Banach Space, Second Order Integro Differential Equations And Optimal Control

In this thesis, we use semigroup of operators theory to consider the second order nonlinear integro-differential systems with unbounded operator on Banach space X, and study optimal control problem of the corresponding system. We discuss the following two classes of second order nonlinear integro-differential equations, that is,(a) Case one:where B is the infinitesimal generator of a C₀ - semigroup in Banach space X, A is a closed linear unbounded operator, and D(B) (?) D(A).(b) Case two:where A = E², E is the infinitesimal generator of a C₀ - group on Banach spaceX, the operator G above (1) and (2) is a nonlinear integral operator dependence of x and x, given byFor (1), in the first, we construct the semigroup generated by the matrix operatorΓ=(?) by means of the semigroup generated by the principal operator B, andinvestigate its property(such as compactness). Secondly, we introduce the reasonable mild solution of the equation (1) and prove the existence and uniqueness of mild solution, we also obtain continuous dependence of the solutions on the initial data. Further, the existence of optimal control problems of systems governed by the equation (1) is presented. In the end, some example is given for demonstration. For (2), in order to conquer the difficulty the unbounded operator matrix can not generate semigroup on X×X, we introduce the suitable interpolation spaces and reasonable mild solution, prove the existence and uniqueness of mild solution of the equation (2), and also obtain continuous dependence of the solutions on the initial data. Further, the existence of optimal control problems of systems governed by the equation (2) is presented. At last, some example is given for demonstration.