Pontryagin Maximum Principle for finite dimensional nonlinear optimal control problems on time scales

In this article we derive a strong version of the Pontryagin Maximum Principle for general nonlinear optimal control problems on time scales in finite dimension. The final time can be fixed or not, and in the case of general boundary conditions we derive the corresponding transversality conditions. Our proof is based on Ekeland's variational principle. Our statement and comments clearly show the distinction between right-dense points and right-scattered points. At right-dense points a maximization condition of the Hamiltonian is derived, similarly to the continuous-time case. At right-scattered points a weaker condition is derived, in terms of so-called stable $\Omega$-dense directions. We do not make any specific restrictive assumption on the dynamics or on the set $\Omega$ of control constraints. Our statement encompasses the classical continuous-time and discrete-time versions of the Pontryagin Maximum Principle, and holds on any general time scale, that is any closed subset of $\R$.