Sparse Jurdjevic–Quinn stabilization of dissipative systems

Marco Caponigro (1) , Benedetto Piccoli (2) , Francesco Rossi (3) , Emmanuel Trélat (4, 5, 6)

1 M2N - Modélisation mathématique et numérique
2 Department of Mathematical Sciences [Camden]
3 LSIS - Laboratoire des Sciences de l'Information et des Systèmes
4 LJLL - Laboratoire Jacques-Louis Lions
5 UPMC - Université Pierre et Marie Curie - Paris 6
6 SU - Sorbonne Université

For control-affine systems with a proper Lyapunov function, the classical procedure Jurdjevic– Quinn (see [21]) gives a well-known and widely used way of designing feedback controls that asymp-totically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated at the same time. In this paper we give sufficient conditions under which this stabilization can be done by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. We thus obtain a sparse version of the classical Jurdjevic–Quinn theorem. We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying feedback, a sampled feedback, and a hybrid hysteresis. We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models.

Domaines: Optimisation et contrôle [math.OC]

Marco Caponigro, Benedetto Piccoli, Francesco Rossi, Emmanuel Trélat. Sparse Jurdjevic–Quinn stabilization of dissipative systems. 2016. ⟨hal-01397843v1⟩