Other research product . Lecture . 2019

CONTROL OF CERTAIN PARABOLIC MODELS FROM BIOLOGY AND SOCIAL SCIENCES

Name: CONTROL OF CERTAIN PARABOLIC MODELS FROM BIOLOGY AND SOCIAL SCIENCES
Keywords: Reaction-diffusion, Control, Steady-states, Constraints, Mathematical Biology, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]

Ruiz-Balet, Domènec; Zuazua, Enrique;

English

Published: 15 Jul 2019

Publisher: HAL CCSD

Country: France

- Summary
- References
  (64)

Abstract

Doctoral; 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.

Subjects

Reaction-diffusion, Control, Steady-states, Constraints, Mathematical Biology, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]

Related Organizations

Université Paris Diderot
France

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Lecture . 2019

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