Pontriagin's maximum principle and the principle of optimality☆

Abstract This paper demonstrates that Pontriagin's Maximum Principle may be derived from the principle of Optimality. It considers a control system described by ẋ = f(x, u, t) where the control vector u is restricted to a closed and bounded set. The optimal control steers the system from an initial state x0 at 0 to a moving target in such a way that the cost of control along the optimal trajectory ∫t0t1f0[x0(τ), u0(τ)dτ is minimized. The optimal control u0 satisfies the maximum principle at each point of the trajectory: max〈g(x), f(x, u)〉 = 〈ḡ(x), f(x, u0)〉 where the vector g(x) is related to ∇Γ where Γ(x0, x) is the minimal cost of going from x0 to x.