New optimality conditions for nonsmooth control problems

• Published in: Journal of global optimization Vol. 57; no. 4; pp. 1465 - 1484
• Main Authors: de Oliveira, Valeriano Antunes, Silva, Geraldo Nunes
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2013; Springer; Springer Nature B.V
• Subjects: Computer Science; Control theory; Equivalence; Lagrange multiplier; Lagrange multipliers; Mathematical programming; Mathematics; Mathematics and Statistics; Maximum principle; Operations Research/Decision Theory; Optimal control; Optimization; Real Functions; Studies; Optimality conditions; Generalized invexity; Nonsmooth optimal control; 90C26; 49K15
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-012-0003-4

This work considers nonsmooth optimal control problems and provides two new sufficient conditions of optimality. The first condition involves the Lagrange multipliers while the second does not. We show that under the first new condition all processes satisfying the Pontryagin Maximum Principle (called MP-processes) are optimal. Conversely, we prove that optimal control problems in which every MP-process is optimal necessarily obey our first optimality condition. The second condition is more natural, but it is only applicable to normal problems and the converse holds just for smooth problems. Nevertheless, it is proved that for the class of normal smooth optimal control problems the two conditions are equivalent. Some examples illustrating the features of these sufficient concepts are presented.