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Nonconvex variational problems related to a hyperbolic equation. (English) Zbl 0956.49001
This paper is concerned with an optimal control problem for hyperbolic equations with Darboux boundary conditions. The state variable $$z$$ takes values in $$\mathbb{R}^n$$. The cost functional is not convex, it involves terms depending on $$\langle z,\nu \rangle$$, where $$\nu$$ is a given direction in $$\mathbb{R}^n$$. The authors first prove an extension of the Lyapunov convexity theorem and next establish the existence of optimal solutions. Bang-bang results are also considered.

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 49J45 Methods involving semicontinuity and convergence; relaxation 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 35A15 Variational methods applied to PDEs
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