Quadratic Growth Conditions in Optimal Control Problems

This paper provides an analysis of weak, Pontryagin and bounded strong minima satisfying a quadratic growth condition for optimal control problems of ordinary differential equations with constraints on initial-final state and pointwise constraints of two types: (a) inequality and equality mixed (control-state) constraints satisfying the hypothesis of uniform linear independence of gradients with respect to control u of active constraints and (b) inequality control constraints satisfying the hypothesis of uniform positive linear independence of gradients of active constraints. 1. Mixed control-state constraints with linearly independent gradients with respect to control In this section we recall results due to N.P. Osmolovskii. We discuss the quadratic conditions [11, Sec. S.2], [16] of a minimum for an optimal control problem with mixed (control-state) equality and inequality type constraints, satisfying the hypothesis of linear independence of gradients with respect to control u of active constraints. The necessary condition for a minimum is that the maximum of the quadratic forms on the set of critical variations is non-negative, where the maximum is taken over the set of multipliers satisfying Pontryagin’s principle (by contrast to abstract optimization problems where the maximum is on the larger set of Lagrange multipliers, see [9, 11]). An appropriate strengthening of this condition turns out to be sufficient for a strong minimum, provided that the admissible controls are bounded by any large constant. We call bounded strong the minimum of this type. The sufficient conditions of bounded strong minimum guarantee the so-called “bounded strong quadratic growth”. 1.1. Pontryagin and bounded strong minima. First order necessary conditions. Let [t0, tf ] denote a fixed time interval. Let U := L∞(t0, tf ;R) and 2000 Mathematics Subject Classification. Primary 49K15; Secondary 90C48. This work started during a visit of the second author at INRIA-Saclay in the year 2007, thanks to an INRIA grant. The second author was supported by grants RFBR 8-01-00685 and NSh-3233.2008.1. c ©2010 J. F. Bonnans, N.P. Osmolovskii