Research Article

Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Nikolai P. Osmolovskii

Systems Research Institute, ul. Newelska 6, 01-447 Warszawa, Poland

Politechnika Radomska, ul. Malczewskiego 20A, 26-600 Radom, Poland

University of Natural Sciences and Humanities in Siedlce, ul. 3 Maja 54, 08-110 Siedlce, Poland. osmolovski@uph.edu.pl

Abstract

Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.

(Received November 28 2009)

(Revised November 27 2010)

(Online publication June 22 2011)

Key Words:

• Pontryagin’s principle;
• critical cone;
• quadratic form;
• second order sufficient condition;
• quadratic growth;
• Hoffman’s error bound

Mathematics Subject Classification:

• 49K15