On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation

Enrique Fernández-Cara (), Dany Nina-Huamán (), Miguel R. Nuñez-Chávez () and Franciane B. Vieira ()
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Enrique Fernández-Cara: Universidad de Sevilla
Dany Nina-Huamán: IME, Univ. Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis
Miguel R. Nuñez-Chávez: IME, Univ. Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis
Franciane B. Vieira: Universidade Federal do Piauí, Campus Universitário Ministro Petrônio Portella

Journal of Optimization Theory and Applications, 2017, vol. 175, issue 3, No 3, 652-682

Abstract: Abstract This paper deals with the analysis of the internal and boundary control of a one-dimensional parabolic partial differential equation with nonlinear diffusion. First, we prove a local null controllability result with distributed controls, locally supported in space. The proof relies on local inversion (more precisely, we use Liusternik’s Inverse Function Theorem), together with some appropriate specific estimates. We also establish a similar result with controls on one side of the boundary. Then, we consider an iterative algorithm for the computation of null controls, we prove the convergence of the iterates, and we perform some numerical experiments.

Keywords: Nonlinear parabolic partial differential equations; Local null controllability; Internal and boundary controls; Numerical solution of null controllability problems; 35K55; 35B37; 93C20 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1190-4

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