Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control

Patricio Guzmán (1) , Swann Marx (2) , Eduardo Cerpa (3)

1 DIM - Departamento de Ingeniera Matematica [Santiago]
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
3 UTFSM - Universidad Tecnica Federico Santa Maria [Valparaiso]

In this paper we stabilize the linear Kuramoto-Sivashinsky equation by means of a delayed boundary control. From the spectral decomposition of the spatial operator associated to the equation, we find that there is a finite number of unstable eigenvalues. After applying the Artstein transform to deal with the delay phenomenon, we design a feedback law based on the pole-shifting theorem to exponential stabilize the finite-dimensional system associated to the unstable eigenvalues. Then, thanks to the inversion of the Artstein transform and the use of a Lyapunov function, we obtain a delayed feedback law that exponential stabilize the original unstable infinite-dimensional system.

Mots clés: Partial differential equations; time delay; global stability; feedback stabilization; transformations; pole assignment; Lyapunov function

Domaines: Automatique; Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Patricio Guzmán, Swann Marx, Eduardo Cerpa. Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control. 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2019, May 2019, Oaxaca, Mexico. ⟨10.1016/j.ifacol.2019.08.013⟩. ⟨hal-02002453⟩