A theoretical investigation of Brockett's ensemble optimal control problems
M. Sc. Jan Bartsch | Germany
This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework.
Following Brockett's formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.
A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations
Dr. Behzad Azmi | Österreichische Akademie der Wissenschaften | Austria
One efficient strategy for dealing with optimal control problems on an
infinite time horizon is the receding horizon framework. In this approach, the solution of an
infinite-horizon problem is approximated through the concatenation of a sequence of finite-horizon optimal controls on overlapping temporal intervals. In this talk, we are concerned with the stabilization of a class of time-varying linear parabolic equations by means of a finite-dimensional receding horizon control (RHC). We discuss the stability and suboptimality of RHC with respect to the different choices of the control costs. Particularly, we consider the case where the squared $\ell_1$-norm as the control cost is chosen. This leads to a nonsmooth infinite-horizon problem which allows a stabilizing control with a low number of active actuators over time. Numerical experiments are also given.
Nash equilibria and bargaining solutions of bilinear quantum game problems
M. Sc. Francesca Calà Campana | University of Würzburg | Germany
This work is devoted to a theoretical and numerical investigation of Nash equilibria for quantum differential games that arise in the case of multiple control functions acting on the same quantum system and having different objectives with non-cooperative character. Existence of a NE is proved and a related Nash's bargaining solution that aims at improving all controls' objectives with respect to the Nash equilibria is discussed.
These problems are solved by a relaxation method combined with a semi-smooth Newton scheme.
Analysis of the Receding Horizon Control method for stabilization problems
Dr. Laurent Pfeiffer | University of Graz | Austria
This talk is dedicated to the analysis of the Receding Horizon Control (RHC) algorithm. The method aims at approximating the solution to optimal control problems on large time-horizons. As it is well-known, it consists in solving a sequence of truncated problems with a small prediction horizon. A control is generated by concatenation of the obtained solutions, each solution being restricted to a sampling time.
The analysis will focus on a class of infinite-horizon stabilization problems of partial differential equations. An error estimate, bringing out the effect of the prediction horizon and the sampling time, will be provided for the distance of the generated control to the optimal one. The use of a terminal cost for the truncated problems will also be discussed.
 Karl Kunisch and Laurent Pfeiffer. The effect of the Terminal Penalty in Receding Horizon Control for a Class of Stabilization Problems. ArXiv preprint, 2018.
 Tobias Breiten and Laurent Pfeiffer. On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems. ArXiv preprint, 2018.