On optimality conditions for optimal control problems with mixed constraints

An optimal control problem for time-varying, nonlinear differential equations with state-dependent control constraints is considered, the data may be nonsmooth. A weak maximum principle for problems with equality mixed constraints and pointwise set constraints imposed only on some components of the control variable is presented. This result is then applied to derive another weak maximum principle for problems with both equality and inequality mixed constraints. The essential assumption is a full rankness condition imposed on the constraints. Additionally, a lemma shows how the results are related to previous contributions.