Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

• 成果类型: Article
• 署名作者: Alvarez, Felipe; Bolte, Jerome; Bonnans, J. Frederic; Silva, Francisco J.
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Institut Polytechnique de Paris; Ecole Polytechnique; Universidad de Chile; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidad de Chile; Universite PSL; Ecole des Hautes Etudes en Sciences Sociales (EHESS); Universite de Toulouse; Universite Toulouse 1 Capitole; Centre National de la Recherche Scientifique (CNRS); Toulouse School of Economics; Sapienza University Rome
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-011-0477-8
• 发表日期: 2012
• 页码: 473-507
• 关键词: point methods; algorithm

摘要

We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L (s) norms (s in [1, +a]), error estimates of the state and the adjoint state in Sobolev spaces W (1,s) (s in [1, +a)) and also error estimates for the value function. For the L (1) norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order where is the (small) penalty parameter.