Space–time spectral collocation method for one-dimensional PDE constrained optimisation

This paper solves an optimal control problem governed by a parabolic PDE. Using Lagrangian multipliers, necessary conditions are derived and then space–time spectral collocation method is applied to discretise spatial derivatives and time derivatives. This method solves partial differential equations numerically with errors bounded by an exponentially decaying function which is dependent on the number of modes of analytic solution. Spectral methods, which converge spectrally in both space and time, have gained a significant attention recently. The problem is then reduced to a system consisting of easily solvable algebraic equations. Numerical examples are presented to show that this formulation has exponential rates of convergence in both space and time.