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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - OCLOC (From Open to Closed Loop Optimal Control of PDEs)

Teaser

Summary

The advanced grant OCLOC focuses on pushing the boundaries of our understanding of optimal control of partial differential equations (PDEs) from the point of view of mathematical analysis and its numerical 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. While the postdocs hired by ERC-support mainly contributed to WP1 and WP4, many papers could also be written and are in various stages of acceptance, which contribute to the goals of WP2 and WP3.
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.
One of the goals of the ERC grant is to shift part of the focus from open loop control of PDEs to closed loop control. 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 shift towards new horizons. This objective is certainly well on its way and the paradigmatic change has been observed in the scientific community.

Work performed

\"In the following, we point to selected papers from the first funding period. Non-smooth sparsity-promoting functionals were analyzed in the context of optimal control in \"\"Stabilization by sparse controls for a class of semilinear parabolic equations\"\" by E. Casas and K. Kunisch. 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 \"\"Nonconvex penalization of switching control of partial differential equations\"\" by C. Clason, K. Kunisch and A. Rund, non-convex functionals were exploited to solve optimal control problems with switching constraints, where at most one component of the control is allowed to be active at any point in time. In \"\"On the monotone and primal-dual active set schemes for l^p-type problems, pâˆˆ (0,1]\"\" by D. Ghilli and K. Kunisch, new numerical schemes for non-convex optimization problems were proposed and applied to optimal control problems, reconstruction of images, and fracture mechanics.
The synthesis of optimal feedback policies in the context of the aforementioned non-smooth/non-convex optimization framework was studied in \"\"Infinite horizon sparse optimal control\"\", by D. Kalise, K. Kunisch, and Z. Rao. The relevance of this work is that it is one of the first attempts to understand the interplay of WP1 with feedback controls and Hamilton-Jacobi-Bellman (HJB) equations in WP4. In particular, it presents a characterization of parsimonious control actions in terms of sparsity (optimal controllers should have as many zero components as possible), and switching (optimal controllers should have only one non-zero component). The analysis of such a transition regime is the subject of current work, and has already generated interesting applications in the context of agent-based models and social dynamics, such as the article \"\"A Boltzmann approach to mean-field sparse feedback control\"\" with G. Albi and M. Fornasier.
Another application driven topic has been the study of real-time synthesis for multiscale agent-based models originating from social dynamics and crowd motion by model predictive control techniques. In the paper \"\"Invisible control of self-organizing agents leaving unknown environments\"\", by G. Albi, M. Bongini, E. Cristiani and D. Kalise, the design of model predictive control strategies to reduce human congestion in evacuation scenarios is presented.
In \"\"Viscosity methods for large deviations estimates of multiscale stochastic processes\"\" by D. Ghilli homogenization of Hamilton-Jacobi-Bellman (HJB) equations was studied with special attention to the financial applications. Nonlocal HJ equations related to discontinuous stochastic processes were analyzed in \"\"On Neumann problems for nonlocal Hamilton-Jacobi equations with dominating gradient terms\"\" by D. Ghilli.
In \"\"Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs\"\" by D. Kalise and K. Kunisch, a computational technique for the solution of high-dimensional HJB equations was presented. Experiments up to dimension 14 are reported.
For the monodomain equation, arising in WP3 in the 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 by D.Souza and K.Kunisch, and boundary stabilizability results were obtained by T.Breiten and K. Kunisch for the 3-D case.\"

Final results

\"Non smoothness optimization problems had already been studied, mainly for L1-type functionals, and algorithms like iteratively reweighted least squares algorithms (IRLS) and the \"\"Fast Iterative Schrinkage Thresholding algorithm\"\" (FISTA) had been developed. The particular new feature of our problems is that they are not only non-smooth but non-convex. Therefore, new analytical and numerical techniques needed to be developed, in part inspired by Newton-type methods. The research carried out on l^p sparse optimization with p<1 revealed that the use of non-smooth non convex functionals is of great practical relevance in diverse situations including imaging and optimal control. Particularly relevant are the applications in continuum mechanics, e.g. brittle or cohesive fracture, where singular behavior is modeled by non-smooth non-convex energies.
The research on model predictive control for crowd motion studied in WP2 showcases the great potential for applicability of model predictive control in different societal challenges related to agent-based models in life and social sciences. Model predictive control techniques are essential to reduce the associated computational burden and achieve real-time computability. This research has attracted considerable attention by European media, including press releases in Germany, Italy, and Austria (https://tinyurl.com/ychpzz5c).
Control problems on multidomain studied in WP4 have become recently an active fields of research and have several applications e.g. to optimal control on networks and hybrid systems. Our new main contributions consist in junction conditions which guarantee the well-posedness of the HJB equation.
The high-dimensional computational solver for HJB developed in WP4 outperforms by a factor of 2 the available toolboxes. The proposed approach is a step towards optimal feedback synthesis of large-scale dynamical systems, such as quadrocopters, or approximate PDE models. Introducing additional tensor analysis techniques may allow to further improve its performance. In the future we plan to extend the approach to HJB-Isaacs equations.
We further plan to keep up our exciting activities investigating expansions of the value functions for optimal control problems, exploiting recent result in the context of receding horizon control and possibly also for turnpike behavior.
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