An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks.

Castro Barbero, Carlos Manuel and Palacios, Francisco and Zuazua Iriondo, Enrique (2008). An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks.. "Mathematical Models & Methods in Applied Sciences", v. 18 (n. 3); pp. 369-416. ISSN 0218-2025. https://doi.org/10.1142/S0218202508002723.

Description

Title: An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks.
Author/s:
  • Castro Barbero, Carlos Manuel
  • Palacios, Francisco
  • Zuazua Iriondo, Enrique
Item Type: Article
Título de Revista/Publicación: Mathematical Models & Methods in Applied Sciences
Date: March 2008
Volume: 18
Subjects:
Freetext Keywords: Inviscid Burgers equation; Optimal control; Numerical approximation; Shocks.
Faculty: E.T.S.I. Caminos, Canales y Puertos (UPM)
Department: Matemática e Informática Aplicadas a la Ingeniería Civil [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 − d inviscid Burgers equation. We first prove the existence of minimizers and, by a -convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so called onesided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms.

More information

Item ID: 2924
DC Identifier: http://oa.upm.es/2924/
OAI Identifier: oai:oa.upm.es:2924
DOI: 10.1142/S0218202508002723
Official URL: http://www.worldscinet.com/m3as/18/1803/S02182025081803.html
Deposited by: Memoria Investigacion
Deposited on: 14 May 2010 09:41
Last Modified: 20 Apr 2016 12:32
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