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An optimal control problem in coefficients for a strongly degenerate parabolic equation with interior degeneracy. (English) Zbl 1373.49002
Summary: We deal with an optimal control problem in coefficients for a strongly degenerate diffusion equation with interior degeneracy, which is due to the nonnegative diffusion coefficient vanishing with some rate at an interior point of a multi-dimensional space domain. The optimal controller is searched in the class of functions having essentially bounded partial derivatives. The existence of the state system and of the optimal control are proved in a functional framework constructed on weighted spaces. By an approximating control process, explicit approximating optimality conditions are deduced, and a representation theorem allows one to express the approximating optimal control as the solution to the eikonal equation. Under certain hypotheses, further properties of the approximating optimal control are proved, including uniqueness in some situations. The uniform convergence of a sequence of approximating controllers to the solution of the exact control problem is provided. The optimal controller is numerically constructed in a square domain.

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35K65 Degenerate parabolic equations
35R30 Inverse problems for PDEs
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