Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods

INRIA Saclay and Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France; Frederic.Bonnans@inria.fr; hermant@cmap.polytechnique.fr

Received: 24 July 2006
Revised: 2 May 2007

Abstract

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. Under some reasonable assumptions, we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity. Those results enable us to design a new continuation algorithm, presented at the end of the paper, that handles automatically changes in the structure of the trajectory.