MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/652819 652819 Research Article A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3 Pop Camelia Pandolfi Anna Mathematics Department “Politehnica” University of Timisoara 300006 Timisoara Romania uvt.ro 2015 3132015 2015 14 12 2014 25 03 2015 3132015 2015 Copyright © 2015 Camelia Pop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ), R2×SO(3) for the ball-plate problem (see ), G5 for the dynamics of the car with two trailers (see ), SL(2,R) for the Maxwell-Bloch system (see ), or SE(2,R)×SO(2) for the dynamics of a rolling penny (see ).

In  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,biG, 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.

[ · , · ] A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9
A 1 0 A 3 - A 2 0 A 6 - A 5 0 A 9 - A 8
A 2 - A 3 0 A 1 - A 6 0 A 4 - A 9 0 A 7
A 3 A 2 - A 1 0 A 5 - A 4 0 A 8 - A 7 0
A 4 0 A 6 - A 5 0 0 0 0 0 0
A 5 - A 6 0 A 4 0 0 0 0 0 0
A 6 A 5 - A 4 0 0 0 0 0 0 0
A 7 A 5 - A 4 0 0 0 0 0 0 0
A 8 A 5 - A 4 0 0 0 0 0 0 0
A 9 A 5 - A 4 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.=Xi=1mAiui,where XG, 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 .

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=120tfc1u12t+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 ). 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 gR9 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  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,QR,e2MNP=0,0,M,0,0,N,0,0,P,M,N,PR,e3MPQ=0,0,0,0,M,0,0,P,Q,M,P,QR,e4MNP=0,M,0,0,N,0,0,P,0,M,N,PR,e5MNP=M,N,P,0,0,0,0,0,0,M,N,PR,e6MNP=M,0,0,N,0,0,P,0,0,M,N,PR,e7MNP=0,M,0,0,N,0,0,P,0,M,N,PR,e8MNP=0,0,0,M,0,N,P,0,0,M,N,PR,e9MNP=M,0,N,P,0,NPM,0,0,0,M,N,PR,e10MNP=0,M,N,0,0,0,0,P,NPM,M,N,PR,e11MNP=0,0,0,NPM,M,0,N,P,0,M,N,PR,e12MNP=M,N,0,NPM,P,0,-NPM,-P,0,M,N,PR,e13MNP=M,N,0,NPM,P,0,-NPM,P,0,M,N,PR.

Let us start to study the nonlinear stability. We have the following results.

Proposition 5.

The equilibrium states e1MNPQ, M,N,PR, are nonlinearlly stable for any M,N,P,QR.

Proof.

We will make the proof using Arnold’s method (see ). Let Fλ,μ,νCR9,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=0iff  λ=μ=ν=0.

If we consider the space (20)X=Ker(dC1)e1MNPQKer(dC2)e1MNPQKer(dC3)e1MNPQ=span100000000,010000000,001000000,then (21)vt2F0,0,0e1MNPQv=a2+b2+c2,for any vX; that is, v=abc000000t, so (22)2F0,0,0e1MNPQX×Xis positive defined.

Proposition 6.

The equilibrium states e3M,P,Q, M,P,QR, are nonlinearlly stable if and only if M,NR,Q=0.

Proof.

We will make the proof using Arnold’s method (see ). Let Fλ,μ,νCR9,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=0iff  λ+μ=P2M2,ν=-PM,Q=0.

If we consider the space (25)Y=KerdHe3MN0KerdC1e3MN0Ker(dC3)e3MNQ=span100000000,010000000,001000000,000100000,000001000,000000100,000000010,000000001,then (26)vt2Fλ,μ,νe3MN0v=λa2+λb2+λc2+μd2+μe2+(1+λ)f2+g2+h2+2νfd+2νhe,for any vX; that is, v=abcd0efght, so (27)2Fλ,μ,νe3MN0Y×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,PR, are nonlinearlly stable for any M,N,PR.

Proof.

We consider in this case the function Fλ,μ,νCR9,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)=0iff  λ=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 ), 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 ) 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 ). 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,x9R9/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 , as well as for the charged top dynamics (see ). Similar problems have been studied for a lot of Lie groups: for SE(2,R)×SO(2) in , for SE(1,1) in , for SO(4) in , for SE(3,R) in , for a specific Lie group G4 in , 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.

Leonard N. E. Averaging and motion control systems on Lie groups [Ph.D. thesis] 1994 College Park, Md USA University of Maryland Jurdjevic V. The geometry of the plate-ball problem Archive for Rational Mechanics and Analysis 1993 124 4 305 328 10.1007/BF00375605 2-s2.0-21344497588 MR1240578 Petrişor C. Some new remarks about the dynamics of an automobile with two trailers Journal of Applied Mathematics 2014 2014 6 809408 10.1155/2014/809408 MR3224377 David D. Holm D. D. Multiple Lie-Poisson structures, reductions, and geometric phases for the Maxwell-Bloch travelling wave equations Journal of Nonlinear Science 1992 2 2 241 262 2-s2.0-0000633938 10.1007/BF02429857 MR1169593 Aron A. Pop C. Puta M. An optimal control problem on the lie group SE(2,R)×SO(2) Boletin de la Sociedad Mexicana 2009 15 2 129 140 MR2757939 Leonard N. E. Stability of a bottom-heavy underwater vehicle Automatica 1997 33 3 331 346 10.1016/s0005-1098(96)00176-8 MR1442552 Chow W.-L. Uber systeme von linearen partiellen differentiagleichungen erster ordnung Mathematische Annalen 1940 117 1 98 105 10.1007/bf01450011 2-s2.0-34250951325 Krishnaprasad P. S. Optimal control and poisson reduction 1993 93-87 College Park, Md, USA Institute for System Research, University of Maryland Hernandez-Bermejo B. Fairen V. Simple evaluation of Casimir invariants in finite-dimensional Poisson systems Physics Letters. A 1998 241 3 148 154 10.1016/s0375-9601(98)00125-x MR1613891 Arnold V. Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid Doklady Natsionalnoy Akademii Nauk 1965 162 5 773 777 Kahan W. Unconventional Numerical Methods for Trajectory Calculation 1993 Lecture Notes Trotter H. F. On the product of semi-groups of operators Proceedings of the American Mathematical Society 1959 10 545 551 10.1090/S0002-9939-1959-0108732-6 MR0108732 Birtea P. Puta M. Tudoran R. Voicu C. Controllability problems in the charged dynamics Systems and Control Letters 2007 56 7-8 512 515 10.1016/j.sysconle.2007.02.001 MR2332002 2-s2.0-34248999399 Biggs R. Remsing C. C. Cost-extended control systems on Lie groups Mediterranean Journal of Mathematics 2014 11 1 193 215 2-s2.0-84895909904 MR3160621 10.1007/s00009-013-0355-0 Aron A. Mos I. Csaky A. Puta M. An optimal control problem on the Lie group SO(4) International Journal of Geometric Methods in Modern Physics 2008 5 3 319 327 10.1142/s0219887808002795 MR2422030 2-s2.0-44949202572 Craioveanu M. Pop C. Aron A. Petrisor C. 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 2009 Bucharest, Romania 68 78 Lâzureanu C. Bînzar T. On a Hamiltonian version of controls dynamic for a drift-free left invariant control system G4 International Journal of Geometric Methods in Modern Physics 2012 9 8 1250065 10.1142/S021988781250065X Agrachev A. A. Sachkov Y. L. Control Theory from the Geometric Viewpoint 2004 87 Berlin, Germany Springer Encyclopaedia of Mathematical Sciences: Control Theory and Optimization II MR2062547 10.1007/978-3-662-06404-7 Hirsch M. W. Smale S. Differential Equations, Dynamical Systems and Linear Algebra 1974 New York, NY, USA Academic Press MR0486784