 that there is exactly one process satisfying this definition and that it is admissible. First, however, an important property of this process is stated in Proposition 1 .

Proposition 1 Suppose that the assumption of Theorem 4 holds and ( i , ti) is defined by (a), (b).

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High-dimensional linear optimal control problems 503

If y* := Djx* and j := D j i then R t ) 4 y*(t) and J ( t i j ( z ) - uf ( 2 ) ) dz 4 0 for all to

t E [ to, t f l .

Proof Denote for i E I j

then Qi(X( t ) - x*( t ) ) = qi ( t ) - Qix*( t ) > 0 for i E I,: and t E Z j . On the other hand,

Qi (2 ( t ) - x*( t ) ) = QiD: r r ( t i j ( t ) - u f ( z ) ) d z for all t and i (3.9) J 10

because 5 = u; for all j' # j. Consequently, by QiDT < 0 for i E IJ: (see (3.3)),

j]: ( i i j lZ) - u f ( z ) ) dz < 0 for all t E Z i and i E 1;

Since

u Z i = (z ' , z") (3.10) i E l j

and i i j ( z ) = 8; 4 u f ( z ) for z .(to, z f ) , and furthermore the inequality j r : ( t i j ( z ) -

u f ( z ) ) d z < 0 holds for all t E ( t o , 2"). From this and from

j ( t ) - y*(t) = D j ( i ( t ) - x*( t ) ) = DjDT ( t i j ( z ) - u f ( z ) ) dz for all t E [ to , t,]

(3.1 1 )

follows j ( t ) C y*(t) for t E [ to , z"]. This inequality is also fulfilled for t E [z", t ,] , because

and t i j (z) = fi: > u f ( z ) for all z E (z" , t,). The remaining assertion of Proposition 1 follows from (3.1 1 ) and j ( t ) 4 y*(t) for all t E [ to , t,]. 0

Proposition 2 Suppose that the assumption of Theorem 4 holds. Then (a), ( b ) define exactly one

process (i, ti) and it is admissible.

Proof First we prove the existence and the uniqueness of the process (i, ti) constructively. Obviously, there are n , - 1 linearly independent vectors y , E R"' with Djyk =0,

k = I , 2, ..., n , - 1. Denote

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Let us consider two vector functions

X + (+= .vf - ( q z ) + x D;u:(z) + DTP;) dz j ' t j

For these we have [ ] E r and [ x : ( A E r for all t E [ to , 41, because Djyk = 0 x - ( t )

implies

slX-(t) = y; [.Y, + 1: (C(.). 1 D: u$(z) j ' t j

and

= y: Fx0 + 1: ( C ( i ) + 1 D: u$ (z) dz j '+j I

for all r E [ to , t,] and k = 1,2, ..., n, - 1. Moreover, since the jth component of the control vector is dominant on 0, there are exactly one t - and one t + from [ to , t,] with

[x..:o]$d for t and

(3.12)

[ I ] $0 for t # t + and

(i) Case t- >, r + . Because y,, k = l , 2 , ..., n, - 1, are linearly independent, r is a two-dimensional plane in which the graph of x - and I+ runs. Thus there exists t* E It+, t - ] such that x - ( t * ) = x + ( t * ) . It is easy to see that there is exactly one process (f, ti) with (a), (b), namely for z' = z" = t*.

(ii) Case t - < t ' . Here t - < t , and t + > to and with that [x-i-,]E~ and

[x:+I] E because of(3.12). Therefore ( b ) and (3.12) require z'= t - and z"=r+

and the process (2, ti) is defined uniquely in [ to , z'] v [ z r ' , t,] by (a), (b). For every i E 1; the system {QT, y , , y,, ..., y, , - , } is linearly independent, because

Djy, = 0 for all k = 1,2, ..., n, - 1 and QiDT < 0 for all i E I,: (see (3.3)) and all y, r . 7

are linearly independent. Consequently, I-no is a continuous curve with

[ - ]ern0 and [ "' ] = [ ] ~ r n p ^ . From the definitions of I- x - ( t - ) X:(z") x + ( t f )

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High-dimensional linear optimal control problems 505

and (a) it follows that E T for all t E [to, tf]. However, (b) requires that

E & for all t E (z', z"). Hence ( i , ii) is also defined uniquely in [z', z"]. (Other

properties of this process may easily be proved.) (i) We now show that (2, ii) also satisfies the control restrictions. For every i E 1): the definition of Zi in (3.8) gives

Hence, by QiDT # 0 (see Remark 3)

Since the jth component of the control vector is dominant on oi for all i E I; (i.e. on &) and (3.10), (3.14) provides

PI: i iij(t) < pi' a.e. in (z', z") (3.15)

Consequently, because tij. = uf for all j' # j and iij(t) for t E (to, 2') and %(t) = for t E (z", r,), the process ( i , 17) satisfies the control restriction b- < ii(t) </If for

all t E [to, t,]. We now verify the state restriction. The definitions of o i , & and ( f , IS) imply for

every i E I; that Qi i ( t ) < qi(t) for t # Z i and Qii(t) = qi(t) for r E Zi, i.e. Qii(t) $ qi(t) for all t E [to, t , ] If i # I J then QiDT 2 0. O n the other hand, Proposition I says that

P I

! .. (iij(z) - uf ( 2 ) ) dz $ 0 for all t E [to, t f ] Therefore (3.9) gives Qi(,f(t) - s*(t)) $ 0,

and with this Qif(t) < Qi.u*(t) < qi(r) for all t E [to, t,] and i 4 I,:. Hence, the state restriction holds also completely.

Consequently, ( f , IS) is an admissible process. 0

Remark 4 In order to determine(i, IS), the sets I-, and T need not be found. First, we calculate

X-, x f , 1 - , t f . If t - 2 t + then we find t* E [ t f , t - ] with x-(t*) = x+(t*), and we have f ( t ) = s - ( t ) and iij(t) = b,: for t E (to, t*), f ( t ) = x + ( t ) and iij(t) = bjf for t E (t*, t f ) (here z' = z" = t*). If r - < t + then we set z' = t-, z" = t + , x(t) - = x- (t) and $(t) = PI: for t E (to, z'), X(t) = x+( t ) and iij(t) = for t E (if', t,). In (z', z"), we have to choose iij so that the trajectory of i lingers on Q (the meeting of the constructed curve begin-

ning in z' with the point [x +:,,,] is guaranteed).

Now we can use the above results for proving Theorem 4.

ProoJ of Theorem 4 For the sake of Proposition 2 and (3.14) we still have to show that

(a) (x*, u*) = ( f , IS) and

(b) mj(t) < 0 for t E (to, 2'). mj(t) = 0 for t E (z', z") and mj(t) > 0 for t E (z", t,).

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High-dimensional linear optimal control problems 507

Corollary 3 If the assumption of Theorem 4 is fulfilled and to < z' < z" < t,, then @,(zl) =

mj(z" + 0) = 0 and

Vj(z, x*(z)) dz - 1 DjQf p i for t E [ to , 2'1 i s / ; 1l.Z')

Qj( t ) = I

?(z , x*(z)) dz + 1 D ~ Q T dpi for t E [zn, t i] icq I I Z " . l )

pi({z'))[Qi(C(zr) + D ~ u ' ) - &(z l ) ] = 0 for all i with Qix*(z') = qi(zl)

p i ( { zn) ) [Qi(C(zn) + DTu") - qi(z0)] = 0 for all i with Qix*(zU) = q,(z")

where v l = [ ~ ; 0; ... v;,lT and vU=[v'; v; ... v;,lT

with fif for i = j pf for i = j

and vf = U: (2 ' ) for i # j u:(zn) for i # j

Therefore, the dominance condition produces the result that pi({z l ) ) = 0 for all i, with Qix*(zf) = qi(zr) and pi({zU}) = 0 for all i, with Qix*(zn) = qi(zU). Consequently, by (2.1), @,(z') = @,(z' + 0) and mj(zt') = mj(zW + 0) , i.e. @,(z') = mj(zu + 0) = 0.

The remaining assertion of this corollary follows from the last equation and (2.1). 0

The lbllowing theorem can be proved in the same way as Theorem 4.

Theorem 5 Let (x*, u*) be an optimal process of (1.1), (1.2). Assume that b ( t , <) < 0 for

[tl E G and If # 0. Moreover, let the jth component of the control vector be

dominant on Pi for all i E If. Then there exist z' and z" with to < z' < z" < ti and

Dj(t) z 0 and uf ( t ) = S f for all t E ( to , z ' )

mj( t ) < 0 and u f ( t ) = 8,: for all t E (z", t,)

@,(I) = 0 and [x:,tJ E ig+ 0 for all t E ( z S j zrf)

For every i E If and almost everywhere in { t E [ to , t , ]JQix*(t ) = q i ( t ) ) the equation

holds.

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. 508 H. X. Phli

In the following, an example of an application of Theorems 4 and 5 is considered.

Exumple 2

jog ( - x i cos (x2/4) +exp (t) u, - tu,) dt-min

For this problem we have

Because A(t, 5) = [: :I, (3.4) implies

Furthermore, D,Q:=-1, D , Q I = 1, DIQT=O, D,Q:=I, D2Q:=0, D,Q:=O, D,Q: = - l and D,Q: = 1. Consequently, by (3.3) 1: = {2,4), 1; = {I), If = (4) and I ; = (3).

Now we can use Theorems 4 and 5 for solving this problem.

First, let j = 2. Clearly, V,(t, 5) > 0 for all E G and I ; + a. Moreover, the [;I second component of the control vector is dominant on the border 0, =

G I - 5 , = Q 3 t = q 3 ( t ) = O

for all 1 0 , I < 2 and t E [O,9]. Therefore the assumption of Theorem 4 holds for j = 2. Thus there exists a subinterval (z;, z;) c (O,9) such that

- 1 for t E (0, z;)

0 for t E ( z ; , z;), x:(t) = 0 for t E (z;, zI;)

1 f o r t ~ ( z ; , 9 )

Since xf (0) = 0, xZ(9) = 5,i: = u: and x: 2 0, it implies that z; = 0 and 2'; = 4. Hence

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0 for t E [O, 4 ) 0 for t E [O,4] u:(t) = and x:(t) = (3.18)

1 for t E [4,9] t - 4 for t E [4 ,9]

(see Fig. 1).

Figure 1

At present, for j = 1 there holds V,(t , 5 ) 1 0 for all

Furthermore, the dominance condition of the first component of the control vector is satisfied on ei for i E 1: = {2 ,4} , because

~ j ~ ( t ) - Q ~ ( C ( t ) + D ; u ~ ) = O ~ ( - 2 , 2 ) =int {wIw=Q2D:ul, lu,I G 2 )

q4(t ) - Q4(C(t) + D;v2) = 0 E [- 1 , I] c ( - 2 , 2) = int { w ( w = Q,D:u,, loll < 2 )

for all Ivzl < 1 and t E [O, 91. Thus the assumption of Theorem 5 holds for j = 1. Therefore there exists a subinterval (z',, 2';) c (O,9) such that

( 2 f o r t ~ ( O , z ' , )

0 for t E int { t E (z', , z;) Ix:(t) = 4 ) u:(t) =

-u;(t) for t E int { t E ( z ' ~ , zr;)Ix:(t) + x ; ( t ) = 6 )

x:(t) = 4 or x:(t) = 6 - x;( t ) for t E (z',, z ; )

Together with 1: = u:, x:(O) = x:(9) = 0 and (3.15), we can calculate i', = 2 and

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z; = 8. It is easy to show that x:(t) = 4 for t E [2, z*], and x:(t) = 6 - xT(t) for t E [z*, 81 for a certain z*, which is a solution of the equation xz(z*) = 2, i.e. z* = 6 (see Remark 4). On that account

r 2 for t E [O, 2) r 2t for t E [O,2]

0 for t E [2,6) for t E [2,6] u:(t) = and x:(t) = (3.19)

- 1 for t E [6,8) - t + 1 0 f o r t ~ [ 6 , 8 ]

1 - 2 f o r t ~ [ 8 , 9 ] 1-2t + 18 for t E [8,9]

The process (x*, u*) defined by (3.18), (3.19) (x* = [x:, x:lT, u* = [u:, u:IT) is admissible with respect to (3.17) and it is the unique process which satisfies the necessary conditions for optimality in Theorems 4 and 5, while the problem (3.17) possesses a t least one optimal process. Hence, (x*, u*) is the unique optimal process of (3.17) (see Fig. 1).

Note that we get the same optimal process for (3.17) if L,(t, <) = I,([) for an arbitrary function I, with I,([) > 0 for all t E [O,9] and L,(t, g) = I,([) for an arbitrary function I, with I,([) < 0 for all t E [O,9]. Once more, we see the advantage of the method of region-analysis for solving some optimal control problems with insufficient information.

We have seen how the method of region-analysis works for some problems in which the state region G has a simple structure with respect to the function y, j = 1,2, ..., n,, i.e. if the assumptions of Theorems 3 o r 4 or 5 are satisfied. Remark 2 also holds for Theorems 4 and 5 and by it the suppositions of these theorems may be weakened.

Remark 5 The assumptions of Theorems 3 ,4 and 5 may be weakened once more if %, If and

I; are replaced by c , c and c: with

c=lf\l and (:=I;\I

where I c {l , 2, ..., n , ) and for every i E I, mi may be represented in terms of x* and u*. Then

c (z , x*(z)) dz + 1 DjQrmi(t) ie(q+"i'-)

The results of this remark may be used for determining the optimal control of a system of hydroelectric power plants o r for solving the control problem considered in the next section.

4. Example: practical application In the present section, the results of the last sections are used to investigate an

inventory problem. We consider the inventory of benzene, whose quantity is denoted by x (barrels). In order to obtain benzene, one can buy u, barrels per unit time of ready-for-use benzene, or u, barrels per unit time of crude petroleum. Clearly, before

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High-din~ensional linear optimal control problems 51 1

using crude petroleum i t must be processed, which yields a variety of products that make up the total cost B, (including the import cost). Because benzene is only one of many useful products from petroleum, its cost B, is a part of B,, i.e. B, < E,. Therefore the total input cost is given by B,u, + B,u,, where B, is the price of the ready-for-use benzene. Let A(x) describe the storage cost, depending on the storage state x , and let p denote the discount rate ( 0 < p c 1 ) . The complete inventory cost in the fixed time interval [to t,] is

Note that the transport cost is included in B, and B,. If c is the demand for benzene which must be covered and if one barrel of crude

petroleum gives d barrels of benzene ( 0 < d < I ) , then the change of the benzene level in the storehouse is described by i ( t ) = -c(t) + u,( t ) + du,(t). Together wirth control and state restrictions, we have to solve the following optimal control problem

S;~XP [ - d l - t o ) ] [ A ( x ( t ) ) + B 1 ( r ) u I ( t ) + B 2 ( t ) ~ , ( t ) l dl-min

i ( t ) = -c( t ) + u , ( t ) + du,(t) 1 (4.1)

Here p , , P,, a , , a,, xo and x, are assigned values. /?, and P , are upper bounds for the import quantity at any time t , determined by the transport capacity or by the delivery capacity of the exporter and so on. a, is the storehouse capacity and a , ( 2 0 ) is the lower bound below which the capacity must not fall (the reliability bound).

For this problem, 11, = 1 , n , = 11, = 2, Lo([, 5 ) = exp [ - p ( t - t o ) ] A([ ) , L , ( I , 5 ) =

exp C - ~ ( t - t o ) ] B , ( t ) , L2(r, 5 ) = exp [ - A t - toll B,(t), C( t ) = -c(t) ,

Q = [t:] = [- :] and 41) = [::'I Now we solve the problem (4.1) under some additional assumptions. First, the

dominance condition

o </I, c c(t) <dB, for all t E [ to , t f ] (4.2)

This inequality implies that (3.7 a) holds for j = 2 and i = 1,2, that is, the second component of the control vector is dominant on the ith border

Secondly, assume a , < x O = x f < a 2

This restraint is not essential, but helps to avoid the consideration of some subordinate cases.

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512 H . X . Phu

Thirdly, we require that the storage cost is strictly monotone increasing with respect to the storage state, i.e.

Ar(C) := 9 > 0 for all 5 E [ a , , a,] d5

Clearly, the last condition holds in practice. Equation (3.4) gives

Furthermore, from Dl Q , = - 1, Dl Q, = 1, D,Q, = -d, D2Q, = d and (3.3), it follows that I; = {I}, I : = (21, I ; = {I) and 1: = ( 2 ) .

Let us solve the above problem for the case with constant input costs B , and B,.

Then V,( t , 5 ) > 0 and V,(t, 5 ) > 0 hold for all E G = [ to , t,] x [ a , , a,]. Because [I1 the second component of the control vector is dominant on &, and I; = { I } , Theorem 4 shows that there exists a subinterval (z; , z ; ) c ( t o , t,) such that

@,(t) < 0, x*( t ) > a , and u $ ( t ) = 0 for t E ( r , , z ;)

@,(t) > 0, x t ( t ) > a , and u f ( t ) = 8, for t 6 ( z ; , t i ) 1 (4.5) 1

) = 0 * a and u l ( t ) = - ( C ( L ) - uf ( t ) ) for t 6 ( z ; , 2';) d J

if (x*, u*) is an optimal process of (4.1) (u* = [u:, u:IT). In the following, we have to determine z;, i';, u: and x*.

Equations (2.1) and (2.2) imply by using (3.1) and (3.4) for (4.1) that

where j = 1,2, p, and p, are certain real numbers and pi is concentrated on { r E [ to , t i ] I.x*(t) = a i } for i = 1, 2. Since a , < x*(r) < a , for t E [ to , 2;) u(z; , t i] (by

the dominance condition (4.2) and (4.5)) and t ( r , 5 ) > 0 for all E G. 4 is strictly

monotone increasing in [to, z;] and (i;, t,]. Now consider three possible cases.

(u) Case B, > B,/d (see Fig. 2) Here, the last equation of (4.6) implies a,(() < ( l /d)@,( t ) for all t E [ to , t,].

Therefore, by ( 4 3 , @, ( t ) < 0 for t E [to, z;] . Because cD, is strictly monotone increasing in ( z ; , t i] , there exists z , E (z';, t,] such that @,( t ) < 0 for t ~ ( z ; , z l ) and @,(t ) > 0 for

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Figure 2. (for E l > BJd)

t E ( z l , t,) and @ , ( z l ) = 0 i f z1 < i f . Consequently, by Theorem 2 >

1 0 for t E [to. z;) 0 for t E [to, z l )

u:( t ) = @(t ) = c(t)/d for t E [z; , 2;) 81 for t [ z i , tr)

P2 for t E [z;, t,]

i X O - j,: 44 .I. for t E [ t o , z;]

for t E (z ; , z;) a1

x*(t) = . ~ - d 8 ~ ( t ~ - t ) - 8 ~ ( t ~ - z ~ ) + c(z)dz f o r t S [ z ; , z ~ l

11

S:' x f - ( j l + dP2)(tf - I ) + j, c(z) dz Ior 9

2

> (4.7)

Note that z; may be equal to z'; and z , may be equal to t,, i.e. the interval (z;, z;) or [z , , t r ) may be empty.

For this case, we still have to determine z , , z; and 2';. Combining (4.3) and (4 .9 , we

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obtain r , < 2; < z'; < ti. For that reason, Corollary 3 gives @,(z;) = @,(z; + 0) = 0 and

(these equations also hold for the next cases, (h ) and (c)). Because pi is concentrated in 0, and a , < x*(t) < a 2 for t E (z;, t,], (4.6) and (4.8)

give

Finally, z, and z; and z'; may be calculated by (4.7) and

z; = z; and x*(z;) > a , or x*(z;) = x*(z;) = a ,

z, = t, or z; < z, < tf and @,(z,) = 0 (4.9)

Thus, the process (x*, u*) is uniquely determined. Because it is admissible and simultaneously the unique process which satisfies the necessary conditions for optim- ality, while (4.1) possesses at least one optimal process, (x*, u*) is the unique optimal process of (4.1) under the conditions (4.2)-(4.4) if B, and B, are constant and B, > BJd.

(b) Case B, < B2/d (see Fig. 3) The last equation of (4.6) implies for this case that @,(t) >(l/d)@,(t) for all

t E [I,, t,]. Therefore, by ( 4 4 , @,(t) > 0 for all t E [z;, t,]. Because @, is strictly monotone increasing in [to, z;], there is a z, E [to, z;) such that a , ( [ ) < 0 for t E [to, z,) and @,(t) > O for t E ( z ~ , ti] and @,(z,) = O if z, > t o . Thus

0 f o r t E [ ~ , , z , ) u: (t) =

PI for t E IZ , , t f l

C O for t E [to, z;)

u:( t ) = -(c(tj - 8 , ) for t E [z;, 2;) 1 : I s 2 for t E [z;, t f l

Y, - 1: c(z) dz for t E [tw z , ]

xo + ( t - z,)p, - ~ ( z ) dz for t E [z, , z;]

a I

I: for t E (z;, z;)

xf - (Dl + dB2)([, - t) + c(z) dz for t E [z;, ti]

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High-dimensional linear optimal control problems 515

Figure 3. (for B, < B,/d).

Furthermore, (4.6) and (4.8) give

V2(2, x*(z)) dz - B1 + V1(z, x*(z)) dz I:' Finally, z , and z; and z; are determined uniquely by the last equation and (4.10), and

- z'r , - , and x*(z;) > a , or x*(z;) =x*(z ';) = a l (4.1 1 )

z , = to or t o < z , <z ; and @ , ( z l ) = 0

The process ( x * , u*) defined uniquely by (4 . l o ) , (4.11) is the unique optimal process of (4.1) under the conditions (4.2)-(4.4) if B , and B , are constant and if Bl < BJd .

(c) Case Bl = BJd (see Fig. 4) For this case, @,(t ) < 0 for t E ( t o , 2;) and @,(t) = 0 for t E (z;, 2;) and @,(t ) > 0 for

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516 H. X . Phli

t E ( z ; , t f ) follow from (4.5) and (4.6). Thus, by the definition of the switching function,

for t E [ to , z ; )

for t E [z; , z ; )

PI for t E [z); , tf]

C O for t E [ t o , z ; )

u z ( t ) = - (c( t ) - u:( t ) ) for t E [z ; , z ; ) i '!l I P 2 for t E [z I ; , tf]

x o - 44 dz ' for t E [ t o , z;)

a1 for t E (z; , z ; )

Figure 4. (for B , = BJd).

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High-dimensional linear optimal coritrol problems 517

where z; and z'; are uniquely defined by Z' , - - Z f f , and x*(z;) > a , or z; < z; and x*(z;) = x*(z;) = a l (4.13)

Analogously, the process (x*, u*) delined by (4.12) and (4.13) is the unique optimal process of (4.1) under (4.2)-(4.4) if B, and B, are constant and Bl = BJd.

We have seen that the problem (4.1) with (4.2)-(4.4) and constant prices B, and B, may be solved completely by means of the theorems proved in $ 5 2 and 3. Note that we can proceed as in Remark 5. After using Theorem 2, the measures p, and p, may be determined as follows.

dp, = 0 for t E [to, t,]

0 for t E [to, z;]

By applying these formulae in (4.6), we obtain

where

Now the problem (4.1) under the assumptions mentioned may be solved for the three cases as above.

Finally we consider briefly the case with varied prices B, and B,. Let us assume here that

A'(() + pBi(t) - B,(t) > 0

for all [I] E G and let (x*, u*) be an optimal process of (4.1) under (4.2)-(4.4), then

( d ) (4.7) and (4.9) hold if E l ( [ ) > B,(t)/d for all t E [to, t,];

(e) (4.10), (4.1 I ) hold if B,(t) < B,(t)/d for all t E [to, t,];

(f) (4.12), (4.13) hold if B, (1) = B,(t)/d for all t E [to, t,].

Thus this problem may also be solved by means of the method of region-analysis when the prices are not constant.

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518 High-dimensional linear optimal control problems

5. Concluding remarks In this paper, we have shown how some high-dimensional linear optimal control

problems can be solved by the method of region-analysis. In order to determine the jth component uf of the optimal control u*, the jth component mj of the switching function @was investigated by means of a suitable function 5 . In particular, the case ?(t, 5) > 0 or 5(t , 5) < 0 in the whole state region G was studied. If the sign of 5 changes in G, then this method may be applied to certain problems. For example, if the dimension of the state and control functions is equal to one, linear optimal control problems may be solved as in Phli (1985 and 1986 b-c).

High-dimensional linear problems with varying signs of 5 will be considered in another paper. Here, however, we can say that this method may be used for estimating the number of switchings. For example, let G' be a subset of int G with ?(t, 5) > 0 or

7 -7

Vj(t , 5 ) < 0 in all G' and E G' for t E [t ' , to ] . Then the jth component u: of the

optimal control u* has aimost,ne switching in [t', t"]. This estimate is important for computing optimal processes (see Sirisena 1974).

If the dominance condition does not hold, then the method of region-analysis can also be used to solve some linear problems (see Phu 1986 a).

ACKNOWLEDGMENT The author thanks Mrs Beate Fiedler for correcting the English of this paper.

REFERENCES ALY, G . M., and CHAN, W. C., 1973, Int. J. Control, 17, 809. ANDERSON, E. J., NASH, P., and PEROLD, A. F., 1983, SIAM JI Control Optim., 21, 758. BELL, D. J., and BOISSARD, M., 1979, Inl. J . Conrrol, 29, 981. BELL, D. J., and TURPIN, K., 1981, Inf. J . Control, 34, 153. BORTINS, R., BOUSTANY, N. M., and POWERS, W. F., 1980, Optimal Control Applic. Methods, 1,279. ~oFFE, A. D., and TICHOMIROV, V. M., 1979, Theorie der Extremalaufgaben (Berlin: VEB Deutscher

Verlag der Wissenschaften). MASASUKE, S., and YOSHIKAZU, S., 1971, Int. J. Control, 13, 131. MCDANELI., J. P., and POWERS, W. F., 1971, SIAM J. Control, 9, 161. PHU, H. X., 1984, Zeitschrft Anal. Anwendungen, 3, 527; 1985, Optimization, 16, 273; 1986 a, Syst.

Control Lett., 8, to be published; 1986 b, Ibid., 8, to be published; 1986 c, Ibid., 8, to be published.

SIRISENA, H. R., 1974, In:. J . Control, 19, 257. SOLIMAN, M. A,, and RAY, W. H., 1972, Int. J . Control, 16, 261. SPREYER, J. L., and JACOBSON, D. H., 1971, J . Math. Analysis Applic., 33, 163.

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