Optimal Control Problems for Nonlinear Hyperbolic Distributed Parameter Systems with Damping Terms

Abstract

Abstract. In this paper we study the quadratic optimal control problems for the nonlinear damped second order evolution equations in Hilbert spaces of Gelfand fivefolds. We prove the existence of optimal controls, and establish the necessary conditions of optimality according to various types of observations by using the transposition method.

Optimal control problems for the equation of motion of membrane with strong viscosity

“…The result enables us to study the optimal control problems associated with (1.2) in the standard manner due to the theory of Lions [5]. We also refer to Ha and Nakagiri [2] for the optimal control problems on second order semilinear equations.…”

Fréchet differentiability of solution mappings for semilinear second order evolution equations

“…The above result can be used to derive the necessary optimality conditions for non-convex cost functionals as studied in [4], and also to derive the local controllability for the control system associated with (5.1).…”

Gâteaux differentiability of solution mappings for semilinear second-order evolution equations

“…Also in [4] only the Lipschitz continuity on the nonlinear term f (t, y) in y is supposed without assuming any compactness nor monotonicity on f (t, y). The well-posedness results and the functional differentiability for the solutions are essentially utilized in Ha and Nakagiri [5] for the study of optimal control problems for (1.1). The differentiability is crucial in studying the optimization problems, and we can extend some of results obtained in Ahmed and Teo [1] and Lions [6], to the nonlinear system involving (1.1) by using the differentiability.…”