A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

Abstract

Abstract In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain L2(H1)−L2(L2) $L^{2}(H^{1})-L^{2}(L ^{2})$ a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain L2(L2)−L2(L2) $L^{2}(L^{2})-L^{2}(L^{2})$ a posteriori error estimates of the approximation solutions for both the state and the control.

Keywords

• Optimal control problem
• Nonlinear parabolic equations
• Variational discretization
• Spectral method
• A posteriori error estimates