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GAUGE SYMMETRIES AND NOETHER CURRENTS IN OPTIMAL CONTROL

DELFIM F. M. TORRES

arXiv:math/0301116v1 [math.OC] 11 Jan 2003

Abstract. We extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. As far as we consider a semi-invariance notion, and the transformation group may also depend on the control variables, the result is new even in the classical context of the calculus of variations.

1. Introduction The study of invariant variational problems

b

Minimize J [x(·)] =

a

L (t, x(t), x(t)) dt ˙

in the calculus of variations was initiated in the early part of the XX century by Emmy Noether who, in?uenced by the works of Klein and Lie on the transformation properties of di?erential equations under continuous groups of transformations (see e.g. [2, Ch. 2]), published in her seminal paper [13, 14] of 1918 two fundamental theorems, now classical results and known as the (?rst) Noether theorem and the second Noether theorem, showing that invariance with respect to a group of transformations of the variables t and x implies the existence of certain conserved quantities. These results, also known as Noether’s symmetry theorems, have profound implications in all physical theories, explaining the correspondence between symmetries of the systems (between the group of transformations acting on the independent and dependent variables of the system) and the existence of conservation laws. This remarkable interaction between the concept of invariance in the calculus of variations and the existence of ?rst integrals (Noether currents) was clearly recognized by Hilbert [6] (cf. [12]). The ?rst Noether theorem establishes the existence of ρ ?rst integrals of the Euler-Lagrange di?erential equations when the Lagrangian L is invariant under a group of transformations containing ρ parameters. This means that the invariance hypothesis leads to quantities which are constant along the Euler-Lagrange extremals. Extensions for the Pontryagin extremals of optimal control problems are available in [19, 21, 20]. The second Noether theorem establishes the existence of k (m + 1) ?rst integrals when the Lagrangian is invariant under an in?nite continuous group of transformations which, rather than dependence on parameters, as in the ?rst theorem, depend

2000 Mathematics Subject Classi?cation. 49K15, 49S05. Key words and phrases. optimal control, Pontryagin extremals, gauge symmetry, second Noether theorem, Noether currents. This research was supported in part by the Optimization and Control Theory Group of the R&D Unit Mathematics and Applications, and the program PRODEP III 5.3/C/200.009/2000. Partially presented at the 5th Portuguese Conference on Automatic Control (Controlo 2002 ), Aveiro, Portugal, September 5–7, 2002. Accepted for publication in Applied Mathematics ENotes, Volume 3.

1

2

DELFIM F. M. TORRES

upon k arbitrary functions and their derivatives up to order m. This second theorem is not as well known as the ?rst. It has, however, some rather interesting implications. If for example one considers the functional of the basic problem of the calculus of variations in the autonomous case,

b

(1)

J [x(·)] =

a

L (x(t), x(t)) dt , ˙

the classical Weierstrass necessary optimality condition can easily be deduced from the fact that the integral (1) is invariant under transformations of the form T = t + p(t), X = x(t), for an arbitrary function p(·) (see [11, p. 161]). The second Noether theorem is related to: (i) parameter invariant variational problems, i.e., problems of the calculus of variations, as in the homogeneous-parametric form, which are invariant under arbitrary transformations of the independent variable t (see [1, p. 266], [11, Ch. 8], [3, p. 179]); (ii) the singular Lagrangians and the constraints in the Hamiltonian formalism, a framework studied by Dirac-Bergmann (see [4, 5]); (iii) the physics of gauge theories, such as the gauge transformations of electrodynamics, electromagnetic ?eld, hydromechanics, and relativity (see [3, pp. 186–189], [11, p. 160], [10], [17]). For example, if the Lagrangian L represents a charged particle interacting with a electromagnetic ?eld, one ?nds that it is invariant under the combined action of the so called gauge transformation of the ?rst kind on the charged particle ?eld, and a gauge transformation of the second kind on the electromagnetic ?eld. As a result of this invariance it follows, from second Noether’s theorem, the very important conservation of charge. The invariance under gauge transformations is a basic requirement in Yang-Mills ?eld theory, an important subject, with many questions for mathematical understanding (cf. [7]). To our knowledge, no second Noether type theorem is available for the optimal control setting. One such generalization is our concern here. Instead of using the original argument [13, 14] of Emmy Noether, which is fairly complicated and depends on some deep and conceptually di?cult results in the calculus of variations, our approach follows, mutatis mutandis, the paper [19], where the ?rst Noether theorem is derived almost e?ortlessly by means of elementary techniques, with a simple and direct approach, and it is motivated by the novelties introduced by the author in [21]. Even in the classical context (cf. e.g. [10]) and in the simplest possible situation, for the basic problem of the calculus of variations, our result is new since we consider symmetries of the system which alter the cost functional up to an exact di?erential; we introduce a semi-invariant notion with some weights λ0 , . . . , λm (possible di?erent from zero); and our transformation group may depend also on x (the control). Our result hold both in the normal and abnormal cases. ˙ 2. The Optimal Control Problem We consider the optimal control problem in Lagrange form on the compact interval [a, b]:

b

Minimize J[x(·), u(·)] =

a

L (t, x(t), u(t)) dt

over all admissible pairs (x(·), u(·)),1

n (x(·), u(·)) ∈ W1,1 ([a, b]; Rn ) × Lr ([a, b]; ? ? Rr ) , ∞

satisfying the control equation x(t) = ? (t, x(t), u(t)) ˙ a.e. t ∈ [a, b] .

1The notation W 1,1 is used for the class of absolutely continuous functions, while L∞ represents

the class of measurable and essentially bounded functions.

GAUGE SYMMETRIES IN OPTIMAL CONTROL

3

The functions L : R × Rn × Rr → R and ? : R × Rn × Rr → Rn are assumed to be C 1 with respect to all variables and the set ? of admissible values of the control parameters is an arbitrary open set of Rr . Associated to the optimal control problem there is the Pontryagin Hamiltonian T H : [a, b] × Rn × ? × R × (Rn ) → R which is de?ned as (2) H(t, x, u, ψ0 , ψ) = ψ0 L(t, x, u) + ψ · ?(t, x, u) . A quadruple (x(·), u(·), ψ0 , ψ(·)), with admissible (x(·), u(·)), ψ0 ∈ R? , and ψ(·) ∈ 0 W1,1 ([a, b]; Rn ) (ψ(t) is a covector 1 × n), is called a Pontryagin extremal if the following two conditions are satis?ed for almost all t ∈ [a, b]: The Adjoint System: ?H ˙ (3) ψ(t) = ? (t, x(t), u(t), ψ0 , ψ(t)) ; ?x The Maximality Condition: (4) H (t, x(t), u(t), ψ0 , ψ(t)) = max H (t, x(t), u, ψ0 , ψ(t)) .

u∈?

The Pontryagin extremal is called normal if ψ0 = 0 and abnormal otherwise. The celebrated Pontryagin Maximum Principle asserts that if (x(·), u(·)) is a minimizer of the problem, then there exists a nonzero pair (ψ0 , ψ(·)) such that (x(·), u(·), ψ0 , ψ(·)) is a Pontryagin extremal. Furthermore, the Pontryagin Hamiltonian along the extremal is an absolutely continuous function of t, t → H (t, x(t), u(t), ψ0 , ψ(t)) ∈ W1,1 ([a, b]; R) , and satis?es the equality dH ?H (5) (t, x(t), u(t), ψ0 , ψ(t)) = (t, x(t), u(t), ψ0 , ψ(t)) , dt ?t for almost all t ∈ [a, b], where on the left-hand side we have the total derivative with respect to t and on the right-hand side the partial derivative of the Pontryagin Hamiltonian with respect to t (cf. [16]. See [18] for some generalizations of this fact). 3. Main Result To formulate a second Noether theorem in the optimal control setting, ?rst we need to have appropriate notions of invariance and Noether current. We propose the following ones. De?nition 3.1. A function C (t, x, u, ψ0 , ψ) which is constant along every Pontryagin extremal (x(·), u(·), ψ0 , ψ(·)) of the problem, (6) C (t, x(t), u(t), ψ0 , ψ(t)) = k , t ∈ [a, b] , for some constant k, will be called a Noether current. The equation (6) is the conservation law corresponding to the Noether current C. De?nition 3.2. Let C m ? p : [a, b] → Rk be an arbitrary function of the independent variable. Using the notation . α(t) = t, x(t), u(t), p(t), p(t), . . . , p(m) (t) , ˙ we say that the optimal control problem is semi-invariant if there exists a C 1 transformation group g : [a, b] × Rn × ? × Rk?(m+1) → R × Rn × Rr , (7) g (α(t)) = (T (α(t)) , X (α(t)) , U (α(t))) ,

4

DELFIM F. M. TORRES

which for p(t) = p(t) = · · · = p(m) (t) = 0 corresponds to the identity transforma˙ tion, g(t, x, u, 0, 0, . . . , 0) = (t, x, u) for all (t, x, u) ∈ [a, b] × Rn × ?, satisfying the equations (8) λ0 · p(t) + λ1 · p(t) + · · · + λm · p(m) (t) ˙ + L (t, x(t), u(t)) + d L (t, x(t), u(t)) dt

d d F (α(t)) = L (g (α(t))) T (α(t)) , dt dt

(9)

d d X (α(t)) = ? (g (α(t))) T (α(t)) , dt dt

for some function F of class C 1 and for some λ0 , . . . , λm ∈ Rk . In this case the group of transformations g will be called a gauge symmetry of the optimal control problem. Remark 3.1. We use the term “gauge symmetry” to emphasize the fact that the group of transformations g depend on arbitrary functions. The terminology takes origin from gauge invariance in electromagnetic theory and in Yang-Mills theories, but it refers here to a wider class of symmetries. Remark 3.2. The identity transformation is a gauge symmetry for any given optimal control problem. Theorem 3.1 (Second Noether theorem for Optimal Control). If the optimal control problem is semi-invariant under a gauge symmetry (7), then there exist k (m + 1) Noether currents of the form ψ0 ?F (α(t))

(i) ?pj 0

+ λi L (t, x(t), u(t)) j

+ ψ(t) ·

?X (α(t)) ?pj

(i) 0

? H(t, x(t), u(t), ψ0 , ψ(t))

?T (α(t)) ?pj

(i) 0

(i = 0, . . . , m, j = 1, . . . , k), where H is the corresponding Pontryagin Hamiltonian (2). Remark 3.3. We are using the standard convention that p(0) (t) = p(t), and the following notation for the evaluation of a term: . (?)|0 = (?)|p(t)=p(t)=···=p(m) (t)=0 . ˙ Remark 3.4. For the basic problem of the calculus of variations, i.e., when ? = u, Theorem 3.1 coincides with the classical formulation of the second Noether theorem if one puts λi = 0, i = 0, . . . , m, and F ≡ 0 in the De?nition 3.2, and the transformation group g is not allowed to depend on the derivatives of the state variables (on the control variables). In §4 we provide an example of the calculus of variations for which our result is applicable while previous results are not. Proof. Let i ∈ {0, . . . , m}, j ∈ {1, . . . , k}, and (x(·), u(·), ψ0 , ψ(·)) be an arbitrary Pontryagin extremal of the optimal control problem. Since it is assumed that to the values p(t) = p(t) = · · · = p(m) (t) = 0 it corresponds the identity gauge ˙ (i) transformation, di?erentiating (8) and (9) with respect to pj and then setting

GAUGE SYMMETRIES IN OPTIMAL CONTROL

5

p(t) = p(t) = · · · = p(m) (t) = 0 one gets: ˙ (10) λi j d ?F (α(t)) d L+ (i) dt dt ?pj =

0

?L ?T (α(t)) (i) ?t ?pj +

+

0

?L ?X (α(t)) · (i) ?x ?pj +L

0

0

?L ?U (α(t)) · (i) ?u ?pj ?? ?X (α(t)) · (i) ?x ?pj ?? ?U (α(t)) · (i) ?u ?pj

d ?T (α(t)) (i) dt ?pj

,

0

(11)

d ?X (α(t)) (i) dt ?pj

=

0

?? ?T (α(t)) (i) ?t ?pj

+

0

0

+

+?

0

d ?T (α(t)) (i) dt ?pj

,

0

with L and ?, and its partial derivatives, evaluated at (t, x(t), u(t)). Multiplying (10) by ψ0 and (11) by ψ(t), we can write: ψ0 (12) ?L ?T (α(t)) (i) ?t ?pj +

0

?L ?X (α(t)) · (i) ?x ?pj

+

0

?L ?U (α(t)) · (i) ?u ?pj d ? λi L j dt +

0

0

d ?T (α(t)) +L (i) dt ?pj +ψ(t) ·

d ?F (α(t)) ? (i) dt ?pj 0

0

?? ?T (α(t)) (i) ?t ?pj

?? ?X (α(t)) · + (i) ?x ?pj 0 d ?T (α(t)) (i) dt ?pj ?

0

?? ?U (α(t)) · (i) ?u ?pj

0

+?

d ?X (α(t)) (i) dt ?pj

= 0.

0

According to the maximality condition (4), the function ψ0 L (t, x(t), U (α(t))) + ψ(t) · ? (t, x(t), U (α(t))) attains an extremum for p(t) = p(t) = · · · = p(m) (t) = 0. Therefore ˙ ψ0 ?L ?U (α(t)) · (i) ?u ?pj + ψ(t) ·

0

?? ?U (α(t)) · (i) ?u ?pj

=0

0

and (12) simpli?es to ψ0 ?L ?T (α(t)) (i) ?t ?pj +

0

?L ?X (α(t)) · (i) ?x ?pj ?

+L

0

d ?T (α(t)) (i) dt ?pj ? λi j

0

0

d ?F (α(t)) (i) dt ?pj +

0

d L dt +?

0

+ψ(t) ·

?? ?T (α(t)) (i) ?t ?pj

?? ?X (α(t)) · (i) ?x ?pj ?

d ?T (α(t)) (i) dt ?pj

0

d ?X (α(t)) (i) dt ?pj

= 0.

0

6

DELFIM F. M. TORRES

Using the adjoint system (3) and the property (5), one easily concludes that the above equality is equivalent to d dt ψ0 ?F (α(t))

(i) ?pj 0

+ ψ0 λi L + ψ(t) · j

?X (α(t))

(i) ?pj 0

?H

?T (α(t)) ?pj

(i) 0

= 0.

Quod erat demonstrandum. 4. Example Consider the following simple time-optimal problem with n = r = 1 and ? = (?1, 1). Given two points α and β in the state space R, we are to choose an admissible pair (x(·), u(·)), solution of the the control equation x(t) = u(t) , ˙ and satisfying the boundary conditions x(0) = α, x(T ) = β, in such a way that the time of transfer from α to β is minimal: T → min . In this case the Lagrangian is given by L ≡ 1 while ? = u. It is easy to conclude that the problem is invariant under the gauge symmetry g (t, x(t), u(t), p(t), p(t), p(t)) ˙ ¨ = p(t) + t, (p(t) + 1)2 x(t), 2¨(t)x(t) + (p(t) + 1)u(t) , ˙ p ˙ i.e., under T = p(t) + t , X = (p(t) + 1)2 x(t) , ˙ U = 2¨(t)x(t) + (p(t) + 1)u(t) , p ˙ where p(·) is an arbitrary function of class C 2 ([0, T ]; R). For that we choose F = p(t), λ0 = λ1 = λ2 = 0, and conditions (8) and (9) follows: d d d L (T, X, U ) T = (p(t) + t) = F + L(t, x(t), u(t)) , dt dt dt d p ˙ ˙ ? (T, X, U ) T = [2¨(t)x(t) + (p(t) + 1) u(t)] (p(t) + 1) dt d d 2 (p(t) + 1) x(t) = X . ˙ = dt dt From Theorem 3.1 the two non-trivial Noether currents (13) (14) ψ0 ? H , 2ψ(t)x(t) ,

are obtained. As far as ψ0 is a constant, the Noether current (13) is just saying that the corresponding Hamiltonian H is constant along the Pontryagin extremals of the problem. This is indeed the case, since the problem under consideration is autonomous (cf. equality (5)). The Noether current (14) can be understood having in mind the maximality condition (4) ( ?H = 0 ? ψ(t) = 0). ?u 5. Concluding Remarks In this paper we provide an extension of the second Noether’s theorem to the optimal control framework. The result seems to be new even for the problems of the calculus of variations. Theorem 3.1 admits several extensions. It was derived, as in the original work by Noether [13, 14], for state variables in an n-dimensional Euclidean space. It can be formulated, however, in contexts where the geometry is not Euclidean (these extensions can be found, in the classical context, e.g. in [8, 9, 15]). It admits also a generalization for optimal control problems which are invariant in a mixed

GAUGE SYMMETRIES IN OPTIMAL CONTROL

7

sense, i.e., which are invariant under a group of transformations depending upon ρ parameters and upon k arbitrary functions and their derivatives up to some given order. Other possibility is to obtain a more general version of the second Noether theorem for optimal control problems which does not admit exact symmetries. For example, under an invariance notion up to ?rst-order terms in the functions p(·) and its derivatives (cf. the quasi-invariance notion introduced by the author in [20] for the ?rst Noether theorem). These and other questions, such as the generalization of the ?rst and second Noether type theorems to constrained optimal control problems, are under study and will be addressed elsewhere. References

[1] R. Courant and D. Hilbert. Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y., 1953. Zbl 0051.28802 MR 16:426a [2] A. Dur?n. Symmetries of di?erential equations and numerical applications. Universidade de a Coimbra, Departamento de Matem?tica, Coimbra, 1999. Zbl 0967.34001 MR 2001f:35024 a [3] I. M. Gelfand and S. V. Fomin. Calculus of variations. Dover Publications, Mineola, NY, 2000. Zbl 0964.49001 [4] X. Gr`cia and J. M. Pons. Canonical Noether symmetries and commutativity properties for a gauge systems. J. Math. Phys., 41(11):7333–7351, 2000. Zbl pre01639412 MR 2001k:70023 [5] M. Guerra. Solu??es generalizadas para problemas L-Q singulares. PhD thesis, Departamento co de Matem?tica, Universidade de Aveiro, 2001. a [6] D. Hilbert. Grundlagen der physik. Math. Ann., 92:258–289, 1924. [7] A. Ja?e and E. Witten. Quantum Yang-Mills theory. Problem Description of the Yang-Mills Existence and Mass Gap Millennium Prize Problem, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI), http://www.claymath.org/prizeproblems/yang mills.pdf. [8] J. Komorowski. A modern version of the E. Noether’s theorems in the calculus of variations. I. Studia Math., 29:261–273, 1968. Zbl 0155.17602 MR 37:799 [9] J. Komorowski. A modern version of the E. Noether’s theorems in the calculus of variations. II. Studia Math., 32:181–190, 1969. Zbl 0176.41402 MR 39:7485 [10] J. D. Logan. On variational problems which admit an in?nite continuous group. Yokohama Math. J., 22:31–42, 1974. Zbl 0295.49023 MR 50:14419 [11] J. D. Logan. Invariant variational principles. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1977. MR 58:18024 [12] D. Lovelock and H. Rund. Tensors, di?erential forms, and variational principles. WileyInterscience [John Wiley & Sons], New York, 1975. MR 57:13703 [13] E. Noether. Invariante variationsprobleme. G¨tt. Nachr., pages 235–257, 1918. o JFM 46.0770.01 [14] E. Noether. Invariant variation problems. Transport Theory Statist. Phys., 1(3):186–207, 1971. English translation of the original paper [13]. Zbl 0292.49008 MR 53:10538 [15] B. F. Plybon. New approach to the Noether theorems. J. Mathematical Phys., 12:57–60, 1971. Zbl 0205.10902 MR 43:2591 [16] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko. The mathematical theory of optimal processes. Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962. Zbl 0882.01027 MR 29:3316b [17] M. A. Tavel. Milestones in mathematical physics: Noether’s theorem. Transport Theory Statist. Phys., 1(3):183–185, 1971. Zbl 0291.49035 MR 53:10537 [18] D. F. M. Torres. A remarkable property of the dynamic optimization extremals. Investiga??o ca Operacional, 22(2):253–263, 2002. [19] D. F. M. Torres. Conservation laws in optimal control. In Dynamics, Bifurcations and Control, volume 273 of Lecture Notes in Control and Information Sciences, pages 287–296. Springer-Verlag, Berlin, Heidelberg, 2002. Zbl pre01819752 MR 1901565 [20] D. F. M. Torres. Conserved quantities along the Pontryagin extremals of quasi-invariant optimal control problems. Proc. 10th Mediterranean Conference on Control and Automation – MED2002 (invited paper), 10 pp. (electronic), Lisbon, Portugal, July 9-12, 2002. [21] D. F. M. Torres. On the Noether theorem for optimal control. European Journal of Control, 8(1):56–63, 2002. ? Departamento de Matematica, Universidade de Aveiro, 3810-193 Aveiro, Portugal E-mail address: delfim@mat.ua.pt URL: http://www.mat.ua.pt/delfim

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52

Approximate

Conformal Invariance

Perturbation to

eld theories, if a Lagrangian possesses certain

1 Introduction

A Note on the scale

Pontryagin extremals,

The following result is the analogue of Version (II) of

A Formulation of

Comments on the

erence

Higher spin extensions of nonabelian