A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 1, March 23 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Rezumat. Un principiu de maxim pentru o problemă vectorială de control optimal. Ca aplicaţie a unei reguli abstracte a multiplicatorilor s-a stabilit în lucrarea [1] un principiu de maxim pentru o problemă vectorială de control optimal guvernată de o ecuaţie integrală de tip Fredholm. Pentru a nu mări excesiv lungimea lucrării [1], demonstraţia acestui principiu a fost acolo doar schiţată. demonstraţia completă. În prezenta lucrare se dă acum 1. Introduction In the paper [1] we have established multiplier rules for so-called weak dynamic multiobjective optimization problems by using a suitable generalization of the derived sets introduced by M. R. Hestenes [2], [3], [4] for scalar optimization problems. Also in that paper we have used the obtained multiplier rules to state necessary conditions for the local solutions of an abstract multiobjective optimal control problem. Furthermore, we have noticed that these very general optimality conditions can yield a maximum principle for a multiobjective optimal control problem governed by an integral equation of Fredholm type (Theorem 5.1 in [1]). But, in order to avoid an excessive length of the paper, in [1] we have limited ourselves only to a sketch of this application. The goal of the present paper is to give the complete proof of this specific maximum principle. Received by the editors: Mathematics Subject Classification. 49K27, 49J22. 25

2 2. Notations Throughout this paper, N is the set of all positive integers, R is the set of all real numbers, for every m N, R m is the usual m-dimensional Euclidean space of all m-tuples v = (v 1,..., v m ) of real numbers. The subset of R m, consisting of all vectors v = (v 1,..., v m ) with v j for each j {1,..., m}, is denoted by R m +. The inner product of two vectors v, w R m is denoted by v, w. If v R m, then v marks its Euclidean norm. Given any number r >, we put B m + (r) = { v R m + v r }. If X Y are normed linear spaces over the same field, then (X, Y) denotes the normed linear space of all continuous linear mappings A : X Y. Given a point x in a normed linear space a number r >, we denote by B(x, r) the closed ball centered at x with radius r. If M is a subset of a normed linear space, then int M designates the interior of M cl M the closure of M. Finally, we mention some notations regarding functions. The Fréchet derivative of a function f of a single variable is denoted by df, while the partial Fréchet derivative with respect to the nth variable of a function f of several variables is denoted by d n f. If x is a point in a linear space X A is a linear mapping from X into another linear space, then Ax denotes the value of A at x. 3. A Necessary Optimality Condition Let X be a Banach space, which does not reduce to its zero-vector, let X be a nonempty open subset of X, let U be a nonempty set, let m 1, m 2 m 3 be positive integers, let f 1 : X U R m1, f 2 : X U R m2, f 3 : X U R m3 be vector-valued functions which are Fréchet differentiable at each point (x, u) in X U with respect to the first variable. Further, let K 1, K 2 K 3 be convex cones in the spaces R m1, R m2 R m3, respectively, satisfying the following assumptions: int K 1, int K 2, K 2 K 3 are closed. (1) 26

3 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM For each i {1, 2, 3}, we define by K i = {w R mi v K i : v, w } the dual cone of K i. Let F : X U X be a function which is Fréchet differentiable at each point (x, u) X U with respect to the first variable, let S be the set defined by S = {(x, u) X U F (x, u) =, f 2 (x, u) K 2, f 3 (x, u) K 3 }. A point (x, u ) X U is said to be a: (i) weakly K 1 -maximal point of f 1 over S if (x, u ) S [f 1 (x, u ) + int K 1 ] f 1 (S) = ; (ii) local weakly K 1 -maximal point of f 1 over S if (x, u ) S if there is a neighbourhood V of x such that [f 1 (x, u ) + int K 1 ] f 1 (S (V U)) =. The problem of finding the weakly K 1 -maximal points of f 1 over S is called a weak multiobjective optimal control problem is expressed in short as (CP) f 1 (x, u) K1 max weakly subject to (x, u) X U, F (x, u) =, f 2 (x, u) K 2, f 3 (x, u) K 3. The introduction of problem (CP) allows one to call the weakly K 1 -maximal points of f 1 over S solutions to problem (CP). By analogy, the local weakly K 1 - maximal points of f 1 over S can be named local solutions to problem (CP). As an application of multiplier rules stated for arbitrary weak dynamic multiobjective optimization problems, in Section 4 of the paper [1] we have derived necessary optimality conditions for the local solutions to problem (CP). One of the theorems given there will be recalled here. In order to formulate shorter this theorem, we put m = m 1 + m 2 + m 3 conceive the corresponding space R m as the product space R m1 R m2 R m3, i.e. any vector v R m is identified with a certain triple (v 1, v 2, v 3 ) R m1 R m2 R m3. In particular, the zero-vector in R m 27

4 is = ( 1, 2, 3 ), where i (i {1, 2, 3}) is the zero-vector in R mi. Further, we consider the vector-valued function f : X U R m defined by f(x, u) = (f 1 (x, u), f 2 (x, u), f 3 (x, u)). By using these notations, the following theorem is valid. THEOREM 1 [1, Theorem 4.6]. Let (x, u ) X U be a local solution to problem (CP) for which the operator A = d 1 F (x, u ) is bijective, let D R m be a non-empty set such that, for all n N all n-tuples (d 1,..., d n ) of points belonging to D, there exist a number r > a function ω 2 : B+(r n ) U satisfying the following conditions: (i) ω 2 () = u ; (ii) for each x X, the function t B+(r n ) F (x, ω 2 (t)) X is continuous on B+(r n ); (iii) the function t B+(r n ) d 1 F (x, ω 2 (t)) (X, X ) is continuous at ; (iv) lim x x sup { d 1 F (x, ω 2 (t)) d 1 F (x, ω 2 (t)) t B n + (r ) } = ; (v) for each x X, the function t B+(r n ) f(x, ω 2 (t)) R m is continuous on B+(r n ); (vi) there is a number a > such that B(x, a) X such that sup { d 1 f(x, ω 2 (t)) x B(x, a), t B n +(r ) } < ; (vii) sup (viii) sup { } F (x, ω 2 (t)) / t t B+(r n ), t < ; { } d 1 f(x, ω 2 (t)) d 1 f(x, u ) / t t B+(r n ), t < ; (ix) lim x x sup { d 1 f(x, ω 2 (t)) d 1 f(x, ω 2 (t)) t B n + (r ) } = ; (x) lim t 1 t [ f(x, ω 2 (t)) f(x, u ) P t d 1 f(x, u )ω (t) ] =, where P t = t 1 d t n d n for all t = (t 1,..., t n ) R n ω (t) = A 1 F (x, ω 2 (t)) for all t B n +(r ). 28

5 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Then there exists a vector (λ 1, λ 2, λ 3) K1 K2 K3 \ {( 1, 2, 3 )} such that sup { d 1, λ 1 + d 2, λ 2 + d 3, λ 3 (d1, d 2, d 3 ) D } f 2 (x, u ), λ 2 =. hold. 4. The Maximum Principle In this section we apply Theorem 1 to derive a maximum principle for a multiobjective optimal control problem governed by an integral equation of Fredholm type. In what follows we suppose that T is a positive number, V is a non-empty subset of a real Banach space V, W is a real Banach space which does not reduce to its zero-vector. Let I denote the interval [, T ], let C(I, W) be the linear space of all continuous functions x : I W endowed with the norm x = max { x(τ) τ I}, let P C(I, V ) be the set of all piecewise continuous functions u : I V that are continuous at continuous on the left at each point belonging to the interval ], T ]. Further, let ϕ i : I W cl V R mi (i {1, 2, 3}) be functions that are continuous, Fréchet differentiable with respect to the second variable such that the mappings d 2 ϕ i : I W cl V (W, R mi ) (i {1, 2, 3}) are continuous, let φ : I I W cl V W 29

6 be a function which is continuous, Fréchet differentiable with respect to the third variable, for which the mapping d 3 φ : I I W cl V (W, W) is continuous has the property that the family { d3 φ(σ, τ,, v) : W (W, W) (σ, τ, v) I I V } is uniformly equicontinuous on each closed bounded subset of W. As in Section 3, let K 1, K 2 K 3 be convex cones in the spaces R m1, R m2 R m3, respectively, satisfying the assumptions specified in (1). The problem we will discuss in this section is: (ECP) subject to ϕ 1 (τ, x(τ), u(τ)) dτ K1 max weakly x C(I, W), u P C(I, V ), x(σ) = φ(σ, τ, x(τ), u(τ)) dτ (σ I), ϕ 2 (τ, x(τ), u(τ)) dτ K 2, ϕ 3 (τ, x(τ), u(τ)) dτ K 3. This problem is a special case of the problem (CP) investigated in the preceding section. To see this, it suffices to define the functions f i : C(I, W) P C(I, V ) R mi (i {1, 2, 3}) by f i (x, u) = ϕ i (τ, x(τ), u(τ)) dτ (i {1, 2, 3}), on the one h, F : C(I, W) P C(I, V ) C(I, W) by 3

7 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM F (x, u)(σ) = x(σ) φ(σ, τ, x(τ), u(τ)) dτ (σ I), on the other h, as well as to take X = X = C(I, W) U = P C(I, V ). Furthermore, it should be emphasized that the functions f i (i {1, 2, 3}) F introduced above are Fréchet differentiable with respect to the first variable. The corresponding partial Fréchet derivatives are given by d 1 f i (x, u)y = d 2 ϕ i (τ, x(τ), u(τ))y(τ) dτ (i {1, 2, 3}), (d 1 F (x, u)y)(σ) = y(σ) d 3 φ(σ, τ, x(τ), u(τ))y(τ) dτ (σ I), for all (x, u) C(I, W) P C(I, V ) all y C(I, W). Thus it makes sense to try to specialize Theorem 1 to problem (ECP). To this end we define the functions ϕ : I W cl V R m f : C(I, W) P C(I, V ) R m by ϕ(τ, w, v) = (ϕ 1 (τ, w, v), ϕ 2 (τ, w, v), ϕ 3 (τ, w, v)), f(x, u) = (f 1 (x, u), f 2 (x, u), f 3 (x, u)), respectively. Then we have f(x, u) = ϕ(τ, x(τ), u(τ)) dτ, d 1 f(x, u)y = d 2 ϕ(τ, x(τ), u(τ))y(τ) dτ for all (x, u) C(I, W) P C(I, V ) all y C(I, W). Taking into account all these assumptions considerations concerning the problem (ECP), we get from Theorem 1 the following result. THEOREM 2 [1, Theorem 5.1]. Let (x, u ) C(I, W) P C(I, V ) be a local solution to problem (ECP) satisfying the following conditions: 31

8 (j) for each y C(I, W) the integral equation x = y + has a unique solution x C(I, W); (jj) there is a number a > such that d 3 φ(, τ, x (τ), u (τ))x(τ) dτ sup { d 2 ϕ(τ, x(τ), v) (τ, x, v) I C(I, W) V, x x a } <. Then there exists a vector such that λ = (λ 1, λ 2, λ 3) K 1 K 2 K 3 \ {( 1, 2, 3 )} max {H(τ, v) v V } = H(τ, u (τ)) for all τ I (2) ϕ 2 (τ, x (τ), u (τ)) dτ, λ 2 =, (3) where I is the set of all points τ ], T ] at which u is continuous, H(τ, ) : V R is the function defined by H(τ, v) = ϕ(τ, x (τ), v) + d 2 ϕ(σ, x (σ), u (σ))h(σ; τ, v) dσ, λ, h( ; τ, v) : I W denotes the solution of the variational equation x = φ(, τ, x (τ), v) + d 3 φ(, t, x (t), u (t))x(t) dt. Proof. At first we notice that the operator A = d 1 F (x, u ) is bijective because of condition (j). Next we construct a subset D of the space R m which satisfies the hypotheses of Theorem 1. For this purpose we associate with each pair (τ, v) I V the following expressions: α(τ, v) = ϕ(τ, x (τ), v) ϕ(τ, x (τ), u (τ)), β(τ, v) = φ(, τ, x (τ), v) φ(, τ, x (τ), u (τ)), d(τ, v) = α(τ, v) + d 1 f(x, u ) A 1 β(τ, v). 32

9 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM After that we put D = {d(τ, v) (τ, v) I V }. Now, let n be any positive integer, let d j = d(τ j, v j ) (j {1,..., n}) be points belonging to D. For each j {1,..., n} we set α j = α(τ j, v j ) β j = β(τ j, v j ). Then we have d j = α j + d 1 f(x, u ) A 1 β j for all j {1,..., n}. Without loss of the generality we can assume that the points d 1,..., d n are in such a manner numbered that < τ 1 τ 2... τ n T. Put τ =. Then choose a number r > satisfying Set r = r/n. r < τ j+1 τ j whenever j {,..., n 1} τ j < τ j+1 (4) [τ j r, τ j ] I for all j {1,..., n}. Next we define a function ω 2 : B n +(r ) P C(I, V ). Fix any point t = (t 1,..., t n ) in B n +(r ), Then we have For each j {1,..., n} we denote It is easily seen that (4) (5) imply t t n n t r. (5) N j = { k N j < k n τ k = τ j } t j if N j = a j = t j + k N j t k if N j. < τ j a j τ j a j + t j T for all j {1,..., n}. (6) When n > 1, then we additionally have τ j a j + t j τ j+1 a j+1 for all j {1,..., n 1}. (7) 33

10 From (6) (7) it follows that the intervals I j (j {1,..., n}), defined by satisfy I j = ]τ j a j, τ j a j + t j ] for every j {1,..., n}, I j I for all j {1,..., n}, I j I k = for all j, k {1,..., n}, j k. These properties of the intervals I j (j {1,..., n}) enable us to define the function ω 2 (t) : I V by v j if τ I j for some j {1,..., n} ω 2 (t)(τ) = u (τ) if τ I \ (I 1... I n ). In view of this definition we obviously have ω 2 (t) P C(I, V ). In what follows we prove that the number r the function ω 2 defined above satisfy the conditions (i) (x) of Theorem 1. In the proofs of some of these conditions we shall use the compact set L = [τ 1 r, τ 1 ]... [τ n r, τ n ], which is enclosed in I. Besides, given any t = (t 1,..., t n ) B n +(r ), we shall need the intervals They satisfy L j = [τ j a j, τ j a j + t j ], where j {1,..., n}. L j [τ j r, τ j ] L for all j {1,..., n}. Indeed, let j be any index in {1,..., n}. Since t j a j, we have L j [τ j a j, τ j ]. On the other h, the inequality a j t t n holds. Consequently, (5) implies a j r, whence [τ j a j, τ j ] [τ j r, τ j ]. Thus we have L j [τ j r, τ j ] L, as claimed. 1 are satisfied. ω 2 () = u. 34 Now we consecutively prove that the conditions (i) (x) occurring in Theorem Condition (i): If t =, then I j = for every j {1,..., n}. Thus we have

11 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Condition (ii): We fix a function x C(I, W). Since the functions (σ, τ) I L φ(σ, τ, x(τ), v j ) W (j {1,..., n}) (σ, τ) I L φ(σ, τ, x(τ), u (τ)) W are continuous on the compact set I L, there exists a number c > such that φ(σ, τ, x(τ), v j ) + φ(σ, τ, x(τ), u (τ)) c (8) for all (σ, τ) I L all j {1,..., n}. Let t 1 = (t 1 1,..., t 1 n) t 2 = (t 2 1,..., t 2 n) be points in B+(r n ). For every j {1,..., n} we put L j1 = [τ j a j1, τ j a j1 + t 1 j], L j2 = [τ j a j2, τ j a j2 + t 2 j], M j = {τ j (1 τ)a j1 τa j2 τ [, 1]}, where a j1 a j2 are the numbers used in the definition of the function ω 2 (t 1 ) ω 2 (t 2 ), respectively. Obviously, we have a j1 a j2 t 1 j t 2 j + k N j t 1 k t 2 k n t 1 t 2 (9) for every j {1,..., n}. Fix any σ I. In virtue of (8) (9) it follows that φ(σ, τ, x(τ), v j ) dτ φ(σ, τ, x(τ), v j ) dτ c(2 a j1 a j2 + t 1 j t 2 j ) L j1 L j2 c(2n + 1) t 1 t 2 φ(σ, τ, x(τ), u (τ)) dτ c a j1 a j2 cn t 1 t 2 M j for all j {1,..., n}. Accordingly, we have τ j φ(σ, τ, x(τ), ω 2 (t 1 )(τ)) dτ τ j φ(σ, τ, x(τ), ω 2 (t 2 )(τ)) dτ τ j 1 φ(σ, τ, x(τ), u (τ)) dτ + τ j 1 φ(σ, τ, x(τ), v j ) dτ φ(σ, τ, x(τ), v j ) dτ M j L j1 L j2 35

12 + k N j L k1 φ(σ, τ, x(τ), v k ) dτ L k2 φ(σ, τ, x(τ), v k ) dτ 2cn(n + 1) t 1 t 2 for every j {1,..., n} such that τ j 1 < τ j. Taking into account that we obtain φ(σ, τ, x(τ), ω 2 (t 1 )(τ)) dτ τ j τ j 1 φ(σ, τ, x(τ), ω 2 (t 1 )(τ)) dτ φ(σ, τ, x(τ), ω 2 (t 2 )(τ)) dτ τ j τ j 1 φ(σ, τ, x(τ), ω 2 (t 1 )(τ)) dτ 2cn 2 (n + 1) t 1 t 2. Since σ I was arbitrarily chosen, this result implies φ(σ, τ, x(τ), ω 2 (t 2 )(τ)) dτ, φ(σ, τ, x(τ), ω 2 (t 2 )(τ)) dτ F (x, ω 2 (t 1 )) F (x, ω 2 (t 2 )) 2cn 2 (n + 1) t 1 t 2. Thus the function t B n +(r ) F (x, ω 2 (t)) C(I, W) is continuous on B n +(r ). Condition (iii): Since the functions (σ, τ) I L d 3 φ(σ, τ, x (τ), v j ) (W, W) (j {1,..., n}) (σ, τ) I L d 3 φ(σ, τ, x (τ), u (τ)) (W, W) are continuous on the compact set I L, there exists a number c > such that d 3 φ(σ, τ, x (τ), v j ) d 3 φ(σ, τ, x (τ), u (τ)) c (1) for all (σ, τ) I L all j {1,..., n}. Let the number ε > be arbitrarily given. Let t B+(r n ) be such that t < ε/(cn). Fix any function y C(I, W) for which y 1. In virtue of (1), the expression 36 g(σ) = [ d3 φ(σ, τ, x (τ), ω 2 (t)(τ)) d 3 φ(σ, τ, x (τ), u (τ)) ] y(τ) dτ

13 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM satisfies for all σ I g(σ) L j d 3 φ(σ, τ, x (τ), v j ) d 3 φ(σ, τ, x (τ), u (τ)) y(τ) dτ On the other h we have c(t t n ) cn t < ε. [ d 1 F (x, ω 2 (t)) d 1 F (x, u ) ] y = max {g(σ) σ I}. Consequently, it follows that [ d 1 F (x, ω 2 (t)) d 1 F (x, u ) ] y < ε. Since y was arbitrarily chosen in C(I, W) such that y 1, we get So we have shown that the function is continuous at. d 1 F (x, ω 2 (t)) d 1 F (x, u ) ε. t B n +(r ) d 1 F (x, ω 2 (t)) (C(I, W), C(I, W)) Condition (iv): Let the number ε > be arbitrarily given. Since the family { d3 φ(σ, τ,, v) : W (W, W) (σ, τ, v) I I V } is uniformly equicontinuous on the set W = { w W w x + 1 }, there is a number δ > such that d 3 φ(σ, τ, w 1, v) d 3 φ(σ, τ, w 2, v) < ε/t (11) for all w 1, w 2 W with w 1 w 2 < δ all (σ, τ, v) I I V. Now fix any x C(I, W) such that x x < min {1, δ}. Then we have x(τ), x (τ) W x(τ) x (τ) < δ for all τ I. Next fix a point t B+(r n ), for short, denote u = ω 2 (t). Then (11) implies [ d 1 F (x, u) d 1 F (x, u) ] y { T = max G(σ, τ)y(τ) dτ } σ I 37

14 max { for all y C(I, W) satisfying y 1, where Consequently, we have G(σ, τ) dτ σ I } ε G(σ, τ) = d 3 φ(σ, τ, x(τ), u(τ)) d 3 φ(σ, τ, x (τ), u(τ)). d 1 F (x, u) d 1 F (x, u) ε. Since t was arbitrarily chosen in B n +(r ), the following inequality is true: Thus we have sup { d 1 F (x, ω 2 (t)) d 1 F (x, ω 2 (t)) t B n + (r ) } ε. lim x x sup { d 1 F (x, ω 2 (t)) d 1 F (x, ω 2 (t)) t B n + (r ) } =. Condition (v): We fix a function x C(I, W). Since the functions τ L ϕ(τ, x(τ), v j ) R m (j {1,..., n}) τ L ϕ(τ, x(τ), u (τ)) R m are continuous on the compact set L, there exists a number c > such that for all τ L all j {1,..., n}. ϕ(τ, x(τ), v j ) + ϕ(τ, x(τ), u (τ)) c (12) Let t 1 = (t 1 1,..., t 1 n) t 2 = (t 2 1,..., t 2 n) be points in B n +(r ). By using the intervals L j1, L j2 M j (j {1,..., n}) that we previously employed to show that condition (ii) is satisfied, it follows from (9) (12) that ϕ(τ, x(τ), v j ) dτ ϕ(τ, x(τ), v j ) dτ L j1 L j2 c(2 a j1 a j2 + t 1 j t 2 j ) c(2n + 1) t 1 t 2 that 38 M j ϕ(τ, x(τ), u (τ)) dτ c a j1 a j2 cn t 1 t 2

15 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM for all j {1,..., n}. Accordingly, we have τ j ϕ(τ, x(τ), ω 2 (t 1 )(τ)) dτ τ j ϕ(τ, x(τ), ω 2 (t 2 )(τ)) dτ τ j 1 ϕ(τ, x(τ), u (τ)) dτ + τ j 1 ϕ(τ, x(τ), v j ) dτ ϕ(τ, x(τ), v j ) dτ M j + k N j L k1 L j1 ϕ(τ, x(τ), v k ) dτ L k2 L j2 ϕ(τ, x(τ), v k ) dτ 2cn(n + 1) t 1 t 2 for every j {1,..., n} such that τ j 1 < τ j. Taking into account that we obtain = τ j τ j 1 f(x, ω 2 (t 1 )) f(x, ω 2 (t 2 )) ϕ(τ, x(τ), ω 2 (t 1 )(τ)) dτ ϕ(τ, x(τ), ω 2 (t 1 )(τ)) dτ ϕ(τ, x(τ), ω 2 (t 2 )(τ)) dτ τ j τ j 1 ϕ(τ, x(τ), ω 2 (t 2 )(τ)) dτ, f(x, ω 2 (t 1 )) f(x, ω 2 (t 2 )) 2cn 2 (n + 1) t 1 t 2. Thus the function t B n +(r ) f(x, ω 2 (t)) R m is continuous on B n +(r ). Condition (vi): Set B(x, a) = { x C(I, W) x x a } c = sup { d 2 ϕ(τ, x(τ), v) τ I, x B(x, a), v V }. Let x be in B(x, a), let t be in B n +(r ). Since the function ω 2 (t) takes its values in V, we have d 2 ϕ(τ, x(τ), ω 2 (t)(τ)) y(τ) d 2 ϕ(τ, x(τ), ω 2 (t)(τ)) y(τ) c y for all τ I all y C(I, W). This result implies d 1 f(x, ω 2 (t))y = d 2 ϕ(τ, x(τ), ω 2 (t)(τ)) y(τ) dτ T sup { d 2 ϕ(τ, x(τ), ω 2 (t)(τ)) y(τ) τ I } ct y 39

16 for all y C(I, W). Hence, we have d 1 f(x, ω 2 (t)) ct. Since x t were arbitrarily chosen in B(x, a) B n +(r ), respectively, it is true that sup { d 1 f(x, ω 2 (t)) x B(x, a), t B n +(r ) } ct. Condition (vii): Since the functions (σ, τ) I L φ(σ, τ, x (τ), u (τ)) W (σ, τ) I L φ(σ, τ, x (τ), v j ) W (j {1,..., n}) are continuous on the compact set I L, there exists a number c > such that φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), v j ) c for all (σ, τ) I L all j {1,..., n}. Then we have = max max { T F (x, ω 2 (t)) [ φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), ω 2 (t)(τ)) ] } dτ σ I { n L j for all t B n +(r ), thus φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), v j ) } dτ σ I c(t t n ) cn t sup { F (x, ω 2 (t)) / t t B n +(r ), t } cn. Condition (viii): Since the functions τ L d 2 ϕ(τ, x (τ), v j ) (W, R m ) (j {1,..., n}) τ L d 2 ϕ(τ, x (τ), u (τ)) (W, R m ) are continuous on the compact set L, there exists a number c > such that d 2 ϕ(τ, x (τ), v j ) d 2 ϕ(τ, x (τ), u (τ)) c 4

17 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM for all τ L all j {1,..., n}. Fix any t B n +(r ). Then we have = = [d 1 f(x, ω 2 (t)) d 1 f(x, u )] y [ d2 ϕ(τ, x (τ), ω 2 (t)(τ)) d 2 ϕ(τ, x (τ), u (τ)) ] y(τ) dτ d2 ϕ(τ, x (τ), ω 2 (t)(τ)) d 2 ϕ(τ, x (τ), u (τ)) dτ L j d2 ϕ(τ, x (τ), v j ) d 2 ϕ(τ, x (τ), u (τ)) dτ c(t t n ) cn t for all y C(I, W) satisfying y 1. This result implies d 1 f(x, ω 2 (t)) d 1 f(x, u ) cn t. Since t was arbitrarily chosen in B n +(r ), we get sup { d 1 f(x, ω 2 (t)) d 1 f(x, u ) / t t B n + (r ), t } cn. Condition (ix): Let the number ε > be arbitrarily given. For each (τ, w) I W we denote g j (τ, w) = d 2 ϕ(τ, x (τ) + w, v j ) d 2 ϕ(τ, x (τ), v j ) (j {1,..., n}), g(τ, w) = d 2 ϕ(τ, x (τ) + w, u (τ)) d 2 ϕ(τ, x (τ), u (τ)). Since the function (τ, w, v) I W cl V d 2 ϕ(τ, w, v) (W, R m ) is continuous, we can apply Lemma 2, given in [5], conclude that lim s sup { g j (τ, w) τ I, w W, w s } = for each j {1,..., n} that lim s sup { g(τ, w) } τ I, w W, w s =. 41

18 Consequently, there is a number δ > such that sup { g j (τ, w) τ I, w W, w δ } < ε/(2t ) sup { g(τ, w) } τ I, w W, w δ < ε/(2t ). Now let x C(I, W) be any function satisfying x x < δ. t B n +(r ). Then we have [d1 f(x, ω 2 (t)) d 1 f(x, ω 2 (t))] y Fix any T = [ d2 ϕ(τ, x(τ), ω 2 (t)(τ)) d 2 ϕ(τ, x (τ), ω 2 (t)(τ)) ] y(τ) dτ d 2 ϕ(τ, x(τ), ω 2 (t)(τ)) d 2 ϕ(τ, x (τ), ω 2 (t)(τ)) dτ d2 ϕ(τ, x(τ), v j ) d 2 ϕ(τ, x (τ), v j ) dτ + d2 ϕ(τ, x(τ), u (τ)) d 2 ϕ(τ, x (τ), u (τ)) dτ sup { g j (τ, x(τ) x (τ)) } { τ I + T sup g(τ, x(τ) x (τ)) } τ I < ε for all y C(I, W) satisfying y 1. This result implies d 1 f(x, ω 2 (t)) d 1 f(x, ω 2 (t)) ε. Since t was arbitrarily chosen in B n +(r ), we have Consequently, it is true that sup { d 1 f(x, ω 2 (t)) d 1 f(x, ω 2 (t)) t B n + (r ) } ε. lim x x sup { d 1 f(x, ω 2 (t)) d 1 f(x, ω 2 (t)) t B n + (r ) } =. 42 Condition (x): We denote P α t = t 1 α t n α n for all t = (t 1,..., t n ) B n +(r ).

19 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM We claim that the function satisfies t B n +(r ) f(x, ω 2 (t)) f(x, u ) P α t R m 1 [ lim t f(x, ω 2 (t)) f(x, u ) P α t ] =. (13) t To prove this, let the number ε > be arbitrarily given. Since the functions τ L ϕ(τ, x (τ), v j ) R m (j {1,..., n}) τ L ϕ(τ, x (τ), u (τ)) R m are continuous on the compact set L, they are uniformly continuous on this set. Thus there exists a number δ > such that for all j {1,..., n} all τ L satisfying τ τ j < δ the following inequalities hold: ϕ(τ, x (τ), v j ) ϕ(τ j, x (τ j ), v j ) < ε/(2n); These inequalities imply ϕ(τ, x (τ), u (τ)) ϕ(τ j, x (τ j ), u (τ j )) < ε/(2n). ϕ(τ, x (τ), v j ) ϕ(τ, x (τ), u (τ)) α j < ε/n (14) for all j {1,..., n} all τ L satisfying τ τ j < δ. Now let t B n +(r ) \ {} be any point such that t < δ/n. Then we have = L j f(x, ω 2 (t)) f(x, u ) P α t [ ϕ(τ, x (τ), v j ) ϕ(τ, x (τ), u (τ)) α j] dτ t 1 A t n A n, (15) where A j = max { ϕ(τ, x (τ), v j ) ϕ(τ, x (τ), u (τ)) α j τ L j } for j {1,..., n}. Next take into consideration that, if τ L j for some j {1,..., n}, then τ lies in L satisfies τ τ j a j t t n n t < δ. (16) 43

20 Consequently, (14) implies A j < ε/n for all j {1,..., n}. In view of this result, we get from (15) that hence f(x, ω 2 (t)) f(x, u ) P α t < ε (t t n )/n ε t, Thus (13) is true, as claimed. Next, we denote 1 [ f(x, ω 2 (t)) f(x, u ) P α t ] < ε. t P β t = t 1 β t n β n for all t = (t 1,..., t n ) B n +(r ). A reasoning similar to that used in the proof of (13) reveals that the function satisfies t B n +(r ) F (x, ω 2 (t)) + P β t C(I, W) 1 [ lim t F (x, ω 2 (t)) + P β t ] =. (17) t Indeed, let the number ε > be arbitrarily given. Since the functions (σ, τ) I L φ(σ, τ, x (τ), v j ) W (j {1,..., n}) (σ, τ) I L φ(σ, τ, x (τ), u (τ)) W are continuous on the compact set I L, they are uniformly continuous on this set. Thus there exists a number δ > such that for all j {1,..., n}, all σ I, all τ L satisfying τ τ j < δ the following inequalities hold: φ(σ, τ, x (τ), v j ) φ(σ, τ j, x (τ j ), v j ) < ε/(2n); φ(σ, τ, x (τ), u (τ)) φ(σ, τ j, x (τ j ), u (τ j )) < ε/(2n). These inequalities imply φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), v j ) + β j (σ) < ε/n (18) for all j {1,..., n}, all σ I, all τ L satisfying τ τ j < δ. 44

21 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Now, let t B n +(r ) \ {} be any point such that t < δ/n. Then we have = L j F (x, ω 2 (t))(σ) + (P β t)(σ) [ φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), v j ) + β j (σ) ] dτ t 1 B 1 (σ) t n B n (σ) (19) for every σ I, where B j (σ) = max { φ(σ, τ, x (τ), u (τ)) φ(σ, τ, x (τ), v j ) + β j (σ) τ Lj } for j {1,..., n}. As before, now take into consideration that if τ L j for some index j {1,..., n}, then τ lies in L satisfies (16). Consequently, (18) implies B j (σ) < ε/n for all j {1,..., n} all σ I. In view of this result, we get from (19) that F (x, ω 2 (t))(σ) + (P β t)(σ) < ε (t t n )/n ε t for all σ I. From this it follows that hence Thus (17) is true, as claimed. From (17) we obtain F (x, ω 2 (t)) + P β t < ε t, 1 [ F (x, ω 2 (t)) + P β t ] < ε. t where 1 [ lim t ω (t) + t t j A 1 β j] =, (2) Obviously, (2) yields ω (t) = A 1 F (x, ω 2 (t)) for all t B n +(r ). 1 [ lim t d1 f(x, u )ω (t) + t Finally, note that the point P t defined by t j d 1 f(x, u ) A 1 β j] =. (21) P t = t 1 d t n d n for all t = (t 1,..., t n ) R n, 45

22 in our case can be written under the form P t = P α t + t j d 1 f(x, u ) A 1 β j. Accordingly, we conclude from (13) (21) that 1 [ lim t f(x, ω 2 (t)) f(x, u ) P t d 1 f(x, u )ω (t) ] =. t Summing up, all the hypotheses of Theorem 1 are fulfilled. By applying this theorem, it follows that there is a vector λ = (λ 1, λ 2, λ 3) K 1 K 2 K 3 \ {( 1, 2, 3 )} satisfying the inequality d(τ, v), λ whenever (τ, v) I V (22) as well as the equality (3). From (22) we obtain (2). Indeed, to see this, we fix any τ I. Since we have A 1 φ(, τ, x (τ), v) = h( ; τ, v) for all v V, it follows that d 1 f(x, u ) A 1 φ(, τ, x (τ), v) = d 2 ϕ(σ, x (σ), u (σ))h(σ; τ, v) dσ. In view of this result, H(τ, ) can be rewritten as follows: H(τ, v) = ϕ(τ, x (τ), v) + d 1 f(x, u ) A 1 φ(, τ, x (τ), v), λ for every v V. Therefore we have H(τ, v) H(τ, u (τ)) = d(τ, v), λ for all v V. In virtue of (22) it follows that H(τ, v) H(τ, u (τ)) for all v V. Consequently, the equality (2) holds, which completes the proof. 46

23 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM References [1] W. W. Breckner, Derived sets for weak multiobjective optimization problems with state control variables, J. Optim. Theory Appl. 93 (1997), [2] M. R. Hestenes, On variational theory optimal control theory, SIAM J. Control 3 (1965), [3] M. R. Hestenes, Calculus of Variations Optimal Control Theory, John Wiley Sons, New York, [4] M. R. Hestenes, Optimization Theory, John Wiley Sons, New York, [5] W. H. Schmidt, Notwendige Optimalitätsbedingungen für Prozesse mit zeitvariablen Integralgleichungen in Banachräumen, Z. Angew. Math. Mech. 6 (198), Faculty of Mathematics Computer Science, Babeş-Bolyai University, 34 Cluj-Napoca, Romania 47