Optimality-based control space reduction for high-dimensional control spaces

Abstract We study Galerkin model reduction for unconstrained linear-quadratic optimal control problems and show that state-space reduction alone already induces a reduced control structure via the optimality conditions. As a result, the solely state-reduced and the combined control- and state-reduced problems are equivalent, allowing fast optimization over a reduced control space without introducing additional approximation error. We derive lower and upper a posteriori error bounds for the optimal control and use them within an online-adaptive algorithm that constructs sufficiently accurate reduced spaces while solving the control problem. Convergence of the algorithm is proven, and numerical results demonstrate that combined control and state-space reduction yields significant speed-ups without loss of accuracy compared to state-space reduction alone.