A Class Of Nonlinear Impulsive Integral Differential Equations And Optimal Control

In this thesis, by using the semigroup theory, we study a class of nonlinear impulsive integral differential equations and optimal control problem of system governed by integral differential equations on Banach space.We at first study a nonlinear impulsive integral differential equations as follows:x (t) = Ax(t) + F(t,x(t), (Gx)(t)),t ∈ (0,T]/Dx(0) = x₀ (1)â–³x(t_i) = J_i(x(t_i)),i = 1,2, ...non a Banach Space X. We study the existence and uniqueness of mild solution, the regularity properties.Furthermore, we turn to consider the following controlled system governed by impulsive integral differential evolution equations:x(t) = Ax{t) + F(t, x(t), (Gx)(t)) + B(t)u(t),t ∈ (0, T]\Dx(0) = x₀, (2)We discuss the existence of optimal control for Bolza problem (P): Find u⁰ ∈ U_ad such thatWherex^u denotes the mild solution of system (2) corresponding to the control u ∈ U_ad. We also study the continuous dependence on initial states and control for the system (2); the necessary conditions of optimality which is the main result of this thesis are presented.At last, we present an example to demonstrate our results.