Workpackage 3

One efficient strategy for dealing with optimal control problems on an infinite time horizon is Model Predictive Control (MPC). In this approach, an infinite horizon optimal control problem is approximated by a sequence of finite horizon problems in a receding horizon fashion. Stability (convergence to steady state) is not generally ensured due to the use of a finite prediction horizon. Thus, in order to ensure the asymptotic stability of the controlled system,  additional terminal cost functions and/or terminal constraints are often needed to be added to the finite horizon problems. In Workpackage 3, we are concerned with the analysis of MPC for controlled systems governed by partial differential equations. As the first goal, we commence with establishing sufficient conditions for infinite dimensional continuous time nonlinear dynamical systems (PDEs) which guarantee the stability of MPC without terminal cost and terminal constraints. In this respect, the stabilizability of the underlying system is the key condition. Based on this condition the suboptimality and stability of MPC are investigated.With relation to Workpackage 2, we investigate the stability of MPC applied to the monodomain equations. The next step will consist in analysing MPC for the stabilization of trajectories.

More recently, sparse controls have attracted the attention of many researchers in the fields of optimal control of partial differential equations.  As the second goal, we investigate MPCs which are spatially and/or temporally sparse. To enhance the sparsity, we shall use  functionals which exhibit temporal  and/or spatial sparsity pattern. This can be achieved with non-smooth analysis techniques and, thus, will rely on the part in Workpackage 1, for which we choose p=1.