joint work with Stefan Klus
We present a framework for feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator describing the dynamics of observables. By introducing a finite number of constant controls, a dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem, which can be solved efficiently when replacing the PDE constraint of the autonomous systems by K-ROMs. The approach is applied to examples of varying complexity.
joint work with Sebastian Sager, Sven Leyffer
We present a gradient descent method for optimal control problems with ordinary or partial differential equations and integer-valued distributed controls. Our method is based on a derivative concept similar to topological gradients. We formulate first-order necessary optimality conditions, briefly sketch a proof of approximate stationarity, and address issues with second derivatives and sufficient optimality conditions. We give numerical results for two test problems based on the Lotka-Volterra equations and the Laplace equation, respectively.
joint work with Christian Kirches
PDE-constrained control problems with distributed integer controls can be relaxed by means of partial outer convexification. Algorithms exist that allow to approximate optimal relaxed controls with feasible integer controls. These algorithms approximate relaxed controls in the weak$^*$ topology of $L^\infty$ and can be interpreted as a constructive transfer of the Lyapunov convexity theorem to mixed-integer optimal control, which yields state vector approximation in norm for compact solution operators of the underlying process. The chain of arguments is demonstrated numerically.