The SCIDiS project aims at the stability and stabilization analysis of infinite dimensional systems using Lyapunov-like methodology. The objective is to provide generic methods for building Lyapunov functionals to stability analysis or to control distributed parameter systems. More precisely, we focus on the existing relationships between Lyapunov methods for time-delay systems characterized by hereditary differential equations and distributed parameter systems characterized by partial differential equations. Both systems give rise to distributed parameter systems, as they have solutions evolving on an infinite-dimensional space. These two classes of infinite dimensional systems appear in a large set of control applications such as in Engineering, in Population dynamics, or in Computer Science etc.
We propose in this project to extend the well-established results for the stability analysis of various classes of time delay systems based on Lyapunov methods to the case of more general class of infinite dimensional systems. Indeed, in the context of linear time invariant systems with constant delays, it is well known that the asymptotic stability implies the existence of a Complete Lyapunov-Krasovskii functional. Unfortunately the computation of the parameters of this functional results from a matrix partial differential equation, hence complicated to solve. Therefore several directions of research have led to the approximation of these parameters using either a discretization method or a polynomial optimization. In a recent work developed by the participants of the project, a new method based on polynomial approximations on Hilbert space has been provided to approximate these parameter. This step is performed through a finite dimensional polynomial approximation of the functional state. The novelty with respect to the existing approaches is the possibility to upper bound the approximation error thanks to Bessel inequality or Parseval identity. This contribution has open a new direction of research whose potential range covers a large class of infinite dimensional systems.
Following this methodology, the SCIDiS project aims at providing accurate robust stability analysis, stabilization and observation methods for infinite dimensional systems including, at a first stage, various classes of systems with constant/time-varying, discrete/distributed delays. The second and more relevant stage of the project is concerned by the extensions and adaptations of these preliminary results to a more general class of linear controlled distributed parameter systems characterized by hyperbolic partial differential equations. The case of distributed parameter systems coupled with ordinary differential equations is also under consideration within the project.
An additional direction of research that will be undertaken in the project is related the problem of controlling Cyber-Physical Systems and more specifically to the robustness of numerical implementations through two aspects. The first problem is related to the implementation of infinite dimensional controller as for instance the Smith predictor. It is indeed well known that this class of controllers is highly sensitive to model imperfections and numerical implementation, questions that we are interested in. The second aspect is related to the sampled-data implementation of control law and/or observers.