An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

• Published in: Journal of optimization theory and applications Vol. 151; no. 2; pp. 260 - 291
• Main Authors: Li, Bin, Yu, Chang Jun, Teo, Kok Lay, Duan, Guang Ren
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.11.2011; Springer Nature B.V
• Subjects: Accuracy; Applications of Mathematics; Approximation; Calculus of Variations and Optimal Control; Optimization; Constraints; Control theory; Dynamical systems; Engineering; Inequalities; Inequality; Mathematics; Mathematics and Statistics; Numerical analysis; Operations Research/Decision Theory; Optimal control; Optimization; Optimization techniques; Penalty function; Studies; Terminals; Theory of Computation; Transformations; Continuous state and control inequality constraints; Control parametrization; Exact penalty function; Optimal control; Terminal state equality constraints; Time scaling transform; Numerical method
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-011-9904-5

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.