Full workshop description flyer (pdf)

**Organizers:**

- Didier Henrion (LAAS-CNRS Univ. Toulouse, FR and Czech Tech. Univ. Prague, CZ)
- Milan Korda (EPFL Lausanne, CH)
- Alexandre Mauroy (Univ. Liège, BE)
- Igor Mezić (Univ. California Santa Barbara, USA)

**Abstract: **

Nonconvex control and optimization problems for nonlinear
dynamical systems can be approached with numerical
methods inspired by operator theory. The workshop is an
opportunity to present for the first time in a unified way two
major operator theoretical approaches to nonlinear dynamical
systems:

- Koopman operator methods for dynamical systems, relying on Galerkin numerical discretization techniques;
- polynomial optimization and optimal control formulated as generalized problems of moments, discretized by hierarchies of convex linear matrix inequalities, and solved numerically with semidefinite programming.

- Morning session focusing more on theoretical background
- 9:00-9:45: General background on convex optimization and polynomial optimization (D. Henrion): We survey linear programming in finite-dimensional cones, with a focus on the semidefinite cone, or cone of non-negative quadratic forms. We then leverage this knowledge to infinite-dimensional cones of non-negative continuous functions and non-negative Borel measures on compact supports, and show its relevance in the context of polynomial optimization.
- 10:00-10:45: General background on dynamical systems (I. Mezić): We introduce the representation of a dynamical system in terms of a linear composition operator (the Koopman operator). We present the formalism in the context of linear systems first, to recover and reinterpret the classical spectral expansion. Then we derive the so-called Koopman Mode Expansion for general nonlinear (and potentially nonsmooth) systems.
- 11:15-12:00: LMI relaxations for generalized problems of moments and their applications (D. Henrion): Using infinite-dimensional linear programming duality, we build Lasserre.s hierarchy of moment-sum-of-squares relaxations for polynomial optimization and motivate its extension to generalized problems of moments in polynomial optimal control.
- 12:15-13:00: Numerical methods for ergodic partition theory (I. Mezić): We concentrate on the first term in the Koopman model analysis, that contains system invariants. We describe all the possible invariants of the dynamical system. We discuss a connection between ergodic sets in dynamical systems and reachable sets in control theory. Then we introduce numerical methods for computation of ergodic sets and ergodic partition.
- Afternoon session focusing more on applications
- 14:00-14:45: Region of attraction computations for polynomial dynamical systems (M. Korda): This part of the workshop describes how to use the concept of occupation measures to characterize the region of attraction as an infinite dimensional linear program in the cone of nonnegative measures. The infinite dimensional linear program is then relaxed to a finite-dimensional semidefinite program providing outer approximations to the region of attraction. We discuss theoretical issues (such as the existence and uniqueness of the solutions to the Liouville.s equation) as well as practical issues such as numerical issues arising when solving the SDP relaxations.
- 15:00-15:45: Global stability analysis for nonlinear systems using the eigenfunctions of the Koopman operator (A. Mauroy): The operator-theoretic framework yields a novel approach to stability analysis, which mirrors the spectral stability analysis of linear systems. In this context, we will show that the existence of specific eigenfunctions of the Koopman operator implies global stability of the attractor. Several systematic methods for computing these eigenfunctions will be presented, which are related to the use of moments in the dual space.
- 16:15-17:00: Invariant set computations for polynomial dynamical systems (M. Korda): This part of the workshop describes how to handle problems on infinite horizon using the previously developed techniques of occupation measures. We describe the discounted infinite-time Liouville.s equation and discuss the relation of its solutions to the trajectories of the controlled system. Then we apply these techniques to characterize the maximum controlled invariant set as well as the value function of an infinite-time discounted optimal control problem as an infinite-dimensional LP in the cone of nonnegative measures. SDP relaxations of this LP provide converging hierarchy of outer approximations to the MCI set and approximations from below to the value function.
- 17:15-18:00: Neuroscience applications: isochrons and isostables (A. Mauroy): The spectral analysis of the Koopman operator reveals geometric properties of the system that play a central role for phase reduction and sensitivity analysis. We will show that these results are of great interest in computational neuroscience: they lead to a novel interpretation of the concept of isochron and enable to extend this notion to excitable neuron models. The results are also instrumental in obtaining new numerical schemes that can be used to compute the highly complex isochrons of bursting neuron models.

Last updated on July 20, 2015.