Section: Research Program
Stabilization of interconnected systems

Linear systems: Analytic and algebraic approaches are considered for infinitedimensional linear systems studied within the inputoutput framework.
In the recent years, the YoulaKu$\stackrel{\u02c7}{\mathrm{c}}$era parametrization (which gives the set of all stabilizing controllers of a system in terms of its coprime factorizations) has been the cornerstone of the success of the ${H}_{\infty}$control since this parametrization allows one to rewrite the problem of finding the optimal stabilizing controllers for a certain norm such as ${H}_{\infty}$ or ${H}_{2}$ as affine, and thus, convex problem.
A central issue studied in the team is the computation of such factorizations for a given infinitedimensional linear system as well as establishing the links between stabilizability of a system for a certain norm and the existence of coprime factorizations for this system. These questions are fundamental for robust stabilization problems [1], [2].
We also consider simultaneous stabilization since it plays an important role in the study of reliable stabilization, i.e. in the design of controllers which stabilize a finite family of plants describing a system during normal operating conditions and various failed modes (e.g. loss of sensors or actuators, changes in operating points). Moreover, we investigate strongly stabilizable systems, namely systems which can be stabilized by stable controllers, since they have a good ability to track reference inputs and, in practice, engineers are reluctant to use unstable controllers especially when the system is stable.

The project aims at developing robust stabilization theory and methods for important classes of nonlinear systems that ensure good controllerperformance under uncertainty and time delays. The main techniques include techniques called backstepping and forwarding, contructions of strict Lyapunov functions through socalled "strictification" approaches [3] and construction of LyapunovKrasovskii functionals [4], [5], [6].

For highly complex systems described in the timedomain and which are submitted to constraints, predictive control seems to be welladapted. This model based control method (MPC: Model Predictive Control) is founded on the determination of an optimal control sequence over a receding horizon. Due to its formulation in the timedomain, it is an effective tool for handling constraints and uncertainties which can be explicitly taken into account in the synthesis procedure [7]. The team considers how mutiparametric optimization can help to reduce the computational load of this method, allowing its effective use on real world constrained problems.
The team also investigates stochastic optimization methods such as genetic algorithm, particle swarm optimization or ant colony [8] as they can be used to optimize any criterion and constraint whatever their mathematical structure is. The developed methodologies can be used by non specialists.