Controllability under positive constraints for quasilinear parabolic PDEs

Miguel R. Nuñez-Chávez

doi:10.3934/mcrf.2021024

Article Contents

2022, Volume 12, Issue 2: 327-341. Doi: 10.3934/mcrf.2021024

This issue Previous Article Solvable approximations of 3-dimensional almost-Riemannian structures Next Article A nonzero-sum risk-sensitive stochastic differential game in the orthant

Controllability under positive constraints for quasilinear parabolic PDEs

Miguel R. Nuñez-Chávez^,

DeustoTech, Fundación Deusto, Av. Universidades, 24, 48007, Bilbao, Basque Country, Spain, Instituto de Matemática e Estatística, Universidade Federal Fluminense, R. Prof. Marcos Waldemar de Freitas, s/n, 24210-201, Niterói, RJ, Brazil

^* Corresponding author: Miguel R. Nuñez-Chávez
^* Corresponding author: Miguel R. Nuñez-Chávez

Received: October 2019

Revised: March 2021

Early access: April 2021

Published: June 2022

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Abstract

This paper deals with the analysis of the internal controllability with constraint of positive kind of a quasilinear parabolic PDE. We prove two results about this PDE: First, we prove a global steady state constrained controllability result. For this purpose, we employ the called "stair-case method". And second, we prove a global trajectory constrained controllability result. For this purpose, we employ the well-known "stabilization property" in $ L^2 $ norms. Furthermore, for both results an important argument is needed: the exact local controllability to trajectories. Then we prove the positivity of the minimal controllability time using arguments of comparison principle. Some additional comments and open problems concerning other systems are presented.

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Mathematics Subject Classification: Primary: 35K59, 93C20; Secondary: 35K15.

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Figure 1. The stair-case method. Trajectory of state-control in blue. Path of steady states in red

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Figure 2. Trajectory of state-control in blue. Target trajectory in red

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Figure 3. Initial datum $ \varphi^T $ for the adjoint system

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