Flow regularity and optimality conditions with controls in Lp

In this paper we study the C-r regularity of the flow of a nonlinear nonautonomous control system with respect to control maps belonging to L-p with p greater than or equal to r. The results obtained are applied to get first- and second-order optimality conditions when the control space is L-p. The problem which we consider is in the Mayer form and includes endpoint constraints. We present first-order necessary conditions for a wide class of control systems. Moreover, we show that the usual second-order sufficient conditions are effective only if the map f that defines the control system is a polynomial of degree two in the control variable and the controls belong to L-2.