Sparse Jurdjevic-Quinn stabilization of dissipative systems

For control-affine systems with a proper Lyapunov function, the classical Jurdjevic–Quinn procedure (see [22]) gives a well-known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated, i.e. nonzero, at the same time. In this paper we give sufficient conditions under which this stabilization can be achieved 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 periodic 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.