EXISTENCE THEOREM FOR A GENERAL BOLZA PROBLEM

An existence theorem is proved for a general Bolza problem covering various types of constrained optimal control problems. This theorem seems to be the most general; it covers some well-known results of other authors. The proof of the theorem is based upon a new growth condition. Under this condition, the integrand of the Bolza problem is allowed to decrease arbitrarily fast in the state variable, but the decrease in the state variable and the increase in the control variable must be consistent in some sense.