OPTIMAL CONTROL OF THE COEFFICIENT FOR FRACTIONAL $P$-LAPLACE EQUATION : APPROXIMATION AND CONVERGENCE (Theory of Evolution Equation and Mathematical Analysis of Nonlinear Phenomena)

Abstract

In [.r5] we studied optimal control problems with regional fractional p-Laplace equation, of order sin(0, 1) and pin[2, infty), as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional p-Laplace operator. The purpose of this note is to provide a roadmap on how to apply the results of [5] to the fractional p-Laplace case. The existence and uniqueness of solutions to the state equation and existence of solutions to the optimal control problem follow using similar arguments as in [5]. We prove that the fractional p-Laplacian approaches the standard p-Laplacian as s approaches 1. In this sense, the fractional p-Laplacian can be considered degenerate like the standard p-Laplacian. The remaining steps' are similar to the regional fractional p-Laplacian case, i.e., introduce an auxiliary state equation and the corresponding control problem and then conclude with the convergence of regularized solutions.