On the Existence of Optimal Control of Differential Systems with After-Effect

This paper deals with the study of the optimal control problem for the objective F (x, u, v) = ∫ T 0 f(x(t), x(t− h), ζ(t), u(t), v(t), t)dt, with x ∈ X, u ∈ U, v ∈ V ; X, U and V being vector spaces, and ζ(t) = ∫ h 0 R(t, τ )x(t− τ ) dτ , subject to the differential equation d dtx(t) = m(x(t), x(t− h), ζ(t), u(t), v(t), t) (0 6 t 6 T ), and the constraints g1(u(t), t) ∈ S1, g2(v(t), t) ∈ S2; n1(x(t), t) ∈ V1, n2(x(t− h), t) ∈ V2; n3(ζ(t), t) ∈ V3 (0 6 t 6 T ), where x(t) ∈ R; ζ(t) ∈ R; u(t) ∈ R; f, m, gi (i = 1, 2), ni (1 6 i 6 3) and the entries to r(t, τ ) : R+ × R+ → L(X,X) are continuously differentiable functions. It is assumed that boundary conditions x(0) = x(T ) = 0 are imposed. Si (i = 1, 2) and Vi (1 6 i 6 3) are convex cones. The existence of a time-optimal control in analytic linear systems is also investigated via an extension of the bang-bang principle.