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From Open to Closed Loop Optimal Control of PDEs

Periodic Reporting for period 4 - OCLOC (From Open to Closed Loop Optimal Control of PDEs)

Reporting period: 2020-07-01 to 2021-12-31

The advanced grant OCLOC focused on pushing the boundaries of our understanding of optimal control of nonlinear partial differential equations (PDEs) from the point of view of mathematical analysis, and its numerical and algorithmic realization.
Non-smoothness and non-convex optimal control problems, receding horizon control, optimal control of reaction diffusion equations, and Hamilton Jacobi Bellman equations are some of the keywords characterizing our research. Major progress was made on all work packages.

Historically optimal control with PDEs as constraints arise in complex technological application areas. More recently they also arise in the biosciences and biomedical engineering. Our work in WP3 on the biodomain equations, modelling the electrical activity of the heart, is one such application area of significant practical relevance.
A mayor goal of the ERC grant was to shift part of the focus from open loop control of PDEs to closed loop control. This objective was certainly well achieved and this paradigmatic change has been well recognized in the scientific community. An important side effect of this procedure is that also the focus and interests of the younger members in the research groups of the PI shifted towards new horizons.

For young researchers, work on these topics is particularily demanding, since it requires knowledge and expertise in a large and diverse number of mathematical disciplines All personnel that were employed have obtained good positions after leaving the team of the ERC grant.

The advances made towards optimal control of PDEs would undoubtedly not have been possible without the grant.
We point to selected research highlights which could be achieved during the grant period. Non-smooth sparsity-promoting functionals were analyzed in the context of optimal stabilization problems. The peculiarity of this work is that it could be verified that the optimal controls completely shut down once they are sufficiently close to a stable steady state. In our work on nonconvex penalization of switching control of partial differential equations, optimal control problems with switching constraints, were analysed. New, primal-dual based, numerical schemes for non-convex optimization problems were proposed and analyzed for image reconstruction and fracture mechanics.
The synthesis of optimal feedback policies in the context of non-smooth/non-convex optimization framework was investigated. The relevance of this work is that it is one of the first attempts to understand the interplay of nonsmooth costs with feedback controls and Hamilton-Jacobi-Bellman (HJB) equations. In particular, it presents a characterization of parsimonious control actions in terms of sparsity, and switching conrtols.
Another application driven topic was the study of real-time synthesis for multiscale agent-based models originating from social dynamics and crowd motion by model predictive control techniques. The design of model predictive control strategies to reduce human congestion in evacuation scenarios was developed. Model predictive control techniques are essential to reduce the associated computational burden and to achieve real-time computability. We validated their use for diverse classes of PDE-constrained optimal control problems. Viscosity methods for large deviations estimates of multiscale stochastic processes for Hamilton-Jacobi-Bellman (HJB) equations were studied with special attention to the financial applications. Nonlocal HJB equations related to discontinuous stochastic processes were analyzed as well.

For the monodomain equation, arising in context of electro-cardiology modelling, lack of exact null-controllability could be established. For the one-dimensional case, however, together with moving controllers, exact null-controllability was verified, and optimal boundary stabilizability results were obtained for the 3-D case case. Further, stabilizability by finite dimensional controls was investigated for this class of equations.In a series of highly technical papers, Taylor expansions for the value function associated to optimal control problems were developed, and their numerical relevance was investigated. Several concepts to weaken the curse of dimensionality when computing optimal feedback gains were proposed and analyzed. They rest, in part, on approximating HJB equations, to the other part on machine learning techniques.
By their very nature, all published papers are beyond the state of the art. I especially point to
the numerical work on solving the HJB equation in high dimensions, which was considered as the high risk component of the grant.
In part this is connected to the fact that during the funding period machine learning and neural network techniques emerged as a promising field. We made use of these
novel technologies and merged them with our goals in WP4.

Selected highlights which significantly go beyond the state of the are specified next.
1. The introduction of neural networks and learning techniques to construct feedback controls for systems governed by PDEs.
2. The use of data driven and machine learning techniques to partially overcome the curse of dimensionality in solving HJB equations.
3. Polynomial and tensor based approximation techniques to partially overcome the curse of dimensionality in solving HJB equations.
4. The analysis of the importance of terminal penalties in the receding horizon approach for optimal control of PDEs.
5. The analysis of optimal control of Navier Stokes equations with measure valued controls.
6. Analysis and algorithmic realisation of primal-dual techniques for nonsmooth nonconvex optimization.
7. Taylor expansions for the value functions of infinite dimensional optimal conrtol problems.