Title and Abstract

- Frédéric Boyer (LS2N, Nantes).

Title: Modelling elements in continuum mechanics: application to continuous robotics

Abstract: The objective of this mini-course is to give an example of modelling of infinite-dimensional physical systems. The example chosen will be the deformable body of continuum mechanics. We will start from the geometrical premises of this discipline, to establish the partial differential equations (p.d.e.) of a  three-dimensional deformable body. We then introduce the theory of Cosserat beams and establish their p.d.e. on the Lie group SE(3). The talk will end with some examples llustrating the application of this model to continuous robotics.

- Sébastien Fueyo (GIPSA-lab, Grenoble)

- Pierre Lissy (Ecole des Ponts, Paris).

Title: Polynomial stability for linear PDEs.

Abstract: The most common notion studied in stability or stabilization of systems (typically to the equilibrium 0) is the notion of exponential stability, which requires an exponential rate of convergence in a certain norm. In the case of linear time independant systems in finite dimension, it is the only reasonable notion of convergence, as for semigroups in the natural energy norm. The goal of this mini-course is to explain  how one can relax the notion of stability in order to allow weaker rates of stbilization. Notably, we will focus on the case $iA-BB^*$ (A a selfadjoint operator, BB^* a collocated feedback). We will provide simple examples where one has access to explicit polynomial decay rates, and then, explain the link between exponential/polynomial stability and observability/quantitative approximate controllability or wavepackets conditions.

- Christophe Prieur (GIPSA-lab, Grenoble)

Title: Nonlinear boundary stabilization of partial differential equations.

Abstract: The goal of this course is to introduce the constructive design methods of stabilizing controllers for infinite-dimensional systems as those modelled by partial differential equations (PDEs). The considered approaches include the possible amplitude constraints on the input controls (as those coming from the presence of a saturation map in the control). The numerically computed controls render the origin of the (possibly) nonlinear PDEs an asymptotically stable equilibrium. Different classes of PDEs will be studied as the reaction-diffusion equations, and some hyperbolic equations, exhibiting a large scope of possible results from local asymptotic stability of the equilibrium to a global exponential stability. Numerical algorithms will be also discussed in this course.

- Marius Tucsnak (Université de Bordeaux) [Cancelled]

Title : Norm and time optimal controls for infinite dimensional linear time invariant systems

Abstract : We present an abstract theory of norm and time optimal control for linear time invariant systems. We first recall some basic concepts and results on infinite dimensional control systems (wellposedness, controllability types, duality). We then focus on two remarkable situations where a Pontryagin type maximum principle holds: systems which are either exact or null controllable in any time. In the first case the maximum principle holds in an almost classical form, in the sense that the multiplier lies in the state space. In the second case, the maximum principle holds in a non-standard form: the multiplier lies in a space which can be much larger than the state space. We next provide sufficient conditions for the existence and possibly optimal controls having the bang-bang property. Finally, we provide applications to various systems described by PDEs: Schrödinger, Kirchhoff (for plates) or parabolic equations.