In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in optimal control problems. Using the notion of $L$-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves. 查看全文>>