Existence of optimal homoclinic orbits

The problem of optimal periodic control is considered from a geometric point of view. The objective is to determine the conditions under which a given optimal control problem admits a homoclinic orbit as an extremal solution. The analysis is performed on the Hamiltonian dynamical system obtained from the application of Pontryagin Maximum Principle. Assuming the existence of nondegenerate control, the existence problem is studied through the dynamical structure of the associated critical Hamiltonian dynamical system. A key tool used in the present development is the application of Morse theory in the context of symplectic geometry. The main result of the paper follows from the study of the critical points of the Hamiltonian function. An application example is provided to illustrate the method.