Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimizer as the solely state-reduced problem. Lower and upper \emph{a posteriori} error bounds for the optimal control and a representation for the error in the optimal function value are provided. These bounds are used in an adaptive algorithm to solve the control problem. We prove its convergence and numerically demonstrate the advantage of combined control and state space reduction.