In this paper we study the structure of the Jacobi system for optimal control problems on time scales. Under natural and minimal invertibility assumptions on the coefficients we prove that the Jacobi system is a time scale symplectic system and not necessarily a Hamiltonian system. These new invertibility conditions are weaker than those considered in the current literature. This shows that the theory of time scale symplectic systems, rather than the theory of linear Hamiltonian systems, is fundamental for optimal control problems. Our results in this paper are new even for the Jacobi equations arising in the time scale calculus of variations and, in particular, for the discrete time calculus of variations and optimal control problems. We also show that nonlinear time scale Hamiltonian systems possess symplectic structure, that is, the Jacobian of the evolution mapping satisfies a time scale symplectic system.

Last change: October 26, 2009. (c) Roman Hilscher