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

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.