New computer algebra methods for the stability and the stabilisation of classes of linear systems

In the study of dynamical systems, a fundamental issue concerns their stability since it is a necessary condition for them to work properly. At the core of the effective study of this property is the problem of checking the stability as well as the problem of computing stabilizing controllers for unstable systems. For the case of time-invariant linear systems, within the frequency domain approach, i.e. considering the transfer function representation of the system, the stability property usually corresponds to the lack of poles of the transfer function in some regions of the complex plane. While such a condition is easy to check for finite dimensional systems, i.e. for systems defined by rational transfer functions, using, for instance, the classical Routh-Hurwitz criterion or its multiple analogues, it becomes rather complicated for infinite-dimensional systems such as time-delay systems, two-dimensional systems, etc, and is often relaxed into simpler but approximate conditions. In the present talk, we provide a new computer algebra framework for testing the (asymptotic, independent of the delay, structural) stability of some classes of infinite-dimensional linear systems such as linear time-delay systems with commensurate delays and two-dimensional discrete linear systems. The idea under this novel symbolic approach first consists in transforming the stability conditions into algebraic conditions. Then, they can be effectively checked by means of certified symbolic-numeric techniques and implementations dedicated to the study of polynomial systems recently developed by the computer algebra community and the speaker in his PhD thesis (e.g., Univariate Rational Representation, search for separating forms, certified root isolation, discriminant variety, critical point methods). Using these techniques, we also show how stabilizing controllers for these classes of systems can be effectively computed. Our approach is illustrated with simple and practical explicit examples.