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

Torres, D.F.M.

English

Repositório Institucional da Universidade de Aveiro

Applied Mathematics E-Notes, 3(2003), 49-57 c? ISSN 1607-2510Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/Gauge Symmetries And Noether Currents In OptimalControl ∗Delfim F. M. Torres†Received 7 June 2002AbstractWe extend the second Noether theorem to optimal control problems which areinvariant under symmetries depending upon k arbitrary functions of the indepen-dent variable and their derivatives up to some order m. As far as we considera semi-invariance notion, and the transformation group may also depend on thecontrol variables, the result is new even in the classical context of the calculus ofvariations.1 IntroductionThe study of invariant variational problemsMinimize J [x(·)] =? baL (t, x(t), x˙(t)) dtin the calculus of variations was initiated in the early part of the XX century byEmmy Noether who, influenced by the works of Klein and Lie on the transformationproperties of differential 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 (first) Noether theorem and the second Noethertheorem, showing that invariance with respect to a group of transformations of thevariables t and x implies the existence of certain conserved quantities. These results,also known as Noether’s symmetry theorems, have profound implications in all physicaltheories, explaining the correspondence between symmetries of the systems (betweenthe group of transformations acting on the independent and dependent variables of thesystem) and the existence of conservation laws. This remarkable interaction betweenthe concept of invariance in the calculus of variations and the existence of first integrals(Noether currents) was clearly recognized by Hilbert [6] (cf. [12]).The first Noether theorem establishes the existence of ρ first integrals of the Euler-Lagrange differential equations when the Lagrangian L is invariant under a group oftransformations containing ρ parameters. This means that the invariance hypothesis∗Mathematics Subject Classifications: 49K15, 49S05.†Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal4950 Gauge Symmetries and Noether Currentsleads to quantities which are constant along the Euler-Lagrange extremals. Extensionsfor the Pontryagin extremals of optimal control problems are available in [19, 21, 20].The second Noether theorem establishes the existence of k (m+ 1) first integralswhen the Lagrangian is invariant under an infinite continuous group of transformationswhich, rather than dependence on parameters, as in the first theorem, depend upon karbitrary functions and their derivatives up to order m. This second theorem is not aswell known as the first. It has, however, some rather interesting implications. If forexample one considers the functional of the basic problem of the calculus of variationsin the autonomous case,J [x(·)] =? baL (x(t), x˙(t)) dt , (1)the classical Weierstrass necessary optimality condition can easily be deduced from thefact 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 the-orem is related to: (i) parameter invariant variational problems, i.e., problems of thecalculus of variations, as in the homogeneous-parametric form, which are invariant un-der 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 Hamiltonianformalism, a framework studied by Dirac-Bergmann (see [4, 5]); (iii) the physics ofgauge theories, such as the gauge transformations of electrodynamics, electromagneticfield, hydromechanics, and relativity (see [3, pp. 186—189], [11, p. 160], [10], [17]). Forexample, if the Lagrangian L represents a charged particle interacting with a electro-magnetic field, one finds that it is invariant under the combined action of the so calledgauge transformation of the first kind on the charged particle field, and a gauge trans-formation of the second kind on the electromagnetic field. As a result of this invarianceit follows, from second Noether’s theorem, the very important conservation of charge.The invariance under gauge transformations is a basic requirement in Yang-Mills fieldtheory, an important subject, with many questions for mathematical understanding(cf. [7]).To our knowledge, no second Noether type theorem is available for the optimalcontrol setting. One such generalization is our concern here. Instead of using the orig-inal argument [13, 14] of Emmy Noether, which is fairly complicated and depends onsome deep and conceptually difficult results in the calculus of variations, our approachfollows, mutatis mutandis, the paper [19], where the first Noether theorem is derivedalmost effortlessly by means of elementary techniques, with a simple and direct ap-proach, and it is motivated by the novelties introduced by the author in [21]. Even inthe classical context (cf. e.g. [10]) and in the simplest possible situation, for the basicproblem of the calculus of variations, our result is new since we consider symmetries ofthe system which alter the cost functional up to an exact differential; we introduce asemi-invariant notion with some weights λ0, . . . ,λm (possible different from zero); andour transformation group may depend also on x˙ (the control). Our result hold both inthe normal and abnormal cases.D. F. M. Torres 512 The Optimal Control ProblemWe consider the optimal control problem in Lagrange form on the compact interval[a, b]:Minimize J [x(·), u(·)] =? baL (t, x(t), u(t)) dtover all admissible pairs (x(·), u(·)) 1(x(·), u(·)) ∈Wn1,1 ([a, b];Rn)× Lr∞ ([a, b];Ω ⊆ Rr) ,satisfying the control equationx˙(t) = ϕ (t, x(t), u(t)) a.e. t ∈ [a, b] .The functions L : R × Rn × Rr → R and ϕ : R × Rn × Rr → Rn are assumed tobe C1 with respect to all variables and the set Ω of admissible values of the controlparameters is an arbitrary open set of Rr.Associated to the optimal control problem there is the Pontryagin HamiltonianH : [a, b]×Rn ×Ω×R× (Rn)T → R which is defined asH(t, x, u,ψ0,ψ) = ψ0L(t, x, u) + ψ · ϕ(t, x, u) . (2)A quadruple (x(·), u(·), ψ0, ψ(·)), with admissible (x(·), u(·)), ψ0 ∈ R−0 , and ψ(·) ∈W1,1 ([a, b];Rn) (ψ(t) is a covector 1×n), is called a Pontryagin extremal if the followingtwo conditions are satisfied for almost all t ∈ [a, b]:The Adjoint Systemψ˙(t) = −∂H∂x(t, x(t), u(t), ψ0, ψ(t)) ; (3)The Maximality ConditionH (t, x(t), u(t),ψ0,ψ(t)) = maxu∈ΩH (t, x(t), u,ψ0,ψ(t)) . (4)The Pontryagin extremal is called normal if ψ0 9= 0 and abnormal otherwise. Thecelebrated Pontryagin Maximum Principle asserts that if (x(·), u(·)) is a minimizer ofthe 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 extremalis an absolutely continuous function of t,t :→ H (t, x(t), u(t),ψ0,ψ(t)) ∈W1,1 ([a, b];R) ,and satisfies the equalitydHdt(t, x(t), u(t),ψ0,ψ(t)) =∂H∂t(t, x(t), u(t),ψ0,ψ(t)) , (5)for almost all t ∈ [a, b], where on the left-hand side we have the total derivative withrespect to t and on the right-hand side the partial derivative of the Pontryagin Hamil-tonian with respect to t (cf. [16]. See [18] for some generalizations of this fact).1The notation W1,1 is used for the class of absolutely continuous functions, while L∞ representsthe class of measurable and essentially bounded functions.52 Gauge Symmetries and Noether Currents3 Main ResultTo formulate a second Noether theorem in the optimal control setting, first we need tohave appropriate notions of invariance and Noether current. We propose the followingones.DEFINITION 1. A function C (t, x, u, ψ0, ψ) which is constant along every Pon-tryagin extremal (x(·), u(·), ψ0, ψ(·)) of the problem,C (t, x(t), u(t), ψ0, ψ(t)) = k , t ∈ [a, b] , (6)for some constant k, will be called a Noether current. The equation (6) is the conser-vation law corresponding to the Noether current C.DEFINITION 2. Let Cm 6 p : [a, b]→ Rk be an arbitrary function of the indepen-dent variable. Using the notationα(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 C1 trans-formation groupg : [a, b]×Rn ×Ω×Rk∗(m+1) → R×Rn ×Rr ,g (α(t)) = (T (α(t)) ,X (α(t)) , U (α(t))) , (7)which for p(t) = p˙(t) = · · · = p(m)(t) = 0 corresponds to the identity transformation,g(t, x, u, 0, 0, . . . , 0) = (t, x, u) for all (t, x, u) ∈ [a, b]×Rn×Ω, satisfying the equationsL (g (α(t)))ddtT (α(t))=?λ0 · p(t) + λ1 · p˙(t) + · · ·+ λm · p(m)(t)? ddtL (t, x(t), u(t))+L (t, x(t), u(t)) +ddtF (α(t)) (8)ddtX (α(t)) = ϕ (g (α(t)))ddtT (α(t)) , (9)for some function F of class C1 and for some λ0, . . . ,λm ∈ Rk. In this case the groupof transformations g will be called a gauge symmetry of the optimal control problem.REMARK 1. We use the term “gauge symmetry” to emphasize the fact that thegroup of transformations g depend on arbitrary functions. The terminology takesorigin from gauge invariance in electromagnetic theory and in Yang-Mills theories, butit refers here to a wider class of symmetries.REMARK 2. The identity transformation is a gauge symmetry for any given opti-mal control problem.THEOREM 1 (second Noether theorem for Optimal Control). If the optimal controlproblem is semi-invariant under a gauge symmetry (7), then there exist k (m+ 1)D. F. M. Torres 53Noether currents of the formψ0?∂F (α(t))∂p(i)j?????0+ λijL (t, x(t), u(t))?+ ψ(t) · ∂X (α(t))∂p(i)j?????0−H(t, x(t), u(t),ψ0,ψ(t))∂T (α(t))∂p(i)j?????0(i = 0, . . . ,m, j = 1, . . . , k), where H is the corresponding Pontryagin Hamiltonian (2).REMARK 3. We are using the standard convention that p(0)(t) = p(t), and thefollowing notation for the evaluation of a term:(∗)|0 .= (∗)|p(t)=p˙(t)=···=p(m)(t)=0 .REMARK 4. For the basic problem of the calculus of variations, i.e., when ϕ = u,Theorem 1 coincides with the classical formulation of the second Noether theorem ifone puts λi = 0, i = 0, . . . ,m, and F ≡ 0 in the Definition 2, and the transformationgroup g is not allowed to depend on the derivatives of the state variables (on the controlvariables). In §4 we provide an example of the calculus of variations for which our resultis applicable while previous results are not.PROOF. Let i ∈ {0, . . . ,m}, j ∈ {1, . . . , k}, and (x(·), u(·),ψ0,ψ(·)) be an arbitraryPontryagin extremal of the optimal control problem. Since it is assumed that to thevalues p(t) = p˙(t) = · · · = p(m)(t) = 0 it corresponds the identity gauge transformation,differentiating (8) and (9) with respect to p(i)j and then setting p(t) = p˙(t) = · · · =p(m)(t) = 0 one gets:λijddtL+ddt∂F (α(t))∂p(i)j?????0=∂L∂t∂T (α(t))∂p(i)j?????0+∂L∂x· ∂X (α(t))∂p(i)j?????0+∂L∂u· ∂U (α(t))∂p(i)j?????0+ Lddt∂T (α(t))∂p(i)j?????0, (10)ddt∂X (α(t))∂p(i)j?????0=∂ϕ∂t∂T (α(t))∂p(i)j?????0+∂ϕ∂x· ∂X (α(t))∂p(i)j?????0+∂ϕ∂u· ∂U (α(t))∂p(i)j?????0+ ϕddt∂T (α(t))∂p(i)j?????0, (11)with L and ϕ, and its partial derivatives, evaluated at (t, x(t), u(t)). Multiplying (10)by ψ0 and (11) by ψ(t), we can write:ψ0?∂L∂t∂T (α(t))∂p(i)j?????0+∂L∂x· ∂X (α(t))∂p(i)j?????0+∂L∂u· ∂U (α(t))∂p(i)j?????054 Gauge Symmetries and Noether Currents+Lddt∂T (α(t))∂p(i)j?????0− ddt∂F (α(t))∂p(i)j?????0− λijddtL?+ψ(t) ·?∂ϕ∂t∂T (α(t))∂p(i)j?????0+∂ϕ∂x· ∂X (α(t))∂p(i)j?????0+∂ϕ∂u· ∂U (α(t))∂p(i)j?????0+ϕddt∂T (α(t))∂p(i)j?????0− ddt∂X (α(t))∂p(i)j?????0?= 0. (12)According to the maximality condition (4), the functionψ0L (t, x(t), U (α(t))) + ψ(t) · ϕ (t, x(t), U (α(t)))attains an extremum for p(t) = p˙(t) = · · · = p(m)(t) = 0. Thereforeψ0∂L∂u· ∂U (α(t))∂p(i)j?????0+ ψ(t) · ∂ϕ∂u· ∂U (α(t))∂p(i)j?????0= 0and (12) simplifies toψ0?∂L∂t∂T (α(t))∂p(i)j?????0+∂L∂x· ∂X (α(t))∂p(i)j?????0+ Lddt∂T (α(t))∂p(i)j?????0− ddt∂F (α(t))∂p(i)j?????0− λijddtL?+ψ(t) ·?∂ϕ∂t∂T (α(t))∂p(i)j?????0+∂ϕ∂x· ∂X (α(t))∂p(i)j?????0+ ϕddt∂T (α(t))∂p(i)j?????0− ddt∂X (α(t))∂p(i)j?????0?= 0 .Using the adjoint system (3) and the property (5), one easily concludes that theabove equality is equivalent toddt?ψ0∂F (α(t))∂p(i)j?????0+ ψ0λijL+ ψ(t) ·∂X (α(t))∂p(i)j?????0−H ∂T (α(t))∂p(i)j?????0?= 0 .4 ExampleConsider the following simple time-optimal problem with n = r = 1 and Ω = (−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 control equationx˙(t) = u(t) ,D. F. M. Torres 55and satisfying the boundary conditions x(0) = α, x(T ) = β, in such a way that thetime of transfer from α to β is minimal:T → min .In this case the Lagrangian is given by L ≡ 1 while ϕ = u. It is easy to conclude thatthe problem is invariant under the gauge symmetryg (t, x(t), u(t), p(t), p˙(t), p¨(t))=?p(t) + t, (p˙(t) + 1)2x(t), 2p¨(t)x(t) + (p˙(t) + 1)u(t)?,i.e., underT = p(t) + t , X = (p˙(t) + 1)2x(t) , U = 2p¨(t)x(t) + (p˙(t) + 1)u(t) ,where p(·) is an arbitrary function of class C2 ([0, T ];R). For that we choose F = p(t),λ0 = λ1 = λ2 = 0, and conditions (8) and (9) follows:L (T,X,U)ddtT =ddt(p(t) + t) =ddtF + L(t, x(t), u(t)) ,ϕ (T,X,U)ddtT = [2p¨(t)x(t) + (p˙(t) + 1)u(t)] (p˙(t) + 1)=ddt?(p˙(t) + 1)2x(t)?=ddtX .From Theorem 1 the two non-trivial Noether currentsψ0 −H , (13)2ψ(t)x(t) , (14)are obtained. As far as ψ0 is a constant, the Noether current (13) is just saying thatthe corresponding Hamiltonian H is constant along the Pontryagin extremals of theproblem. 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 themaximality condition (4) (∂H∂u = 0⇔ ψ(t) = 0).5 Concluding RemarksIn this paper we provide an extension of the second Noether’s theorem to the optimalcontrol framework. The result seems to be new even for the problems of the calculusof variations.Theorem 1 admits several extensions. It was derived, as in the original work byNoether [13, 14], for state variables in an n-dimensional Euclidean space. It can beformulated, however, in contexts where the geometry is not Euclidean (these extensionscan be found, in the classical context, e.g. in [8, 9, 15]). It admits also a generaliza-tion for optimal control problems which are invariant in a mixed sense, i.e, which are56 Gauge Symmetries and Noether Currentsinvariant under a group of transformations depending upon ρ parameters and upon karbitrary functions and their derivatives up to some given order. Other possibility is toobtain a more general version of the second Noether theorem for optimal control prob-lems which does not admit exact symmetries. For example, under an invariance notionup to first-order terms in the functions p(·) and its derivatives (cf. the quasi-invariancenotion introduced by the author in [20] for the first Noether theorem). These and otherquestions, such as the generalization of the first and second Noether type theorems toconstrained optimal control problems, are under study and will be addressed elsewhere.Acknowledgment. This research was supported in part by the Optimization andControl Theory Group of the R&D Unit Mathematics and Applications, and the pro-gram PRODEP III 5.3/C/200.009/2000. Partially presented at the 5th PortugueseConference on Automatic Control (Controlo 2002), Aveiro, September 5—7, 2002.References[1] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. IntersciencePublishers, Inc., New York, N.Y., 1953.[2] A. Dura´n, Symmetries of differential equations and numerical applications, Uni-versidade de Coimbra, Departamento de Matema´tica, Coimbra, 1999.[3] I. M. Gelfand and S. V. Fomin, Calculus of variations. Dover Publications, Mine-ola, NY, 2000.[4] X. Gra`cia and J. M. Pons, Canonical Noether symmetries and commutativityproperties for gauge systems. J. Math. Phys., 41(11)(2000), 7333—7351.[5] M. Guerra, Soluc¸o˜es generalizadas para problemas L-Q singulares. PhD thesis,Departamento de Matema´tica, Universidade de Aveiro, 2001.[6] D. 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A modern version of the E. Noether’s theorems in the calculus of variations II,

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Gauge symmetries and Noether currents in optimal control

Gauge symmetries and Noether currents in optimal control

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