Optimal control of multiphase free boundary problems for nonlinear parabolic equations

Abstract

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty }$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.

arXiv comment27 pages. arXiv admin note: text overlap with arXiv:1508.00290

References in corpus (5)

• On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I.Well-posedness and Convergence of the Method of Lines
• On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences
• Optimal Control of the Multiphase Stefan Problem
• Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems
• Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems

Cited by in corpus (1)

• Optimal Stefan Problem