The generalized maximum principle and the singular problem

The extremal properties of a differential system with an integral cost criterion that does not satisfy the strong Legendre condition are treated. The prototype of such a situation is given by an affine system on an analytic manifold M with X₀, . . ., X_m analytic vector fields, and a quadratic cost criterion f(x,u). Such a system is singular if the Hessian matrix is only semipositive definite. In that situation, the question of optimizing ∫f always leads to instantaneous jump directions which act as constraints on the space of regular optimal trajectories of the system. A strengthened form of the maximum principle for such systems is presented. Since this principle contains Lie brackets of arbitrarily high orders, it is termed the generalized maximum principle. For linear quadratic systems, the generalized principle contains all of the necessary information for the resolution of optimal trajectories