Space−time spectral approximations of a parabolic optimal control problem with an L2‐norm control constraint

This article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an L2$$ {L}^2 $$ ‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method.