An Optimal Control Problem in Coefficients for a Strongly Degenerate Parabolic Equation with Interior Degeneracy

Gabriela Marinoschi (), Rosa Maria Mininni () and Silvia Romanelli ()
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Gabriela Marinoschi: Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy
Rosa Maria Mininni: Università degli Studi di Bari Aldo Moro
Silvia Romanelli: Università degli Studi di Bari Aldo Moro

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 1, No 3, 56-77

Abstract: Abstract 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.

Keywords: Optimal control; Optimality conditions; Degenerate diffusion equations; Interior degeneracy; Coefficient identification; Eikonal equation; 49J20; 49K20; 35K65; 35R30 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1077-4

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