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.
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Acknowledgements
The authors thank the reviewers for the extremely careful lecture of the paper and for the pertinent observations and suggestions made in an earlier version. G.M. acknowledges the support of INdAM-GNAMPA, Italy, for May 2015 and of the grant CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0027. R.M.M. and S.R. acknowledge the support of INdAM-GNAMPA.
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Communicated by Roland Herzog.
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Marinoschi, G., Mininni, R.M. & Romanelli, S. An Optimal Control Problem in Coefficients for a Strongly Degenerate Parabolic Equation with Interior Degeneracy. J Optim Theory Appl 173, 56–77 (2017). https://doi.org/10.1007/s10957-017-1077-4
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DOI: https://doi.org/10.1007/s10957-017-1077-4
Keywords
- Optimal control
- Optimality conditions
- Degenerate diffusion equations
- Interior degeneracy
- Coefficient identification
- Eikonal equation