ABSTRACT
Introduction We shall derive second order sufficient optimality conditions (SSC) for an optimal control problem for nonlinear ODEs, subject to pure state
constraints of order one. We follow the approach based on the HamiltonJacobi inequality which was used in [8] for multidimensional control problems. For one-dimensional problems we are able to obtain results more constructive than those in [8]. We continue the line of [6], where mixed control-state constraints were considered. As in [6], we try to obtain a SSC in a possibly weak form which takes into account the strongly active constraints. In the whole analysis, the regularity of the control as well as of the associated Lagrange multipliers plays a crucial role. We consider the case where the control is a continuous function, and we impose constraint qualifications which ensure existence, uniqueness and regularity of the normal Lagrange multipliers. By these constraints qualifications, the analysis is confined to the so-called first order state constraints [9]. The relevant regularity results obtained in [4] are recalled in Section 1. Note that in the regularity analysis a Lagrangian in the socalled indirect form is more convenient, while in the analysis of sufficiency in Sections 2 and 3 a Lagrangian in the direct form is used (see [9] for the definitions).
