Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint

<p style='text-indent:20px;'>This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm error for the state and co-state but also <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.</p>

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