Some characterizations of optimal trajectories in control theory

Several characterizations of optimal trajectories for the classical Mayer problem in optimal control are presented. The regularity of directional derivatives of the value function is studied. For example, it is shown that for smooth control systems the value function V is continuously differentiable along an optimal trajectory x:[t₀, 1]→Rⁿ, provided V is differentiable at the initial point (t ₀, x(t₀)). Then the upper semicontinuity of the optimal feedback map is deduced. The authors also address the problem of optimal design, obtaining sufficient conditions for optimality. Finally, it is shown that the optimal control problem can be reduced to a viability problem