Nonnegative control of finite-dimensional linear systems

Jérôme Lohéac (1) , Emmanuel Trélat (2, 3) , Enrique Zuazua (4, 5, 6)

1 CRAN - Centre de Recherche en Automatique de Nancy
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 CaGE - Control And GEometry
4 FAU - Friedrich-Alexander Universität Erlangen-Nürnberg = University of Erlangen-Nuremberg
5 DEUSTO - Universidad de Deusto
6 UAM - Universidad Autónoma de Madrid

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.

Mots clés: Temps minimal; Nonnegative control; Contrôles positifs; Impulsions de Dirac; Minimal time; Dirac impulse

Domaines: Optimisation et contrôle [math.OC]

Jérôme Lohéac, Emmanuel Trélat, Enrique Zuazua. Nonnegative control of finite-dimensional linear systems. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2021, 38 (2), pp.301-346. ⟨10.1016/j.anihpc.2020.07.004⟩. ⟨hal-02335968⟩