Model Predictive Control for the Fokker-Planck Equation

• Applicant: Professor Dr. Lars Grüne
• Subject Area: Mathematics
• Term: from 2014 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 264433583

For the optimal control of stochastic control systems two different approaches appear in the literature. The first, more traditional approach considers the optimization of expected values or higher moments along future trajectories. The second approach considers the control of the probability density function (PDF) according to some objective, which in the simplest case consists of controlling the PDF to a prespecified reference PDF. The latter approach is in several respects more powerful and versatile, as it allows to shape the entire distribution of the future trajectories opposed to influencing only some moments. However, it is also more difficult to realize. As the PDF can be expressed via the solution of the Fokker-Planck equation - a parabolic partial differential equation (PDE) - the optimal control of PDFs can be posed as a (deterministic) optimal control problems with PDE constraints. Due to the recent progress in optimal control of PDEs as well as in model predictive control, these methods open a feasible way to the control of PDFs, as illustrated by promising results in recent papers by Annunziato and Borzì. Model predictive control (MPC) is the method of choice due to its ability to split up an optimal control problem on a long or infinite time horizon into a series of optimal control problems on shorter horizons which are thus much easier to solve. Two of the key issues in the analysis of MPC schemes are stability and performance. While "stability" expresses the fact that the controlled PDF converges to a desired reference PDF, "performance" measures the loss of optimality of the MPC approach with respect to the true optimal solution on the long or infinite horizon. The latter is particularly important if the reference PDF is not given a priori and encoded in a tracking type functional but implicitly derived from a more general optimization objective - a setting which recently attracted considerable attention under the name of economic MPC. The objective of the theoretical part of this proposal is to derive rigorous statements about stability and performance of Fokker-Planck based MPC. The key challenges are the difficulties introduced by the infinite dimensional PDE setting and the identification of meaningful structural properties of the underlying stochastic control system allowing for such rigorous results.At the core of each MPC scheme a fast and reliable numerical algorithm is needed in order to compute the solutions of the short horizon subproblems. To this end, the development of efficient numerical methods based on suitable optimality conditions will complement the theoretical investigations as a second objective of this proposal. Besides constituting a contribution in its own right, the availability of efficient numerical codes will also serve for the simulation based identification of suitable assumptions for our theoretical results and for verifying the efficiency of the MPC schemes derived in this proposal.

• DFG Programme: Research Grants
• Cooperation Partner: Professor Alfio Borzi, Ph.D.