Making use of a computer algebra system, we define computational tools to identify symmetries and conservation laws in optimal control

Gouveia, Paulo D.F.

Torres, Delfim F.M.

Biblioteca Digital do IPB

2 P.D.F. Gouveia, D.F.M. Torreshandle the problem of determining symmetries and conservation laws in opti-mal control, since they can perform differentiations, simplifications, and solvedifferential equations, in a reliably way. We illustrate our approach with aconcrete optimal control problem borrowed from the literature.2. Conservation Laws in Optimal ControlThe optimal control problem consists to minimize an integral functional,I[x(·),u(·)] =∫ baL(t,x(t),u(t))dt , (2.1)subject to a control system described by ordinary differential equations,x˙(t) = ϕ(t,x(t),u(t)) , (2.2)together with appropriate boundary conditions on the values of x(a) andx(b). The Lagrangian L(·, ·, ·) is a real function, assumed to be continuouslydifferentiable in [a, b]×Rn×Rm; t ∈ R the independent variable; x : [a, b]→ Rnthe vector of state variables; u : [a, b] → Ω ⊆ Rm, Ω an open set, the vectorof controls, assumed to be piecewise continuous functions; andϕ : [a, b]× Rn × Rm → Rnthe velocity vector, assumed to be a continuously differentiable vector func-tion. We propose a computational method that permits to obtain conservationlaws for a given optimal control problem. Our method is based in the versionof Noether’s theorem established in [1]. Let us consider a one-parameter groupof C1 transformations hs : [a, b]× Rn × Rm × R× Rn → R× Rn × Rm × Rn,hs(t,x,u, ψ0,ψ) =(hst (t,x, ψ0,ψ),hsx(t,x, ψ0,ψ), hsu(t,x,u, ψ0,ψ),hsψ(t,x,u, ψ0,ψ)), (2.3)which reduces to the identity transformation when the parameter s vanishes:h0t= t, h0x = x, h0u = u, h0ψ = ψ. Associated with a one-parameter groupof transformations (2.3), we introduce the infinitesimal generatorsT (t,x, ψ0,ψ) =∂∂shst∣∣∣∣s=0, X(t,x, ψ0,ψ) =∂∂shsx∣∣∣∣s=0,U(t,x,u, ψ0,ψ) =∂∂shsu∣∣∣∣s=0, Ψ (t,x,u, ψ0,ψ) =∂∂shsψ∣∣∣∣s=0. (2.4)Emmy Noether was the first who established a relation between the existenceof invariance transformations of the problem and the existence of conservationlaws [4]. Since the work pioneered by Noether, several definitions of invarianceComputation of Conservation Laws in Optimal Control 3have been introduced for the problems of the calculus of variations (see e.g.[4, 6]); and for the problems of optimal control (see e.g. [1, 5, 7]). All thesedefinitions are given with respect to a one-parameter group of transformations(2.3). Although written in a different way one gets, in terms of the generators(2.4), a necessary and sufficient condition of invariance that, essentially, coin-cide to all those definitions. For this reason, here we define invariance directlyin terms of the generators (2.4).Definition 1. ([1]) We say that an optimal control problem (2.1)-(2.2) isinvariant under (2.4) or, equivalently, that (2.4) is a symmetry of the problem,if, and only if, condition∂H∂tT +∂H∂x·X +∂H∂u·U +∂H∂ψ·Ψ −ΨT · x˙−ψT ·dXdt+HdTdt= 0 (2.5)holds, with H the Hamiltonian:H(t,x,u, ψ0,ψ) = ψ0L(t,x,u) + ψT· ϕ(t,x,u) . (2.6)A symmetry is an intrinsic property of the optimal control problem (2.1)-(2.2)(an intrinsic property of the corresponding Hamiltonian (2.6)), and does notdepend on the Pontryagin extremals. If one restricts attention to the quadru-ples (x(·),u(·), ψ0,ψ(·)) that satisfy the Pontryagin maximum principle, onearrives to Noether’s theorem.Theorem 1. (Noether’s theorem) If (2.4) is a symmetry of problem (2.1)-(2.2), thenψ(t)T ·X(t,x(t), ψ0,ψ(t)) −H(t,x(t),u(t), ψ0,ψ(t))× T(t,x(t), ψ0,ψ(t))= const (2.7)is a conservation law.We remark that Noetherian conservation laws (2.7) only depend on the gen-erators T and X of a symmetry (T,X,U ,Ψ ) (2.4).3. Computation of Conservation LawsThe conservation laws we are looking for are obtained substituting in (2.7)the components T and X of a symmetry of the problem. We develop theMaple procedure Noether to do such calculations for us. The input to thisprocedure is: the Lagrangian L and the velocity vector ϕ, that define theoptimal control problem (2.1)-(2.2) and the respective Hamiltonian H ; anda symmetry, or a family of symmetries, obtained by means of our proce-dure Symmetry. The output of Noether is the correspondent conservationlaw (2.7). The reader can find the Maple files with the definitions of theSymmetry and Noether procedures, that constitute our Maple package, athttp://www.mat.ua.pt/delfim/maple.htm.4 P.D.F. Gouveia, D.F.M. TorresThe non-trivial part resides in the determination of the symmetries ofthe problem. Our algorithm for determining the infinitesimal generators (2.4)is based on the necessary and sufficient condition of invariance (2.5). Thekey to do the calculations consists in observing that when we substitute theHamiltonian H and its partial derivatives in the invariance identity (2.5),then the condition become a polynomial in x˙ and ψ˙, and one can equal thecoefficients to zero. Let us see how it works in more detail.Substituting H and its partial derivatives into (2.5), and expanding thetotal derivatives, one can write equation (2.5) as a polynomial in the 2n deriva-tives x˙ and ψ˙:(∂H∂tT +∂H∂x·X +∂H∂u·U +∂H∂ψ· Ψ +H∂T∂t−ψT ·∂X∂t)+(−ΨT +H∂T∂x−ψT ·∂X∂x)· x˙+(H∂T∂ψ−ψT ·∂X∂ψ)· ψ˙ = 0 . (3.1)The terms in (3.1) which involve derivatives with respect to vectors are ex-panded in line-vectors or in matrices, depending, respectively, if the functionis a scalar or a vectorial one.Given an optimal control problem, defined by a Lagrangian L and a controlsystem (2.2), we determine the infinitesimal generators T ,X, U and Ψ , whichdefine a symmetry for the problem, by the following method. Equation (3.1)is a differential equation in the 2n+m+1 unknown functions T , X1, . . . , Xn,U1, . . . , Um, Ψ1, . . . , and Ψn. This equation must hold for all x˙1, . . . , x˙n, ψ˙1,. . . , ψ˙n, and therefore all the coefficients of polynomial (3.1) must vanish.The system of equations obtained from (3.1) is a system of 2n+ 1 partialdifferential equations with 2n+m+1 unknown functions (so, in general, thereexists not a unique symmetry but a whole family of symmetries). Although asystem of partial differential equations, its resolution is possible because thesystem is linear with respect to the unknown functions and their derivatives.However, when dealing with optimal control problems with several state andcontrol variables, the number of calculations is big enough, and the help ofthe computer is more than welcome. Our Maple procedure Symmetry doesall the job for us. Since system is homogeneous, we always have, as trivialsolution, (T,X,U ,Ψ ) = 0. When the output of Symmetry coincides with thetrivial solution, that means that the optimal control problem does not admita symmetry.Summarizing: given an optimal control problem (2.1)-(2.2) we obtain con-servation laws, in an automatic way, through two steps:(i) with our procedure Symmetry we obtain the invariance symmetries ofthe problem;(ii) using the obtained symmetries as input to procedure Noether, basedon Theorem 1, we obtain the correspondent conservation laws.Computation of Conservation Laws in Optimal Control 54. An Illustrative ExampleIn order to show the functionality and the usefulness of the developed routines,we apply our Maple package to a concrete optimal control problem found in theliterature. The calling syntaxes of our procedures are Symmetry(L,ϕ, t, x, u)and Noether(L,ϕ, t, x, u, S), with the respective inputs: Lagrangian L, veloc-ity vector ϕ, name t of the independent variable, list of names x of the statevariables, list of names u of the control variables, and the set S of infinitesimalgenerators for function Noether (the output of Symmetry).Example. Let us consider the following problem:∫ba(u1(t)2 + u2(t)2)dt −→ min,x˙1(t) = u1(t) cosx3(t),x˙2(t) = u1(t) sinx3(t),x˙3(t) = u2(t),where the control system serves as model to the kinematics of a car [3, Ex-ample 18, p. 750]. In this case the optimal control problem has three statevariables (n = 3) and two controls (m = 2). With Maple definitions> L:=u[1]^2+u[2]^2; phi:=[u[1]*cos(x[3]),u[1]*sin(x[3]),u[2]];L := u12 + u22ϕ := [u1 cos (x3) , u1 sin (x3) , u2]our procedure Symmetry determines the infinitesimal invariance generators ofthe optimal control problem under consideration:> Symmetry(L, phi, t, [x[1],x[2],x[3]], [u[1],u[2]]);{T = C2, X1 = −C1x2 + C3, X2 = C1x1 + C4, X3 = C1,U1 = 0, U2 = 0, Ψ1 = −C1ψ2, Ψ2 = C1ψ1, Ψ3 = 0}(Ci, i = 1, . . . , 4, are arbitrary constants). The family of Conservation Lawsassociated with the generators just obtained is easily obtained through ourprocedure Noether (the sign of percentage (%) is an operator used in Mapleto represent the result of the previous command):> Noether(L, phi, t, [x[1],x[2],x[3]], [u[1],u[2]], %);(−C1x2(t) + C3)ψ1(t) + (C1x1(t) +C4)ψ2(t) +C1ψ3(t)−„ψ0`u1(t)2 + u2(t)2´+ u1(t) cos (x3(t))ψ1(t) + u1(t) sin (x3(t))ψ2(t) + u2(t)ψ3(t)«C2 = constThe obtained conservation law depends on four parameters. With the substi-tutions C1 = 1 and C2 = C3 = C4 = 0 we obtain the conservation law6 P.D.F. Gouveia, D.F.M. Torres> subs(C[1]=1,C[2]=0,C[3]=0,C[4]=0, %);−x2(t)ψ1(t) + x1(t)ψ2(t) + ψ3(t) = constwhich corresponds to the symmetry group of planar (orientation-preserving)isometries given in [3, Example 18, p. 750].We recall that any problem of the calculus of variations can always berewritten as an optimal control problem, so we can also apply our Mapleprocedures Symmetry and Noether to obtain variational symmetries andconservation laws in the classical context of the calculus of variations.AcknowledgmentsThis work was partially supported by the PhD grant PRODEP III/5.3/2003of Paulo Gouveia, and by FCT via funds from the R&D unit CEOC.References[1] D.S. Djukic. Noether’s theorem for optimum control systems. Internat. J. Con-trol, 1(18), 667 – 672, 1973.[2] J.W. Grizzle and S.I. Marcus. The structure of nonlinear control systems pos-sessing symmetries. IEEE Trans. Automat. Control, 30(3), 248 – 258, 1985.[3] Ph. Martin, R.M. Murray and P. Rouchon. Flat systems. In: Mathematicalcontrol theory, Part 1, 2 (Trieste, 2001), ICTP Lect. Notes, VIII, 705 – 768(electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002.[4] W. Sarlet and F. Cantrijn. Generalizations of Noether’s theorem in classicalmechanics. SIAM Rev., 23(4), 467 – 494, 1981.[5] D.F.M. Torres. On the Noether theorem for optimal control. European Journalof Control, 8(1), 56 – 63, 2002.[6] D.F.M. Torres. Proper extensions of Noether’s symmetry theorem for nonsmoothextremals of the calculus of variations. Communications on Pure and AppliedAnalysis, 3(3), 491 – 500, 2004.[7] D.F.M. Torres. Quasi-invariant optimal control problems. Portugaliæ Mathe-matica, 61(1), 97 – 114, 2004.

Computation of conservation laws in optimal control

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Computation of conservation laws in optimal control

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