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

作者：Marinoschi Gabriela^*; Mininni Rosa Maria; Romanelli Silvia

来源：Journal of Optimization Theory and Applications, 2017, 173(1): 56-77.

DOI：10.1007/s10957-017-1077-4

• Optimal control
• Optimality conditions
• Degenerate diffusion equations
• Interior degeneracy
• Coefficient identification
• Eikonal equation

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.

• 出版日期2017-4