Symplectic structure of Jacobi systems on time scales
| Authors | |
|---|---|
| Year of publication | 2010 |
| Type | Article in Periodical |
| Magazine / Source | International Journal of Difference Equations |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | Time scale; Jacobi equation; Euler--Lagrange equation; Time scale symplectic system; Linear Hamiltonian system; Discrete symplectic system; Pontryagin weak maximum principle; Quadratic functional; Nonlinear Hamiltonian system |
| Description | 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 variation 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. |
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