A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.
1. Introduction
The optimal control problems arise naturally in a lot of mathematics, engineering, or economics areas. An important class is the control problems on matrix Lie groups due to the possibility to study them from Poisson geometry point of view. Many dynamical systems have as configuration space a Lie group, such as SO(3) for the spacecraft dynamics (see [1]), R2×SO(3) for the ball-plate problem (see [2]), G5 for the dynamics of the car with two trailers (see [3]), SL(2,R) for the Maxwell-Bloch system (see [4]), or SE(2,R)×SO(2) for the dynamics of a rolling penny (see [5]).
In [6] it was shown that the dynamics of the underwater vehicle problem can be viewed as Lie-Poisson dynamics, considering G=SO(3)×R3×R3 and group multiplication in G defined by (1)R1,a1,b1·R2,a2,b2=R1R2,R1a2+a1,R1b2+b1,for any Ri,ai,bi∈G, i=1,2.
The underwater vehicle dynamics has Lie-Poisson form on g∗, the dual of the Lie algebra of G.
The goal of our paper is to study an optimal control problem on the Lie group G and to point out some of its dynamical and geometrical properties.
2. The Lie Group <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M21"><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mn>3</mml:mn><mml:mo mathvariant="bold">)</mml:mo><mml:mo mathvariant="bold">×</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo mathvariant="bold">×</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and Its Lie-Poisson Structure
Let us consider the Lie group G=SO(3)×R3×R3, its Lie algebra g, and the dual of the Lie algebra g∗.
A basis of the Lie algebra is given by the following matrices: (2)A1=000000-1001000000,A2=00100000-10000000,A3=0-100100000000000,A4=0001000000000000,A5=0000000100000000,A6=0000000000010000.
The Lie algebra structure of g is given by Table 1, where {A1,A2,A3,A4,A5,A6,A7,A8,A9} is the canonical basis of the Lie algebra g.
[·,·]
A1
A2
A3
A4
A5
A6
A7
A8
A9
A1
0
A3
-A2
0
A6
-A5
0
A9
-A8
A2
-A3
0
A1
-A6
0
A4
-A9
0
A7
A3
A2
-A1
0
A5
-A4
0
A8
-A7
0
A4
0
A6
-A5
0
0
0
0
0
0
A5
-A6
0
A4
0
0
0
0
0
0
A6
A5
-A4
0
0
0
0
0
0
0
A7
A5
-A4
0
0
0
0
0
0
0
A8
A5
-A4
0
0
0
0
0
0
0
A9
A5
-A4
0
0
0
0
0
0
0
Now, a general left invariant drift-free control system on G with fewer controls than state variables can be written in the following form: (3)X.=X∑i=1mAiui,where X∈G, ui, i=1,2,…,m, are the controls, and m<9.
In all the following we will concentrate on the following left invariant, drift-free control system on G with 3 controls:(4)X.=XA1u1+A2u2+A3u3+A5u5+A7u7.
Then, we have the following.
Proposition 1.
System (4) is controllable.
Proof.
Since the span of the set of Lie brackets generated by A1,A2,A3,A5,A7 coincides with g, the proposition is a consequence of Chow’s Theorem [7].
3. An Optimal Control Problem for System (<xref ref-type="disp-formula" rid="EEq2.1">4</xref>)
Let J be the cost function given by (5)Ju1,u2,u3,u5,u7=12∫0tfc1u12t+c2u22t+c3u32t+c5u52t+c7u72tdtc1>0,c2>0,c3>0,c5>0,c7>0.
Then we have the following.
Proposition 2.
The controls that minimize J and steer system (4) from X=X0 at t=0 to X=Xf at t=tf are given by (6)u1=1c1x1,u2=1c2x2,u3=1c3x3,u5=1c5x5,u7=1c7x7,where xi’s are solutions of(7)x˙1=1c3-1c2x2x3-1c5x5x6,x˙2=1c1-1c3x1x3+1c7x7x9,x˙3=1c2-1c1x1x2+1c5x4x5-1c7x7x8,x˙4=-1c2x2x6+1c3x3x5,x˙5=1c1x1x6-1c3x3x4,x˙6=-1c1x1x5+1c2x2x4,x˙7=-1c2x2x9+1c3x3x8,x˙8=1c1x1x9-1c3x3x7,x˙9=1c1x1x8+1c2x2x7.
Proof.
Let us apply Krishnaprasad’s theorem (see [8]). It follows that the optimal Hamiltonian is given by (8)H(x1,x2,x3,x5,x7)=12x12c1+x22c2+x32c3+x52c5+x72c7.It is in fact the controlled Hamiltonian H¯ given by (9)H¯x1,x2,x3,x5,x7=x1u1+x2u2+x3u3+x5u5+x7u7-12c1u12+c2u22+c3u32+c5u52+c7u72,which is reduced to g∗ via Poisson reduction. Here g-∗ is g∗≃R9 together with the minus Lie-Poisson structure generated by the following matrix: (10)Π-=0-x3x20-x6x50-x9x8x30-x1x60-x4x90-x7-x2x10-x5x40-x8x700-x6x5000000x60-x4000000-x5x400000000-x9x8000000x90-x7000000-x8x70000000.Then the optimal controls are given by (11)u1=1c1x1,u2=1c2x2,u3=1c3x3,u5=1c5x5,u7=1c7x7,where xi’s are solutions of the reduced Hamilton’s equations given by (12)x˙1,x˙2,x˙3,x˙4,x˙5,x˙6,x˙7,x˙8,x˙9t=Π-·∇H,which are nothing else other than the required equations (7).
Proposition 3.
Dynamics (7) has the following Hamilton-Poisson realization: (13)R9,Π-,H,where (14)Π-=0-x3x20-x6x50-x9x8x30-x1x60-x4x90-x7-x2x10-x5x40-x8x700-x6x5000000x60-x4000000-x5x400000000-x9x8000000x90-x7000000-x8x70000000,H(x1,x2,x3,x5,x7)=12x12c1+x22c2+x32c3+x52c5+x72c7.
Proof.
Indeed, it is not hard to see that dynamics (7) can be put in the following equivalent form: (15)x˙1,x˙2,x˙3,x˙4,x˙5,x˙6t=Π-·∇H,as required.
Via Bermejo-Feiren’s technique [9] we are led immediately to the following.
Proposition 4.
The functions C1,C2, and C3, given by (16)C1=12x42+x52+x62,C2=12x72+x82+x92,C3=x1x4+x2x5+x3x6,are Casimirs of our configuration.
The goal of our paper is to study some geometrical and dynamical properties for system (7).
4. Stability
Let us consider now c1=c2=c3=c5=c7=1. Using MATHEMATICA, we can see that the equilibrium states of our dynamics (7) are(17)e1MNPQ=(0,0,0,M,0,N,0,P,Q),M,N,P,Q∈R,e2MNP=0,0,M,0,0,N,0,0,P,M,N,P∈R,e3MPQ=0,0,0,0,M,0,0,P,Q,M,P,Q∈R,e4MNP=0,M,0,0,N,0,0,P,0,M,N,P∈R,e5MNP=M,N,P,0,0,0,0,0,0,M,N,P∈R,e6MNP=M,0,0,N,0,0,P,0,0,M,N,P∈R,e7MNP=0,M,0,0,N,0,0,P,0,M,N,P∈R,e8MNP=0,0,0,M,0,N,P,0,0,M,N,P∈R,e9MNP=M,0,N,P,0,NPM,0,0,0,M,N,P∈R,e10MNP=0,M,N,0,0,0,0,P,NPM,M,N,P∈R,e11MNP=0,0,0,NPM,M,0,N,P,0,M,N,P∈R,e12MNP=M,N,0,NPM,P,0,-NPM,-P,0,M,N,P∈R,e13MNP=M,N,0,NPM,P,0,-NPM,P,0,M,N,P∈R.
Let us start to study the nonlinear stability. We have the following results.
Proposition 5.
The equilibrium states e1MNPQ, M,N,P∈R, are nonlinearlly stable for any M,N,P,Q∈R∗.
Proof.
We will make the proof using Arnold’s method (see [10]). Let Fλ,μ,ν∈C∞R9,R given by (18)Fλ,μ,ν=H+λ2C1+νC2+ν2C3=12x12+x22+x32+x52+x72+λ2(x42+x52+x62)+μ2(x72+x82+x92)+νx7x4+x8x5+x9x6.Follwing Arnold’s method, we have successively(19)∇Fλ,μ,νe1MNPQ=0iffλ=μ=ν=0.
If we consider the space (20)X=Ker(dC1)e1MNPQ∩Ker(dC2)e1MNPQ∩Ker(dC3)e1MNPQ=span100000000,010000000,001000000,then (21)vt∇2F0,0,0e1MNPQv=a2+b2+c2,for any v∈X; that is, v=abc000000t, so (22)∇2F0,0,0e1MNPQ∣X×Xis positive defined.
Proposition 6.
The equilibrium states e3M,P,Q, M,P,Q∈R, are nonlinearlly stable if and only if M,N∈R∗,Q=0.
Proof.
We will make the proof using Arnold’s method (see [10]). Let Fλ,μ,ν∈C∞R9,R given by (23)Fλ,μ,ν=12C2+λ2H+μ2C1+νC3=12(x72+x82+x92)+λ2x12+x22+x32+x52+x72+μ2(x42+x52+x62)+νx7x4+x8x5+x9x6.Follwing Arnold’s method, we have successively(24)∇Fλ,μ,νe3MPQ=0iffλ+μ=P2M2,ν=-PM,Q=0.
If we consider the space (25)Y=KerdHe3MN0∩KerdC1e3MN0∩Ker(dC3)e3MNQ=span100000000,010000000,001000000,000100000,000001000,000000100,000000010,000000001,then (26)vt∇2Fλ,μ,νe3MN0v=λa2+λb2+λc2+μd2+μe2+(1+λ)f2+g2+h2+2νfd+2νhe,for any v∈X; that is, v=abcd0efght, so (27)∇2Fλ,μ,νe3MN0∣Y×Yis positive defined; so, the equilibrium states e3MN0 are nonlinearly stables.
Using similar arguments, we are led to the following result.
Proposition 7.
The equilibrium states e5M,N,P, M,N,P∈R, are nonlinearlly stable for any M,N,P∈R∗.
Proof.
We consider in this case the function Fλ,μ,ν∈C∞R9,R given by (28)Fλ,μ,ν=12C1+λ2H+μ2C2+νC3=12(x42+x52+x62)+λ2x12+x22+x32+x52+x72+μ2(x72+x82+x92)+ν2x7x4+x8x5+x9x6.Now, we have successively(29)∇F(e5M,N,P)=0iffλ=0;
if μ>0 and ν=0, then ∇2F(e5M,N,P)∣X×X>0, where (30)X=span001000000,000100000,000001000,000000100,000000010,000000001,so, via Arnold’s method (see [10]), the equilibrium states e5M,N,P are nonlinearly stables.
Remark 8.
Unfortunately, for the rest of equilibrium states the energy methods do not work. The stability problem must be approached with other tehniques, and it is still open.
5. Numerical Integration of Dynamics (<xref ref-type="disp-formula" rid="EEq3.1">7</xref>)
Kahan’s integrator (see [11]) for (7) can be written in the following form:(31)x1n+1-x1n=h21c3-1c2x3n+1x2n+x2n+1x3n-h2c5x5n+1x6n+x6n+1x5n,x2n+1-x2n=h21c1-1c3x1n+1x3n+x3n+1x1n+h2c7x7n+1x9n+x9n+1x7n,x3n+1-x3n=h21c2-1c1x1n+1x2n+x2n+1x1n+h2c5x4n+1x5n+x5n+1x4n-h2c7x7n+1x8n+x8n+1x7n,x4n+1-x4n=-h2c2x2n+1x6n+x6n+1x2n+h2c3x3n+1x5n+x5n+1x3n,x5n+1-x5n=h2c1x1n+1x6n+x6n+1x1n-h2c3x4n+1x3n+x3n+1x4n,x6n+1-x6n=-h2c1x1n+1x5n+x5n+1x1n+h2c2x2n+1x4n+x4n+1x2n,x7n+1-x7n=-h2c2x2n+1x9n+x9n+1x2n+h2c3x3n+1x8n+x8n+1x3n,x8n+1-x8n=h2c1x1n+1x9n+x9n+1x1n-h2c3x3n+1x7n+x7n+1x3n,x9n+1-x9n=h2c1x1n+1x8n+x8n+1x1n+h2c2x2n+1x7n+x7n+1x2n.
A long but straightforward computation or using eventually MATHEMATICA leads us to the following.
Proposition 9.
Kahan’s integrator (31) has the following properties:
It is not Poisson preserving.
It does not preserve the Casimirs C1,C2,C3 of our Poisson configuration (g,Π-).
It does not preserve the Hamiltonian H of our system (7).
We will discuss now the numerical integration of dynamics (7) via the Lie-Trotter integrator (see [12]). In the beginning, let us observe that the Hamiltonian vector field XH splits as follows: (32)XH=XH1+XH2+XH3+XH5+XH7,where (33)H1=12c1x12,H2=12c2x23,H3=12c3x32,H5=12c5x52,H7=12c7x72.
Their corresponding integral curves are, respectively, given by (34)x1(t)x2(t)x3(t)x4(t)x5(t)x6(t)x7(t)x8(t)x9(t)=Aix1(0)x2(0)x3(0)x4(0)x5(0)x6(0)x7(0)x8(0)x9(0),i=1,2,3,5,7,where(35)A1=1000000000cosatsinat0000000-sinatcosat0000000001000000000cosatsinat0000000-sinatcosat0000000001000000000cosatsinat0000000-sinatcosat,1111111111111111111h11111j11111111111111111a=x1(0)c1,A2=cosbt0-sinbt000000010000000sinbt0cosbt000000000cosbt0-sinbt000000010000000sinbt0cosbt000000000cosbt0-sinbt000000000000000sinbt0cosbt,111111111111111j111111111111111111111111111b=x2(0)c2,A3=cosctsinct0000000-sinctcosct0000000001000000000cosctsinct0000000-sinctcosct0000000001000000000cosctsinct0000000-sinctcosct0000000001,1111111111111111g1111111111111111111111111c=x3(0)c3,A4=10000-dt000010000000001dt00000000100000000010000000001000000000100000000010000000001,d=x5(0)c5,A5=10000000001000000et001000-et00000100000000010000000001000000000100000000010000000001,e=x7(0)c7.
Then the Lie-Trotter integrator is given by(36)x1n+1x2n+1x3n+1x4n+1x5n+1x6n+1=A1A2A3A4A5x1nx2nx3nx4nx5nx6n;that is,(37)x1n+1=x1ncosbtcosct-dtx6ncosbtcosct+x3n(-sinbt+cosbtsinct)+dtx4n(-sinbt+cosbtsinct)-etx8n-sinbt+cosbtsinct,x2n+1=x1ncosctsinatsinbt-cosatsinct-dtx6ncosctsinatsinbt-cosatsinct+x3n(cosatcosct+cosbtsinat+sinatsinbtsinct)+dtx4ncosatcosct+cosbtsinat+sinatsinbtsinct-etx8ncosatcosct+cosbtsinat+sinatsinbtsinct,x3n+1=x1n(cosatcosctsinbt+sinatsinct)-dtx6n(cosatcosctsinbt+sinatsinct)+x3n(cosatcosbt-cosctsinat+cosatsinbtsinct)+dtx4ncosatcosbt-cosctsinat+cosatsinbtsinct-etx8ncosatcosbt-cosctsinat+cosatsinbtsinct,x4n+1=x4ncosbtcosct-x6nsinbt+x5ncosbtsinct,x5n+1=x6ncosbtsinat+x4n(cosctsinatsinbt-cosatsinct)+x5ncosatcosct+sinatsinbtsinct,x6n+1=x6ncosatcosbt+x4n(cosatcosctsinbt+sinatsinct)+x5n-cosctsinat+cosatsinbtsinct,x7n+1=x7ncosbtcosct-x9nsinbt+x8ncosbtsinct,x8n+1=x9ncosbtsinat+x7n(cosctsinatsinbt-cosatsinct)+x8ncosatcosct+sinatsinbtsinct,x9n+1=x9ncosatcosbt+x7n(cosatcosctsinbt+sinatsinct)+x8n-cosctsinat+cosatsinbtsinct.
Now, using MATHEMATICA we obtain the following properties.
Proposition 10.
The Lie-Trotter integrator (37) has the following properties:
It preserves the Poisson structure Π-.
It preserves the Casimirs C1,C2,C3 of our Poisson configuration (g,Π-).
It does not preserve the Hamiltonian H of our system (7).
Its restriction to the coadjoint orbit (Ok,ωk), where (38)Ok=x1,x2,x3,x4,x5,x6,x7,x8,x9∈R9/x42+x52+x62=const,x1x4+x2x5+x3x6=const,x72+x82+x92=const
and ωk is the Kirilov-Kostant-Souriau symplectic structure on Ok gives rise to a symplectic integrator.
6. Conclusion
The paper presents a left invariant controllable system on the Lie group G=SO(3)×R3×R3; this arises naturally from the study of the buoyancy’s dynamics for which the Lie group represents the phase space [6], as well as for the charged top dynamics (see [13]). Similar problems have been studied for a lot of Lie groups: for SE(2,R)×SO(2) in [5], for SE(1,1) in [14], for SO(4) in [15], for SE(3,R) in [16], for a specific Lie group G4 in [17], and so on. For all these examples, the Poisson geometry approach gives the geometric frame of the study and provides specific methods to obtain stability results, numerical integration using Poisson or non-Poisson integrators, or the existence of different type of periodic orbits (see [18, 19]).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
LeonardN. E.JurdjevicV.The geometry of the plate-ball problemPetrişorC.Some new remarks about the dynamics of an automobile with two trailersDavidD.HolmD. D.Multiple Lie-Poisson structures, reductions, and geometric phases for the Maxwell-Bloch travelling wave equationsAronA.PopC.PutaM.An optimal control problem on the lie group SE(2,R)×SO(2)LeonardN. E.Stability of a bottom-heavy underwater vehicleChowW.-L.Uber systeme von linearen partiellen differentiagleichungen erster ordnungKrishnaprasadP. S.Optimal control and poisson reduction199393-87College Park, Md, USAInstitute for System Research, University of MarylandHernandez-BermejoB.FairenV.Simple evaluation of Casimir invariants in finite-dimensional Poisson systemsArnoldV.Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluidKahanW.TrotterH. F.On the product of semi-groups of operatorsBirteaP.PutaM.TudoranR.VoicuC.Controllability problems in the charged dynamicsBiggsR.RemsingC. C.Cost-extended control systems on Lie groupsAronA.MosI.CsakyA.PutaM.An optimal control problem on the Lie group SO(4)CraioveanuM.PopC.AronA.PetrisorC.An optimal control problem on the special Euclidean group SE(3,R)Proceedings of the International Conference of Differential Geometry and Dynamical Systems (DGDS '09)October 2009Bucharest, Romania6878LâzureanuC.BînzarT.On a Hamiltonian version of controls dynamic for a drift-free left invariant control system G_{4}AgrachevA. A.SachkovY. L.HirschM. W.SmaleS.