Sufficient Conditions for a Strong Minimum in Singular Control Problems

This paper derives the conditions guaranteeing that a singular extremal that joins fixed endpoints provides a strong minimum for the independent time variable. For the nonsingular case, Weierstrass has shown that the extremal must be embedded in a field. The principal conditions that imply the existence of a field for a nonsingular problem with an n-dimensional state vector ${\bf x}$ and a scalar control variable u are that ${{\partial ^2 H} / {\partial u^2 }}$ is unequal to zero and that the $n \times (n + 1)$ matrix $[{{\partial {\bf x}(t)} / {\partial \lambda (t_0 )}},\dot {\bf x}(t)]$ has rank n. Here t is the time, H is the generalized Hamiltonian, and $\lambda $ is the adjoint vector. This paper shows that under proper assumptions the field concept can be extended to the singular case. The condition on ${{\partial ^2 H} / {\partial u^2 }}$ is replaced by \[ \frac{\partial } {{\partial u}}\frac{{d^2 }} {{dt^2}}\frac{{\partial H}} {{\partial u}} \ne 0. \] The above matrix, whose first n columns are ob...