Final-state approximate control for the heat equation

• Ano de defesa: 2018
• Autor(a) principal: Flores, Marlon Michael López
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade do Estado do Rio de Janeiro; Centro de Tecnologia e Ciências::Faculdade de Engenharia; Brasil; UERJ; Programa de Pós-Graduação em Engenharia Mecânica
• Palavras-chave em Português: Mechanical engineering; Linear differential equations; Temperature control; Galerkin, Methods; Mathematical models; Engenharia mecânica; Equações diferenciais lineares; Controle de temperatura; Galerkin, Métodos; Modelos matemáticos; ENGENHARIAS::ENGENHARIA MECANICA::FENOMENOS DE TRANSPORTE
• Link de acesso: http://www.bdtd.uerj.br/handle/1/16985

In this work, two types of open-loop control problems are addressed in connection with the linear heat equation in rectangular domains with Dirichlet type boundary conditions in which the control function (depending only on time) constitutes a source term. In both cases, the main objective is to impose a prescribed state (temperature distribution) at the final instant of a given time-interval. Control signals are to be selected on the basis of two optimization problems, one unconstrained and the other one involving constraints on the maximum magnitudes of the values taken by the control signals on the time-interval in question. Both problems have the same quadratic cost-functional. Approximations for the optimal control signals are obtained on the basis of finite-dimensional Galerkin approximation for the linear heat equation. As a consequence, the resulting optimal control signals can be effectively computed. Numerical results for the 1D and 2D linear heat equations are presented to illustrate the results mentioned above. On the basis of the results obtained for the linear heat equation, a heuristic linearization scheme is introduced to address final-state control problems for the non-linear heat equation. This scheme rests on a piecewise linearization of the finite-dimensional, non-linear ODEs corresponding to Galerkin approximations of the non-linear heat equation. Some numerical results are also presented to illustrate this heuristic linearization scheme for the 1D non-linear heat equation.