Optimal Control Problems with Mixed Constraints

• Published in: SIAM journal on control and optimization Vol. 48; no. 7; pp. 4500 - 4524
• Main Authors: Clarke, Francis, de Pinho, M. R.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Applied sciences; Calculus of variations; Calculus of variations and optimal control; Calibration; Computer science; control theory; systems; Control theory; Control theory. Systems; Differential equations; Exact sciences and technology; Formulations; Hypotheses; Inequalities; Mathematical analysis; Mathematics; Optimal control; Optimization; Optimization and Control; Sciences and techniques of general use; Studies; Differential equation; Differential system; nonsinooth analysis; Global local method; mixed constraints; Qualification; 49K21; Nonsmooth analysis; Algebraic system; necessary conditions; Optimal control; Equality constraint; Variational calculus; Optimal control (mathematics); 49K15; Inequality
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/090757642

We develop necessary conditions of broad applicability for optimal control problems in which the state and control are subject to mixed constraints. We unify, subsume, and significantly extend most of the results on this subject, notably in the three special cases that comprise the bulk of the literature: calculus of variations, differential-algebraic systems, and mixed constraints specified by equalities and inequalities. Our approach also provides a new and unified calibrated formulation of the appropriate constraint qualifications, and shows how to extend them to nonsmooth data. Other features include a very weak hypothesis concerning the type of local minimum, nonrestrictive hypotheses on the data, and stronger conclusions, notably as regards the maximum (or Weierstrass) condition. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. This leads to local, intermediate, and global versions of the necessary conditions, according to how the hypotheses are formulated. [PUBLICATION ABSTRACT]