Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem

We extend the DuBois-Reymond necessary optimality condition and Noether's first theorem to variational problems of Herglotz type with time delay. Our results provide, as corollaries, the DuBois-Reymond necessary optimality condition and the first Noether theorem for variational problems with time delay recently proved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619-630]. Our main result is also a generalization of the first Noether-type theorem for the generalized variational principle of Herglotz proved in [Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261-273].


1.
Introduction. It is well-known that the classical variational principle is a powerful tool in various disciplines such as physics, engineering and mathematics. However, the classical variational principle cannot describe many important physical processes. In 1930 Herglotz [22] proposed a generalized variational principle with one independent variable, which generalizes the classical variational principle. As reported in [14,15,16], the principle of Herglotz gives a variational description of nonconservative processes, even when the Lagrangian is autonomous, something that cannot be done with the classical approach. For the importance to include nonconservativism in the calculus of variations, we refer the reader to [26].
Note that (1) represents a family of differential equations: for each function x a different differential equation arises. Therefore, z depends on x, a fact that can be made explicit by writing z{x; t} (or z(t, x(t),ẋ(t))), but for brevity and convenience of notation it is usual to write simply z(t). Observe that Herglotz's variational problem reduces to the classical fundamental problem of the calculus of variations (see, e.g., [12]) if the Lagrangian L does not depend on the variable z: iḟ then we obtain the classical variational problem Herglotz proved that a necessary condition for a trajectory x to be an extremizer of the generalized variational problem z(b) → extr subject to (1)-(2) is given by ∂L ∂x (t, x(t),ẋ(t), z(t)) − d dt ∂L ∂ẋ (t, x(t),ẋ(t), z(t)) + ∂L ∂z (t, x(t),ẋ(t), z(t)) ∂L ∂ẋ (t, x(t),ẋ(t), z(t)) = 0, (3) t ∈ [a, b]. Herglotz called (3) the generalized Euler-Lagrange equation [20,21,29].
Observe that for the classical problem of the calculus of variations one has ∂L ∂z = 0, and the differential equation (3) reduces to the classical Euler-Lagrange equation: A generalized Euler-Lagrange differential equation for Herglotz-type higher-order variational problems was recently proved in [32]. It is well-known that the notions of symmetry and conservation law play an important role in physics, engineering and mathematics [34]. The interrelation between symmetry and conservation laws in the context of the calculus of variations is given by the first Noether theorem [28]. The first Noether theorem, usually known as Noether's theorem, guarantees that the invariance of a variational integral under a group of transformations depending smoothly on a parameter ǫ implies the existence of a conserved quantity along the Euler-Lagrange extremals. Such transformations are global transformations. Noether's theorem explains all conservation laws of mechanics, e.g., conservation of energy comes from invariance of the system under time translations; conservation of linear momentum comes from invariance of the system under spatial translations; while conservation of angular momentum reflects invariance with respect to spatial rotations. The first Noether theorem is nowadays a well-known tool in modern theoretical physics, engineering and the calculus of variations [36]. Inexplicably, it is still not well-known that the famous paper of Emmy Noether [28] includes another important result: the second Noether theorem [35]. Noether's second theorem states that if a variational integral has an infinitedimensional Lie algebra of infinitesimal symmetries parameterized linearly by r arbitrary functions and their derivatives up to a given order m, then there are r identities between Euler-Lagrange expressions and their derivatives up to order m. Such transformations are local transformations because can affect every part of the system differently.
In the last decades, Noether's theorems have been formulated in various contexts: see [2,3,5,8,9,10,11,19,24,25,27,33,35,37] and references therein. Since the variational problem proposed by Herglotz defines a functional, whose extrema are sought, by a family of differential equations, we cannot apply directly the important classical Noether's theorems to this kind of variational problems. During more than 70 years, the generalization of the two Noether theorems to variational problems of Herglotz type remained an open question. The problem was solved in the beginning of XXI century by Georgieva and Guenther [13,14,15,16,17].
The main goal of the present work is to generalize Noether's first theorem to variational problems of Herglotz type with time delay. Variational problems with time delay arguments were first introduced in 1964 by El'sgol'c [6], who derived an Euler-Lagrange type condition for variational problems with time delay. Since then, several authors have worked on various aspects of variational problems with time delay arguments (see [1,18,23,30,31] and references therein). It has been proved that variational systems with time delay play an important role in the modeling of phenomena in various applied fields. However, only recently Frederico and Torres generalized the important Noether's first theorem to optimal control problems with time delay [11]. Here we generalize Herglotz's problem by considering the following variational problem with time delay. Throughout the text, τ denotes a real number such that 0 ≤ τ < b − a; the partial derivative of L with respect to its ith argument is denoted by ∂ i L. To simplify notation, we write z{x; t} τ := z(t, x(t),ẋ(t), x(t − τ ),ẋ(t − τ )).
Problem (P). Determine the trajectories x ∈ C 2 ([a − τ, b], R) satisfying given conditions where β is a fixed real number and δ ∈ C 2 ([a − τ, a], R) is a given initial function, that extremize (minimize or maximize) the value of the functional z{x; b} τ , where z satisfies the differential equatioṅ subject to the initial condition z(a) = γ, where γ is a fixed real number. The Lagrangian L is assumed to satisfy the following hypotheses: , z(t)) for i = 2, . . . , 6 are differentiable for any admissible trajectory x.
Observe that problem z(b) → extr subject to (4)-(5) reduces to the classical fundamental problem of the calculus of variations with time delay if the Lagrangian L does not depend on z. Also note that problem (P ) reduces to the generalized variational problem of Herglotz when τ = 0.
The structure of the paper is as follows. In Section 2 we review some preliminaries about the generalized variational calculus (without time delay). In particular, we recall the notion of invariance and the first Noether theorem for variational problems of Herglotz type. Our main results are given in Section 3: a generalized Euler-Lagrange necessary optimality condition (Theorem 3.3), a DuBois-Reymond necessary optimality condition (Theorem 3.5), and Noether's first theorem for variational problems of Herglotz type with time delay (Theorem 3.9), are proved. We end with an illustrative example of our results in Section 4. The results of the paper are trivially generalized for the case of vector functions x : [a − τ, b] → R n , n ∈ N, but for simplicity of presentation we restrict ourselves to the scalar case.

2.
Review of Noether's first theorem for variational problems of Herglotz type. For the convenience of the reader, we present here the definition of generalized extremal, the definition of invariance of functional z, defined byż = L(t, x,ẋ, z) and z(a) = γ, and we recall Noether's first theorem for the generalized variational problem of Herglotz type. For simplicity of notation, we introduce the operator ·, · defined by . (2) and the boundary conditions x(a) = α and x(b) = β, for some fixed real numbers α, β, then x satisfies the generalized Euler-Lagrange equation Definition 2.2 (Generalized extremals-cf. [15]). The solutions x ∈ C 2 ([a, b], R) of the generalized Euler-Lagrange equation (6) are called generalized extremals.
Consider a one-parameter group of infinitesimal transformations on R 2 , in which ǫ is the parameter and φ and ψ are invertible C 1 functions such that φ(t, x, 0) = t and ψ(t, x, 0) = x. The infinitesimal representation of transformations (7) is given byt where σ and ξ denote the first degree coefficients of ǫ. Explicitly, Definition 2.3 (Invariance-cf. Proposition 3.1 of [15]). The one-parameter group of transformations t = φ(t, x, ǫ) x = ψ(t, x, ǫ) leave the functional z, defined byż = L(t, x,ẋ, z) and z(a) = γ for some fixed real We now prove the following useful result.
Lemma 2.4 (Necessary condition for invariance). If the functional z = z{x; t} defined byż(t) = L(t, x(t),ẋ(t), z(t)) and z(a) = γ, for some fixed real number γ, is invariant under the one-parameter group of transformations (7), then Proof. Note that and by multiplying both sides of the equality by dt dt we have, by the chain rule, that Now, differentiating with respect to ǫ and setting ǫ = 0, we find, by definition of invariance, that Since we are supposing the initial condition z(a) to be fixed (z(a) = γ), thenz(ā) is also fixed (z(ā) =γ) and hence d dǫ (z(ā)) and because σ(a, x) = 0, we can write that dz dǫ Theorem 2.5 (Noether's first theorem for variational problems of Herglotz type [15]). If functional z = z{x; t} defined byż = L (t, x(t),ẋ(t), z(t)) and z(a) = γ, for some fixed real number γ, is invariant under the one-parameter group of transformations (7), then is conserved along the generalized extremals, where λ(t) := e − t a ∂4L x,z (θ)dθ .
3. Main results. We prove some important results for variational problems of Herglotz type with time delay: a generalized Euler-Lagrange necessary optimality condition (Theorem 3.3), a DuBois-Reymond necessary optimality condition (Theorem 3.5) and a Noether's first theorem for variational problems of Herglotz type with time delay (Theorem 3.9). To simplify the presentation, we suppress most of the arguments and the following notation is used throughout: .
Definition 3.2 (Admissible variation). We say that The following result gives a necessary condition of Euler-Lagrange type for an admissible function x to be an extremizer of the functional z{x; b} τ , where z is defined by (4)-(5).
, then x satisfies the following generalized Euler-Lagrange equations with time delay: is a solution to problem (P ) and let η be an admissible variation. Let ǫ be an arbitrary real number and define ζ : Obviously, ζ(a) = 0 and, since x is an extremizer, we conclude that ζ(b) = 0. Observe thaṫ , which means thaṫ Consequently, ζ is solution of the first order linear differential equatioṅ Hence, ζ satisfies the equation Applying the change of variable s = t + τ in the second integral and recalling that From the fundamental lemma of the calculus of variations (see, e.g., [12]), we conclude that

Integration by parts gives
for a ≤ t ≤ b − τ , proving equation (8). Now, if we restrict ourselves to those admissible variations η such that and from the fundamental lemma of the calculus of variations we conclude that , proving equation (9). In order to simplify expressions, and in agreement with Theorem 3.3, from now on we use the notation The following theorem gives a generalization of the DuBois-Reymond condition for classical variational problems [4] and generalizes the Dubois-Reymond condition for variational problems with time delay of [11].
, then x satisfies the following equations: Proof. In order to prove equation (11), let t ∈ [a, b − τ ] be arbitrary. Note that Canceling symmetrical terms, we get Observe that, by hypothesis (10), the last integral is null and by substitution of the Euler-Lagrange equation (8)  Using hypothesis (10) in the right hand side of the last equation, we conclude that Condition (11) follows from the arbitrariness of t ∈ [a, b − τ ]. In order to prove equation (12),

Cancelling symmetrical terms, the previous equation becomes
Substituting the Euler-Lagrange equation (9) and using the hypothesis (10) in the last integral, we conclude that Condition (12) follows from the arbitrariness of t ∈ [b − τ, b].

Remark 2.
For the classical variational problem and for the variational problem of Herglotz (without delayed arguments), the hypothesis (10) is trivially satisfied.
There is an inconsistency in the proof of the DuBois-Reymond equations for the classical variational problem with time delay recently obtained in [11]: the proof is correct if we suppose that along any extremal with time delay for all t ∈ [a − τ, b − τ ]. Such condition is trivially satisfied for the examples presented in [7,11].
Before presenting the extension of the famous Noether's first theorem to variational problems of Herglotz type with time delay, we introduce the definition of invariance and give two useful necessary conditions for invariance. Definition 3.6 (Invariance with time delay). The one-parameter group of invertible leave the functional z defined by (4) Defining h(t) := dz dǫ (t) ǫ=0 , +Lσ. (15) Next we prove that d dǫ dx dt ǫ=0 =ξ −ẋσ.
Substituting (17) into (15), we geṫ Therefore, h satisfies a first order differential equation whose solution is Finally, since functional z defined by (4)- (5) is invariant under the one-parameter group of transformations (13), h ≡ 0 by Lemma 3.7 and we obtain (14).
The next result establishes an extension of the celebrated Noether first theorem to variational problems of Herglotz type with time delay.
From the arbitrariness of t ∈ [a, b] we conclude that for all t ∈ [a, b]. Then, equation (22) becomes for t ∈ [a + τ, b]. Using integration by parts, one has Observe that the terms in ξ inside the integral are null because x satisfies the Euler-Lagrange equation on [a, b − τ ] and that, from the DuBois-Reymond equation (11), the sum of the remaining terms of the integral is zero. This leads to for every t ∈ [a + τ, b], which means that Using integration by parts, we get t2 t1 = 0. Observe that the terms in ξ inside the integral are null because x satisfies the Euler-Lagrange equation (9) and that, from the DuBois-Reymond equation (12), the sum of the remaining terms of the integral is zero. This leads to This ends the proof of our main result. Remark 4. Our first Noether-type theorem is a generalization of Noether's first theorem for the classical variational problem of Herglotz type presented in [15], that is, Theorem 2.5 is a corollary of Theorem 3.9.
Our results provide generalizations of the variational results with time delay presented in [11]. If the Lagrangian L in the definition of z, (4), does not depend on z, then ∂ 6 L ≡ 0 and λ(t) ≡ 1. In that case, problem (P ) reduces to the classical variational problem with time delay. The Euler-Lagrange equations, the DuBois-Reymond conditions and Noether's first theorem with time delay obtained by Frederico and Torres in [11] are particular cases of Theorem 3.3, Theorem 3.5 and Theorem 3.9, respectively. In what follows we use the notation Corollary 1 (See [11]). If x is an extremizer to the functional b a L(t, x(t),ẋ(t), then x satisfies the Euler-Lagrange equations a ≤ t ≤ b − τ , and Corollary 2 (Cf. [11]). If x is an extremizer to the functional (24) and , then x satisfies the DuBois-Reymond equations d dt Corollary 3 (Cf. [11]). If functional (24) is invariant in the sense of Definition 2.3, then the quantities For this problem, Euler-Lagrange optimality conditions (8)-(9) given by Theorem 3.3 assert that ẋ(t) −ẍ(t) = 0, t ∈ [0, 1], 0 = 0, t ∈ [1,2].