Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

Jean-Michel,; Coron,; Jesús,; Ildefonso,; Díaz, Jesús Ildefonso Díaz; Abdel-Malek,; Drici,; Tommaso,; Mingazzini,

doi:10.1007/978-3-642-41401-5_8

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…Note that for this model, due to the degeneration of the diffusivity when m ∼ 0, not even the boundary controllability property without constraints is known (see [6,14]).…”

Section: Preliminary Resultsmentioning

confidence: 99%

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Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.

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“…Note that for this model, due to the degeneration of the diffusivity when m ∼ 0, not even the boundary controllability property without constraints is known (see [6,14]).…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…However, there are several other relevant linear and nonlinear diffusion operators whose analysis is even more challenging. This is the case, for instance, when considering the porous medium equation [119,87,19,43]:…”

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confidence: 99%

Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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“…State of the art. In [13], Coron, Diáz, Drici & Mignazzini prove the nullcontrollability of the porous medium equation set on (0, 1) using Dirichlet boundary controls on both ends as well as a scalar forcing control. A control on one end can also be used as long as the other boundary condition is a Neumann one.…”

Section: Introductionmentioning

confidence: 99%

Null-controllability of perturbed porous medium gas flow

Geshkovski

2020

ESAIM: COCV

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In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.

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Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

Cited by 2 publications

References 13 publications

Control of reaction-diffusion models in biology and social sciences

Control of reaction-diffusion models in biology and social sciences

Null-controllability of perturbed porous medium gas flow

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