| Summary: | 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.
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