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IMA Journal of Mathematical Control and Information 1999 16(2):147-163; doi:10.1093/imamci/16.2.147
© 1999 by Institute of Mathematics and its Applications
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Variational conditions and conjugate points for the fixed-endpoint control problem

JAVIER F. ROSENBLUETH

IIMAS-UNAM Apartado Postal 20-726, Mexico DF 01000, Mexico

For the fixed-endpoint optimal-control problem, with the set of constraints on the state and control functions arbitrary, we provide in this paper a definition of conjugate point together with a set of sufficient conditions for weak and storng local optimality. These conditons, which are a strengthening of known necessary conditions, involve the second variation with respect to the Hamiltonian, and are obtained directly without any reference to possible extended definitions of conjugate points, fields of extremals, solutions of a certain Riccati equation, or Hamilton-Jacobi theory. In other words, the proof of sufficiency is self-contained in optimal-control terms. On the other hand, the notion of conjugate point is derived by a characterization of the positivity of the second variation on a set of nontrivial admissible variations, and the definition we give is ‘proper’ in the sense that the corresponding Jacobi conditions play, for necessity and sufficiency, the same role as in the classical theory of calculus of variations.


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