The conjugate points sufficient conditions for an optimal control

The paper concerns an application of the idea of field theory and the conce pt of "concourse of flights" to the sufficient optimality conditions for th e optimal control problems stated in terms of focal and conjugate points. T he concept of concourse of flights was begun by Young (1969), and later ext ended by Nowakowski (1988). In the paper the definition of a focal and conj ugate point of a field of extremals is given. Using these concepts, we prov e that the existence of a field of extremals without conjugate points impli es the existence of concourse of flights and consequently we obtain the sec ond order sufficient conditions for the generalized problem of Bolza. Anoth er approach to the concept of focal and conjugate points is given by Zeidan (1983, 1984).