Relaxation of control systems under state constraints

In this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control. I n particular, we show that the Mangasarian-Fromowitz constraint qualificati on allows us to rewrite constrained systems as differential inclusions with locally Lipschitz right-hand side. Then the Filippov-Wazewski relaxation t heorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions. If, besides agreeing with the above constraints, the state has to remain in a control-independent set K, then the Mangasarian-Fr omowitz condition cannot hold. This case is investigated as well by means o f a condition on the feasible velocities on the boundary of K.