Minimax Optimal Control Problems for a Semilinear Elastic Beam Equations with Memory
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This paper is concerned withthe minimax optimal control problems of a semilinear elastic beam equation with memory. The existence of an optimal solution for distributed control with a fixed disturbance variable $v$ given as an initial velocity, i.e., the existence problem of the $(P_{v})$ problem, is proven, and the existence of a minmax optimal solution that does not constrain the initial disturbance, i.e., the existence of an optimal solution for the $(P)$ problem is proven. Finally we derive the necessary optimality conditions for optimal solutions ofthe $(P_{v})$ and $(P)$ problems in form of Pontryagin's maximum principle.
Abstract
This paper is concerned withthe minimax optimal control problems of a semilinear elastic beam equation with memory. The existence of an optimal solution for distributed control with a fixed disturbance variable $v$ given as an initial velocity, i.e., the existence problem of the $(P_{v})$ problem, is proven, and the existence of a minmax optimal solution that does not constrain the initial disturbance, i.e., the existence of an optimal solution for the $(P)$ problem is proven. Finally we derive the necessary optimality conditions for optimal solutions ofthe $(P_{v})$ and $(P)$ problems in form of Pontryagin's maximum principle.