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Optimal control of non-smooth wave equations. (English) Zbl 1505.35264

Summary: This paper analyzes the optimal control of an abstract Cauchy problem in nonlinear wave phenomena with nonlinearities assumed to be Lipschitz continuous, monotone, and directionally differentiable. We develop a mathematical optimization theory, including existence and first order optimality conditions. The existence is established by proposing a specific convergence property for the governing state system on the basis of the semigroup theory, the energy balance equality, and monotonicity arguments. Thereafter, optimality conditions of Bouligand type and of weak stationarity type are established by means of the directional differentiability of the control-to-state mapping and a regularization approach. A concrete application in nonlinear acoustic wave phenomena is discussed in the final part of the paper.

MSC:

35L60 First-order nonlinear hyperbolic equations
34K35 Control problems for functional-differential equations
35Q93 PDEs in connection with control and optimization
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