On the Constancy of the Pontryagin Hamiltonian for Autonomous Problems

We provide a new, simpler, and more direct proof of the well known fact that for autonomous optimal control problems the Pontryagin extremals evolve on a level surface of the respective Pontryagin Hamiltonian. Given sets Ω ⊆ R and F ⊆ R, constants a < b, and two continuous functions L(x, u) : R × Ω → R and φ(x, u) : R ×Ω → R with continuous derivatives with respect to x, we define the autonomous optimal control problem as the minimization or maximization of the cost functional I [x(·), u(·)] = ∫ b a L (x(t), u(t)) dt, called the performance index, among all the solutions of the vector differential equation ẋ(t) = φ (x(t), u(t)) for almost all t ∈ [a, b], subject to the boundary conditions (x(a), x(b)) ∈ F . The state trajectory x(·) is a n-vector absolutely continuous function and the control u(·) is a r-vector measurable and bounded function satisfying the control constraint u(t) ∈ Ω: x(·) ∈ W1,1 ([a, b];R ), u(·) ∈ L∞ ([a, b]; Ω). The problem is denoted by (P ). The celebrated Pontryagin maximum principle [5], which is a first-order necessary optimality condition for optimal control, provides a generalization of the classical calculus of variations first-order necessary optimality conditions. It asserts that the minimizers or maximizers of the optimal control problems are to be found among the Pontryagin extremals. Definition. Let us associate to the optimal control problem (P ) the HamiltonianH defined by H(x, u, ψ0, ψ) = ψ0L(x, u)+ψ ·φ(x, u). A quadruple (x(·), u(·), ψ0, ψ(·)), where ψ0 ≤ 0 is a constant and ψ(·) a n-vector absolutely continuous function with domain [a, b], is called a Pontryagin extremal if it satisfies the control system ẋ(t) = ∂H ∂ψ (x(t), u(t), ψ0, ψ(t)); the adjoint system ψ̇(t) = − ∂H ∂x (x(t), u(t), ψ0, ψ(t)); and the maximality conditionH (x(t), u(t), ψ0, ψ(t)) = maxv∈ΩH (x(t), v, ψ0, ψ(t)). In the present note we are interested in the following well-known result [5]. Theorem. If (x(·), u(·), ψ0, ψ(·)) is a Pontryagin extremal of (P ), then (1) H(x(t), u(t), ψ0, ψ(t)) ≡ constant , t ∈ [a, b] . The Theorem has several important applications. In classical mechanics, (1) corresponds to conservation of energy (cf. e.g. [4]); in economics to the Ramsey rule for optimal saving or to the constancy of the welfare measure of national income (cf. e.g. [6]); while in the calculus of variations it corresponds to the second Erdmann necessary optimality condition (cf. e.g. [2]). Although the Theorem is Date: March 3, 2003. 2000 Mathematics Subject Classification. 49K15.