Existence of solutions for a class of nonlinear control problems

We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N_ux in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN_u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⁿ, with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.