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∞ bound, and W 1,1 2 -energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.2010 Mathematics Subject Classification. 35R30, 35R35, 35K20, 35Q93, 49J20, 65M06, 65M12, 65M32, 65N21.Key words and phrases. Inverse multidimensional multiphase Stefan problem, Quasilinear parabolic PDE with discontinuous coefficients, optimal control, Sobolev spaces, method of finite differences, discrete optimal control problem, energy estimate, embedding theorems, weak compactness, convergence in functional, convergence in control, maximal monotone graph.• the following inequalities are satisfied: lim sup ε→0 J * (ε) ≥ J * , lim inf ε→0 J * (−ε) ≤ J * , (3.16) where J * (±ε) = inf GR±ε J (g).Lemma 3.3. [5] The mappings P n , Q n satisfy the conditions of Lemma 3.2.

An artificial neural network‐based method for the optimal control problem governed by the fractional parabolic equation

“…▪ Theorem 5. 3 The state trajectory of model (39) converges to an equilibrium point for any initial point w(0) ∈ R n . In particular, when Ω e has unique equilibrium point, the model (39) is globally asymptotically stable with any initial point w(0) ∈ R n .…”

“…Example 6. 3 We consider the fractional optimal control problem with the following exact solution of the state function…”

“…a space time spectral method is used to solve this problem in Reference [2]. Recently Abdulla and Cosgrov proposed a solution for the multiphase Stefan type free boundary problem as optimal control of parabolic equation [3]. Steinbach et al used space–time finite element methods on fully unstructured simplicity space–time meshes for the numerical solution of parabolic optimal control problems [4].…”

Optimal Control of Singular Parabolic PDEs Modeling Multiphase Stefan-type Free Boundary Problems

“…In earlier paper [7] the method was applied to one dimensional multiphase Stefan problem. In [8] the results of [7] are extended to general second order parabolic free boundary problems in space dimension one. The goal of this paper is to extend the method and results of [9] to optimal control of singular PDE modeling multiphase Stefan-type free boundary problems for the general second order parabolic operators.…”