Optimal control of fractional semilinear PDEs
It provides a theoretical foundation for optimal control of fractional PDEs, which is important for applications in anomalous diffusion and nonlocal phenomena, but the results are incremental extensions of existing PDE control theory.
This paper establishes well-posedness and optimality conditions for optimal control of semilinear fractional PDEs with spectral and integral fractional diffusion operators, proving boundedness of solutions and Lipschitz continuity of the solution map under minimal regularity assumptions.
In this paper we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order $2s$ with $s \in (0,1)$. We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.