Noether's theorem for higher-order variational problems of Herglotz type

We approach higher-order variational problems of Herglotz type from an optimal control point of view. Using optimal control theory, we derive a generalized Euler-Lagrange equation, transversality conditions, a DuBois-Reymond necessary optimality condition and Noether's theorem for Herglotz's type higher-order variational problems, valid for piecewise smooth functions.

It is clear to see that Herglotz's problem (H 1 ) reduces to the classical fundamental problem of the calculus of variations (see, e.g., [12]) if the Lagrangian L does not depend on the z variable: ifż(t) = L(t, x(t),ẋ(t)), t ∈ [a, b], then (H 1 ) is equivalent to the classical variational problem b a L(t, x(t),ẋ(t))dt −→ extr. ( Herglotz proved that a necessary optimality condition for a pair (x(·), z(·)) to be an extremizer of the generalized variational problem (H 1 ) is given by The equation (2) is known as the generalized Euler-Lagrange equation.
Observe that for the classical problem of the calculus of variations (1) one has ∂L ∂z = 0 and equation (2) reduces to the classical Euler-Lagrange equation In [6] we have introduced higher-order variational problems of Herglotz type and obtained a generalized Euler-Lagrange equation and transversality conditions for these problems. In particular, we considered the problem of determining the trajectories x(·) such that We proved that if a pair (x(·), z(·)) is an extremizer of the higher-order problem (H n ), then it satisfies the higher-order generalized Euler-Lagrange equation and the transversality conditions ψ j (b) = ψ j (a) = 0, for j = 1, . . . , n, where x, z n (t) , j = 1, . . . , n. While in [6] the admissible functions are x(·) ∈ C 2n ([a, b]; R m ) and z(·) ∈ C 1 ([a, b]; R), here we consider (H n ) in the wider class of functions x(·) ∈ P C n ([a, b]; R m ) and z(·) ∈ P C 1 ([a, b]; R).
One of the most important results in optimal control theory is Pontryagin's maximum principle proved by Pontryagin et al. in [5]. This principle provides conditions for optimization problems with differential equations as constraints. The maximum principle is still widely used for solving problems of control and other problems of dynamic optimization. Moreover, basic necessary optimality conditions from classical calculus of variations follow from Pontryagin's maximum principle.
One of the problems of optimal control, in Bolza form, is the following one: with some initial condition on ; Ω), with Ω ⊆ R r an open set. In the literature of optimal control, x and u are frequently called the state and control variables, respectively, while φ is known as the payoff or salvage term. Note that the classical problem of the calculus of variations (1) is a particular case of problem (P ) with φ(x) ≡ 0, g(t, x, u) = u and Ω = R m . In this work we show how the results on the higher-order variational problem of Herglotz (H n ) obtained in [6] can be generalized by using the theory of optimal control. The technique used consists in rewriting the generalized higher-order variational problem of Herglotz (H n ) as a standard optimal control problem (P ), and then to apply available results of optimal control theory. For the first-order case we refer the reader to [7].
The paper is organized as follows. In Section 2 we present the necessary concepts and results from optimal control theory: Pontryagin's maximum principle (Theorem 2.1); the DuBois-Reymond condition of optimal control (Theorem 2.3); and the Noether theorem of optimal control (Theorem 2.5). Our main results are given in Section 3: we extend the higher-order Euler-Lagrange equation and the transversality conditions for problem (H n ) found in [6] to admissible functions x(·) ∈ P C n ([a, b]; R m ) and z(·) ∈ P C 1 ([a, b]; R) (Theorem 3.3); we obtain a DuBois-Reymond necessary optimality condition for problem (H n ) (Theorem 3.5); and we generalize the Noether theorem to higher-order variational problems of Herglotz type (Theorem 3.7). We end with Section 4 of conclusions.
2. Preliminaries. We begin this section by stating the well known Pontryagin's maximum principle, which is a first-order necessary optimality condition.
; Ω) is a solution to problem (P ) with the initial condition x(a) = α, α ∈ R, then there exists ψ ∈ P C 1 ([a, b]; R m ) such that the following conditions hold: • the optimality condition • the adjoint system • and the transversality condition where the Hamiltonian H is defined by Definition 2.2 (Pontryagin extremal to (P )). A triplet (x(·), u(·), ψ(·)) with x ∈ P C 1 ([a, b]; R m ), u ∈ P C([a, b]; Ω) and ψ ∈ P C 1 ([a, b]; R m ) is called a Pontryagin extremal to problem (P ) if it satisfies the optimality condition (3), the adjoint system (4) and the transversality condition (5).
Theorem 2.3 (DuBois-Reymond condition of optimal control [5]). If (x(·), u(·), ψ(·)) is a Pontryagin extremal to problem (P ), then the Hamiltonian (6) satisfies the equality The famous Noether theorem [4] is another fundamental tool of the calculus of variations [11], optimal control [8,9,10] and modern theoretical physics [1]. It states that when an optimal control problem is invariant under a one parameter family of transformations, then there exists a corresponding conservation law: an expression that is conserved along all the Pontryagin extremals of the problem (see [8,9,10] and references therein). Here we use Noether's theorem as found in [8], which is formulated for problems of optimal control in Lagrange form, that is, for problem (P ) with φ ≡ 0. In order to apply the results of [8] to the Bolza problem (P ), we rewrite it in the following equivalent Lagrange form: Before presenting the Noether theorem for the optimal control problem (P ), we need to define the concept of invariance. Here we apply the notion of invariance found in [8] to the equivalent optimal control problem (7). In Definition 2.4 we use the little-o notation.
Definition 2.4 (Invariance of problem (P ) cf. [8]). Let h s be a one-parameter family of invertible C 1 maps Problem (P ) is said to be invariant under transformations h s if for all (x(·), u(·)) the following two conditions hold: for some constant ξ; The next result can be easily obtained from the Noether theorem proved by Torres in [8] and Pontryagin's maximum principle (Theorem 2.1).

NOETHER'S THEOREM FOR H-O PROBLEMS OF HERGLOTZ 5
3. Main results. We begin by introducing some definitions for the higher-order variational problem of Herglotz (H n ).
We now present a necessary condition for a pair (x(·), z(·)) to be a solution (extremizer) to problem (H n ). The following result generalizes [6] by considering a more general class of functions. To simplify notation, we use the operator ·, · n , n ∈ N, defined by x, z n (t) := (t, x(t),ẋ(t), . . . , x (n) (t), z(t)). When there is no possibility of ambiguity, we sometimes suppress arguments.
Observe that the optimality condition (15) implies that ψ n = −ψ z ∂L ∂u and that the adjoint system (16) implies that for j = 2, . . . , n, ψ z = −ψ z ∂L ∂z . Hence, ψ z is solution of a first-order linear differential equation, which is solved using an integrand factor to find that ψ z (t) = ke − t a ∂L ∂z dθ with k a constant. From the last transversality condition in (17), we obtain that k = e