Minimax optimal control problems for an extensible beam equation with uncertain initial velocity

Abstract

This paper is devoted to the problem of minimax optimal control problems of an extensible beam equation with distributed controls and initial velocity disturbances (or noises). The existence of optimal solutions for distributed control with fixed disturbance, namely the (P_{v}) problem, and the existence of minimax optimal solutions to (P) problem without restricting the initial disturbance are proved. We derive the necessary optimality conditions for optimal solutions of the (P_{v}) and (P) problems in the form of Pontryagin’s maximum principle.