COUPLED POINTS IN OPTIMAL-CONTROL THEORY

In this note, we introduce the concept of coupled point for an optimal control problem where both state endpoints are allowed to vary. This definition leads to the extension of the theory of conjugate points to the optimal-control setting. Under suitable controllability assumptions, weaker than those previously considered, we show that the nonexistence of coupled points in the open interval (a,b) is a necessary condition for weak local optimality. This result generalizes the ones of the same kind known from the calculus of variations. In the special case, when one or both state endpoints are fixed, the notion of coupled point is more general than those of focal or conjugate point to which it reduces when the two-sided controllability assumption of [6] is satisfied.