Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity

Abstract

In this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation.

Keywords

• weak solution
• parabolic equation
• variable order of nonlinearity
• noncoercive problem
• compensated compactness technique