  An optimal control problem with a time-parameter is considered. The
functional to be optimized includes the maximum over time-horizon reached by a
function of the state variable, and so an $L^\infty$-term. In addition to the
classical control function, the time at which this maximum is reached is
considered as a free parameter. The problem couples the behavior of the state
and the control, with this time-parameter. A change of variable is introduced
to derive first and second-order optimality conditions. This allows the
implementation of a Newton method. Numerical simulations are developed, for
selected ordinary differential equations and a partial differential equation,
which illustrate the influence of the additional parameter and the original
motivation.
