• shareshare
  • link
  • cite
  • add
auto_awesome_motion View all 4 versions
Other research product . Lecture . 2019


Ruiz-Balet, Domènec; Zuazua, Enrique;
Published: 15 Jul 2019
Publisher: HAL CCSD
Country: France
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.

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
64 references, page 1 of 7

Z T Z L Z T Z L Z T Z L 0

[1] A. Audrito, Bistable reaction equations with doubly nonlinear di usion, Discrete Contin. Dyn. Syst.-A 39 (2019), no. 6, 2977{3015. [OpenAIRE]

[2] A. Audrito and J.L. Vazquez, The sher-kpp problem with doubly nonlinear di usion, J. Di erential Equations 263 (2017), no. 11, 7647{7708.

[3] N.H. Barton, The e ects of linkage and density-dependent regulation on gene ow, Heredity 57 (1986), no. 3, 415.

[4] N.H. Barton and M. Turelli, Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of allee e ects, Am. Nat. 178 (2011), no. 3, E48{E75.

[5] N. Bellomo, Modeling complex living systems: A kinetic theory and stochastic game approach, Springer Science & Business Media, 2008.

[6] H. Berestycky, Le nombre de solutions de certains problemes semi-lineaires elliptiques, J. Funct. Anal. 40 (1981), no. 1, 1 { 29.

[7] H. Berestycky and P.L. Lions, Some applications of the method of super and subsolutions, Bifurcation and Nonlinear Eigenvalue Problems, Lecture Notes in Mathematics 782,, Springer-Verlag, Berlin, 1980, pp. 16 { 41.

[8] K. Berthier, S. Piry, J.F. Cosson, P. Giraudoux, J.C. Folt^ete, R. Defaut, Truchetet D., and X. Lambin, Dispersal, landscape and travelling waves in cyclic vole populations, Ecol. Lett. 17 (2014), 53{64. [OpenAIRE]

[9] U. Biccari, M. Warma, and E. Zuazua, Controllability of the one-dimensional fractional heat equation under positivity constraints, Preprint (2019). [OpenAIRE]

Funded by
Interactions of Control, Partial Differential Equations and Numerics
  • Funder: French National Research Agency (ANR) (ANR)
  • Project Code: ANR-16-ACHN-0014
EC| ConFlex
Control of flexible structures and fluid-structure interactions
  • Funder: European Commission (EC)
  • Project Code: 765579
  • Funding stream: H2020 | MSCA-ITN-ETN
Dynamic Control and Numerics of Partial Differential Equations
  • Funder: European Commission (EC)
  • Project Code: 694126
  • Funding stream: H2020 | ERC | ERC-ADG