Continuous Dependence of the Minimum of the Bolza Type Functional on the Initial Data in Nonlinear Optimal Control Problems with Distributed Delay

Let Rx be the n-dimensional vector space of points x = (x1, . . . , xn)T , where T is the sign of transposition. Suppose that O ⊂ Rx, V ⊂ Ru are open sets and X0 ⊂ O, P ⊂ Rp are compact sets. Let the (1+n)-dimensional function F (t, x, u, p) = (f0, f)T be continuous on the set I×O×V ×P and continuously differentiable with respect to x ∈ O, where I = [t0, t1]. By ∆(V ) we denote collection of compact sets U ⊂ V . Let 0 ≤ σ1 < σ2 be a given number and let Φ be the set of initial functions φ(t) ∈ O, t ∈ [t0 − σ2, t0]. Next, let Ω(U) be the set of measurable control functions u(t) ∈ U , t ∈ I, where U ∈ ∆(V ) and let Q be the set of continuous scaler functions q(t, x), (t, x) ∈ I ×O. To each μ = (σ, x0, φ, p) ∈ Λ = [σ1, σ2]×X0×Φ×P we put into correspondence the controlled differential equation with distributed delay