MATHEMATICAL CONTROL AND doi:10.3934/mcrf.2018034 RELATED FIELDS Volume 8, Number 3&4, September & December 2018 pp. 777–787 FEEDBACK STABILIZATION WITH ONE SIMULTANEOUS CONTROL FOR SYSTEMS OF PARABOLIC EQUATIONS C˘ at˘ alin-George Lefter ∗ and Elena-Alexandra Melnig Faculty of Mathematics, University “Al. I. Cuza” Ia¸ si, Romania Octav Mayer Institute of Mathematics, Romanian Academy, Ia¸ si Branch Dedicated to Professor Jiongmin Yong on the occasion of his 60th anniversary Abstract. In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction–diffusion systems are discussed. 1. Introduction. In this paper we study the local feedback stabilization of systems of parabolic equations in a domain Ω ⊂ R n , with only one internally distributed control, supported in a bounded subdomain ω ⊂⊂ Ω. Because of the reduced number of controls, these systems may not be small time local controllable or, more precisely, the linearized system is not controllable. In fact, controllability for the linearized system is usually an argument, via appropriate algebraic Riccati or Lyapunov equations, to construct stabilizing feedbacks. Controllability of linear parabolic equations with internally distributed controls supported in a subdomain was established by O.Yu.Imanuvilov using appropriate global Carleman estimates for the adjoint equation (see [9] for an introduction to the field). Local controllability of nonlinear equations or systems may be deduced from controllability of the linearized system. In such a situation the nonlinear system is small time local controllable. Controllability of systems by a reduced number of controls is a challenging prob- lem and positive results may be obtained under appropriate conditions on the cou- pling terms, as it is the case for example in the phase field models studied in [1]. When a good coupling is not verified for the linearized system, an issue to exploit the nonlinearity is the return method of J.-M.Coron, which is linearization along special solutions, constructed in such a way to fulfill coupling requirements for the linearized system (see [6, 7, 8]). In this paper the strategy for stabilization is, in some sense, similar to the one in [11] and is based not on the controllability of the linearized system but on its approximate controllability. In fact, exact controllability for the linearized system 2010 Mathematics Subject Classification. 35K40, 35K57, 93D15, 93B52, 93B18. Key words and phrases. Reaction-diffusion equations, unique continuation, feedback stabiliza- tion, Lyapunov equation. The second author was supported by a grant of the Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0011. * Corresponding author: C˘ at˘alin-George Lefter (catalin.lefter@uaic.ro). 777