We consider control problems for the variational inequality describing a single degree
of freedom elasto-plastic oscillator. We are particularly interested in finding the "critical excitation",
i.e., the lowest energy input excitation that drives the system between the prescribed initial and
final states within a given time span. This is a control problem for a state evolution described by
a variational inequality. We obtain Pontryagin’s necessary condition of optimality. An essential
difficulty lies with the non continuity of adjoint variables.