An extension of the conjugate point condition to the case of variable end points

The authors introduce the definition of coupled points for a quadratic functional in the calculus of variations, where the endpoints are constrained to affine subspaces. This definition plays the same role played by the definition of local or conjugate point for quadratic functionals in the calculus of variations where one or both endpoints are fixed. They show that the nonexistence of a point coupled with b in (a,b) is a necessary condition for the functional to be positive semidefinite. The definition involves adding to the functional a penalty term which is quadratic in η(c) and vanishes in the classical case. The authors extend the definition of coupled points to the linear regulator problem, where both endpoints can vary in subspaces, and they introduce the appropriate normality conditions needed to obtain a necessary condition analogous to the one obtained in the calculus of variations