  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 $\R$.
