NECESSARY OPTIMALITY CONDITIONS FOR AN INTEGRO-DIFFERENTIAL BOLZA PROBLEM VIA DUBOVITSKII-MILYUTIN METHOD

. In the paper, we derive a maximum principle for a Bolza problem described by an integro-diﬀerential equation of Volterra type. We use the Dubovitskii-Milyutin approach.

One can show that the set of directions of decrease of functional F at a point x 0 and the set of feasible directions for the set Q at a point x 0 are open cones as well as the set of tangent directions to a set Q at a point x 0 is a cone. Now, assume X is locally convex and consider the problem The central result of the Dubovitskii-Milyutin approach is the following theorem.
Theorem 2.4 (Dubovitski-Miljutin). Let x 0 be a local minimum point for problem (4). Assume that the cone K 0 of directions of decrease of functional F at x 0 is nonempty and convex, cones K i , i = 1, ..., n, of feasible directions for the sets Q i at x 0 are nonempty and convex and the cone K n+1 of tangent directions for the set Q n+1 at x 0 is nonempty and convex. Then there exist functionals f i ∈ K * i , i = 0, 1, ..., n + 1, not all identically zero, such that The introduced cones have nice characterizations in some situations. Proposition 1. If X is a Banach space, F : X → R -a functional differentiable at x 0 in Frechet sense, then the cone K d of directions of decrease of functional F at x 0 has the form Theorem 2.5 (Lusternik). If X, Y are Banach spaces, P : X → Y -operator of class C 1 , P (x 0 ) = 0 and Im P (x 0 ) = Y , then the cone K t of tangent directions for the set {x ∈ X; P (x) = 0} at x 0 is a subspace of the form Proposition 2. If Q is a convex set in linear topological space E, then the cone K f of feasible directions for the set Q at a point x 0 ∈ Q is convex and has the form We shall also use the following characterizations of the conjugate cones.
Proposition 3. Let E be a linear topological space, f ∈ E and Then Proposition 4. Let Q be a convex closed set in linear topological space E and x 0 ∈ Q. If IntQ = ∅, then We shall also use the following lemma (see [9]). Lemma 2.6. If X, Y are Banach spaces and Λ : X → Y is linear bounded operator such that Im Λ = Y , then (ker Λ) ⊥ = Im Λ * where Λ * : Y → X is adjoint operator to Λ and (ker Λ) ⊥ is the set of linear continuous functionals on X vanishing on ker Λ.
The set (ker Λ) ⊥ is named the anulator of the subspace ker Λ.
3. Bolza problem. Let us consider the following problem with the set of solutions AC 2 0 = AC 2 0 (J, R n ) (set of absolutely continuous functions possessing squared integrable derivatives, vanishing at t = 0) and the sets of functional parameters (controls) consisting of essentially bounded functions taking their values in the sets M ⊂ R m , N ⊂ R r , respectively. On the sets M , N we assume that they are convex closed with non-empty interiors.
3.1. Differential properties of F . In paper [5], an integro-differential equation of Volterra type of fractional order α ∈ (0, 1) (with derivatives in Riemann-Liouville sense) is investigated. Results on the existence and uniqueness of a solution have been obtained there as well as continuous differentiability property of the mapping assigning the corresponding solution to any functional parameters. Our initial problem (1) is a limit case of the system considered in [5], for α = 1. The same proof as that of Lemma 4.1 from [5] (with α = 1) gives the following theorem.
is continuously differentiable in Gataux (equivalently, in Frechet) sense and the mapping In an analogous way as in [5, proofs of Lemma 4.2 and Lemma 6.1] (see also [7], [8, Proof of Lemma 7]) one can obtain for t ∈ J a.e., x ∈ R n , v ∈ R r , then the partial differential F x (x, u, v) : AC 2 0 → L 2 n given by is bijective.
Corollary 1. Under assumptions of Theorem 3.2, the differential ∂F (x, u, v) : In a similar way, as in the case of Theorem 3.1, one obtains Proposition 5. If the function f 0 : for t ∈ J a.e., x ∈ R n , u ∈ R m , v ∈ R r , then the functional is differentiable in Frechet sense and its differential at a point (x, u, v) is given by

Clearly, we have
Proposition 6. If the function g : R n → R is continuously differentiable, then the mapping is differentiable in Frechet sense and the differential at a point (x, u, v) is given by Corollary 2. If assumptions of Propositions 5 and 6 are satisfied, then the functional F 0 given by (3) is differentiable in Frechet sense and its differential ∂F 0 (x, u, v) at a point (x, u, v) is given by 4. Maximum principle.

Conjugate cones. Let us fix a point (x
Using the results of Sections 2, 3 we assert that the cone K d of directions of decrease of functional F 0 at (x * , u * , v * ) has the form Of course, this set is convex. Moreover, it is nonempty if and only if ∂F 0 (x * , u * , v * ) = 0. When K d = ∅, then the conjugate cone K * d is given by Now, let us consider the cone K t of tangent directions for the set at the point (x * , u * , v * ). From the Lusternik theorem it follows that Of course, K t is a nonempty subspace. Consequently, . From the Lemma 2.6 it follows that (ker ∂F (x * , u * , v * )) ⊥ = Im((∂F (x * , u * , v * )) * ). So, To finish this section, let us write the constraint (u, v) ∈ U × V in the form Proposition 2) the cone K f of feasible directions for the set Q at the point (x * , u * , v * ) ∈ Q is the following ; ρ > 0}. Consequently, it is nonempty and convex. From Proposition 4 it follows that conjugate cone K * f has the form r is a local minimum point for the problem under consideration, then there exist functionals g 0 ∈ K * d , g 1 ∈ K * f , g 2 ∈ K * t , not all identically zero, such that g 0 + g 1 + g 2 = 0.