2025/06/23 by Mohamed Camil Belhadjoudja, Belhadjoudja, Mohamed Camil, Mohamed Maghenem +5
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods
paper · pdf · doi:10.48550/arxiv.2506.18634
For quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow up in open loop, i.e., under null boundary conditions, we provide an estimate of the region of attraction under cubic feedback laws applied at the boundary, using boundary measurements. We guarantee: 1-L 2 and H 1 exponential stability of the origin with an estimate of the region of attraction. 2-Convergence of the H 2 and the C 1 norms of the solutions to zero. 3-Existence and uniqueness of complete classical solutions. 4-Positivity of the solutions starting from positive initial conditions. Unlike existing approaches, our framework handles nonlinear state-dependent diffusion, convection, and (destabilizing) reaction. The cubic terms are used to enlarge our estimate of the region of attraction. The size of the region of attraction is shown, in many cases, to grow unboundedly as diffusion increases. Finally, our controllers can be implemented as Neumann, Dirichlet, or mixed-type boundary conditions.