| Abstract: |
| We study control problems governed by nonlinear ordinary differential equations of the form
$$
\dot x(t) = f(x(t),u(t),v(t)) + \sum_{\alpha=1}^m g_{\alpha}(x(t),u(t))\dot u_{\alpha}(t),\quad \text{for} \ t\in [0,T],
$$
where $x:[0,T] \to \mathbb{R}^n$ is the {\em state variable,} $u:[0,T] \to \mathbb{R}^m$ is the {\em impulsive control} and $v:[0,T] \to \mathbb{R}^l$ is the {\em ordinary control.}
The control $u$ is allowed to be a (discontinuous) bounded variation function, which gives the system an impulsive character. For this class of equations, we adopt the concept of {\em graph completion solution,} that was introduced by A. Bressan and F. Rampazzo in the 90`s.
We consider an optimal control problem in the Mayer form, with general control and final state constraints, for which we prove a maximum principle and higher-order necessary conditions in terms of the adjoint state and the Lie brackets of the involved vector fields. |
|