Optimal control of linear retarded systems in Banach spaces

Abstract

This paper deals with standard optimal control problems, namely, the fixed time integral convex cost problem and the time optimal control problem for linear retarded systems in Banach spaces. For the basis of optimal control theory the fundamental solution is constructed and a variation of constant formula of (mild) solutions is established. After the controlled system description and the formulation of optimal control problems are given, the retarded adjoint system is introduced. For the integral convex cost problem two existence theorems of optimal controls and necessary conditions of optimality are given. These conditions are precisely characterized by the solution of retarded adjoint system. The “pointwise” maximum principle for time varying control domain is derived from the optimality conditions. The bang-bang principle is also established for the terminal value cost problem under some regularity condition of the adjoint system. For the time optimal control problem to a target set an existence theorem is shown. In the case where the target set has interior, the maximum principle and the bang-bang principle are established for the time optimal control. Finally, a convergence theorem of time optimal controls to a point target set is given. This paper also contains illustrative examples which give technologically important control problems.