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

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 $L^2\left(H^1\right)-L^2\left(L^2\right)$ a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain $L^2\left(L^2\right)-L^2\left(L^2\right)$ a posteriori error estimates of the approximation solutions for both the state and the control.