Nonlinear Impulsive Integrodifferential Systems In Banach Space And Optimal Control

In this thesis, we using semigroup theory to consider the nonlinear impulsive integro-differential system and optimal control on Banach space X. We discussed the following three classes of nonlinear impulsive integro-differential equations, that is,(a) time-varying nonlinear impulsive integro-differential equations(b) nonlinear impulsive integro-differential equations of mixed type(c) time-varying nonlinear impulsive integro-differential equations of mixed typewhere∨= {t₁, t₂,…, t_n} (?) (0, T), 0 < t₁ < t₂ <…< t_n < T, A is the infinitesimal generator of a Co-semigroup, {A(t),t∈[0,T]} is a family of closed densely define linear operator, and G, S are nonlinear integral operators given byIt is obvious that operator S is much different from G. J_i(i = 1, 2,…, n) is a nonlinear map andΔx(t_i) = x(t_i + 0) - x(t_i - 0) = x(t_i + 0) - x(t_i). This represents the jump in the state x at time t_i, with J_i determining the size of the jump at time t_i.For (1), we give some generalized Gronwall inequalities with including singularity, impulse. Then introduce the reasonable mild solution of the equation (1) and prove the existence and uniqueness of mild solutions. The existence of optimal control problems of systems governed by the equation (1) is presented. The necessary conditions of optimality for optimal control problems arising in systems governed by nonlinear impulsive integro-differential systems was presented. Lastly, some example is given for demonstration.For (2), we give some generalized Gronwall inequalities with impulse and integral of mixed type and a generalized Ascoli-Arzela theorem. Then introduce the reasonable mild solution of the equation (2) and prove the existence of mild solutions. The existence of optimal control problems of systems governed by the equation (2) is presented. Lastly, some example is given for demonstration.For (3), we give some generalized Gronwall inequalities with including singularity, impulse and integral of mixed type. Then introduce the reasonable mild solution of the equation (3) and prove the existence of mild solutions. The existence of optimal control problems of systems governed by the equation (3) is presented. Lastly, some example is given for demonstration.