Analysis and approximations of an optimal control problem for the Allen-Cahn equation

by   Konstantinos Chrysafinos, et al.

The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters k, h respectively in terms of the parameter ϵ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon 1/ ϵ. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of ϵ. These estimates and a suitable localization technique, via the second order condition (see <cit.>), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations


page 1

page 2

page 3

page 4


Error estimates for the numerical approximation of a non-smooth quasilinear elliptic control problem

In this paper, we carry out the numerical analysis of a non-smooth quasi...

Bilinear optimal control for the fractional Laplacian: error estimates on Lipschitz domains

We adopt the integral definition of the fractional Laplace operator and ...

Semilinear optimal control with Dirac measures

The purpose of this work is to analyze an optimal control problem for a ...

Solution of the Optimal Control Problem for the Cahn-Hilliard Equation Using Finite Difference Approximation

This paper is concerned with the designing, analyzing and implementing l...

An efficient jet marcher for computing the quasipotential for 2D SDEs

We present a new algorithm, the efficient jet marching method (EJM), for...

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

Based on relative energy estimates, we study the stability of solutions ...

Please sign up or login with your details

Forgot password? Click here to reset