A second-order condition that strengthens Pontryagin's maximum principle

Abstract

We derive a second-order necessary condition for optimal control problems defined by ordinary differential equations with endpoint restrictions. This condition, based on a second-order restricted minimization test, bears a somewhat similar relation to the Weierstrass E-condition (the Pontryagin maximum principle) as the Legendre and Jacobi conditions bear to the Euler-Lagrange equation. Specifically, in the context of relaxed controls, the E-condition for free endpoint problems asserts that if a function achieves its minimum over a convex set Q at some point q then its one-sided directional derivatives at q into Q are nonnegative. Our new condition, when applied to the special case of free endpoint problems, corresponds to the observation that if such a one-sided directional derivative at q is 0 then the corresponding second directional derivative is nonnegative. This new condition effectively supplements the Pontryagin maximum principle over the singular regimes of “weakly” normal extremals that are candidates for either a relaxed or an ordinary restricted minimum. Like some other second-order methods, this condition is global over the control set but, unlike the other tests, it is also global over time. A number of examples illustrate its use and behavior.