CONTROL OF CERTAIN PARABOLIC MODELS FROM BIOLOGY AND SOCIAL SCIENCES

Domènec Ruiz-Balet (1) , Enrique Zuazua (2)

1 Fundación Deusto
2 BCAM - Basque Center for Applied Mathematics

These lecture notes address the controllability under relevant state constraints of reaction-diffusion equations. Typically the quantities modeled by reaction-diffusion equations in socio-biological contexts (e.g. population, concentrations of chemicals, temperature, proportions etc) are positive by nature. The uncontrolled models intrinsically preserve this nature thanks to the maximum principle. For this reason, any control strategy for such systems has to preserve these state constraints. We restrict our study in the case of scalar equations with monostable and bistable nonlinearities. The presence of constraints produces new phenomena such as a possible lack of controllability, or existence of a minimal controllability time. Furthermore , we explain general ways for proving controllability under state constraints. Among different strategies, we discuss how to use traveling waves and connected paths of steady states to ensure controllability. We devote particular attention to the construction of such connected paths of steady-states. Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to provide intuition to the reader.

Mots clés: Reaction-diffusion; Control; Steady-states; Constraints; Mathematical Biology

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

Domènec Ruiz-Balet, Enrique Zuazua. CONTROL OF CERTAIN PARABOLIC MODELS FROM BIOLOGY AND SOCIAL SCIENCES. Doctoral. China. 2019. ⟨hal-02453829v1⟩