Duality and sufficiency for fractional control problems with invexity

The control problem is to transfer the state variable from an initial state x0 at t0 to a final state xf at tf so as to minimize a given functional, subject to constraints on the control and state variables. To demonstrate this definition, the primal problem, (P) Minimize ∫t(0)t(f) (f(t,x,u)/k(t,x,u))dt. subject to x(t0) = x0, x(tf) = xf, G(t,x,u) = x′, R(t,x,u)≥0, where f(t,x,u) and k(t,x,u) are real functions, was studied.