Necessary conditions for some optimal control problems with state-dependent control constraints

Necessary conditions of optimality in the form of maximum principles are proved for some optimal control problems with state-dependent control constraints. The authors consider problems with constraints of the form x˙(t)=f¹(t,x(t))+f²(t,u(t)) 0=b¹(t,x(t))+b²(t,u(t)) together with end point constraints and pointwise “set constraints” on the control variable. An optimality condition in the form of a strong maximum principle is derived under a convexity hypothesis. The authors highlight through example the importance of convexity for the validity of their maximum principle. Moreover, the authors show that without such hypothesis no weak version of their maximum principle is valid