Research Article  Open Access
Camelia Pop, "A DriftFree Left Invariant Control System on the Lie Group ", Mathematical Problems in Engineering, vol. 2015, Article ID 652819, 9 pages, 2015. https://doi.org/10.1155/2015/652819
A DriftFree Left Invariant Control System on the Lie Group
Abstract
A controllable driftfree system on the Lie group is considered. The dynamics and geometrical properties of the corresponding reduced Hamiltonâ€™s equations on are studied, where is the minus LiePoisson structure on the dual space of the Lie algebra of . 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 for the spacecraft dynamics (see [1]), for the ballplate problem (see [2]), for the dynamics of the car with two trailers (see [3]), for the MaxwellBloch system (see [4]), or 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 LiePoisson dynamics, considering and group multiplication in defined by for any , .
The underwater vehicle dynamics has LiePoisson form on , the dual of the Lie algebra of .
The goal of our paper is to study an optimal control problem on the Lie group and to point out some of its dynamical and geometrical properties.
2. The Lie Group and Its LiePoisson Structure
Let us consider the Lie group , its Lie algebra , and the dual of the Lie algebra .
A basis of the Lie algebra is given by the following matrices:
The Lie algebra structure of is given by Table 1, where is the canonical basis of the Lie algebra .

Now, a general left invariant driftfree control system on with fewer controls than state variables can be written in the following form: where , , , are the controls, and .
In all the following we will concentrate on the following left invariant, driftfree control system on with 3 controls:
Then, we have the following.
Proposition 1. System (4) is controllable.
Proof. Since the span of the set of Lie brackets generated by coincides with , the proposition is a consequence of Chowâ€™s Theorem [7].
3. An Optimal Control Problem for System (4)
Let be the cost function given by
Then we have the following.
Proposition 2. The controls that minimize and steer system (4) from at to at are given by where â€™s are solutions of
Proof. Let us apply Krishnaprasadâ€™s theorem (see [8]). It follows that the optimal Hamiltonian is given by It is in fact the controlled Hamiltonian given by which is reduced to via Poisson reduction. Here is together with the minus LiePoisson structure generated by the following matrix: Then the optimal controls are given by where â€™s are solutions of the reduced Hamiltonâ€™s equations given by which are nothing else other than the required equations (7).
Proposition 3. Dynamics (7) has the following HamiltonPoisson realization: where
Proof. Indeed, it is not hard to see that dynamics (7) can be put in the following equivalent form: as required.
Via BermejoFeirenâ€™s technique [9] we are led immediately to the following.
Proposition 4. The functions , and , given by 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 . Using MATHEMATICA, we can see that the equilibrium states of our dynamics (7) are
Let us start to study the nonlinear stability. We have the following results.
Proposition 5. The equilibrium states , , are nonlinearlly stable for any .
Proof. We will make the proof using Arnoldâ€™s method (see [10]). Let given by Follwing Arnoldâ€™s method, we have successivelyIf we consider the space then for any ; that is, , so is positive defined.
Proposition 6. The equilibrium states , , are nonlinearlly stable if and only if .
Proof. We will make the proof using Arnoldâ€™s method (see [10]). Let given by Follwing Arnoldâ€™s method, we have successivelyIf we consider the space then for any ; that is, , so is positive defined; so, the equilibrium states are nonlinearly stables.
Using similar arguments, we are led to the following result.
Proposition 7. The equilibrium states , , are nonlinearlly stable for any .
Proof. We consider in this case the function given by Now, we have successivelyif and , then , where so, via Arnoldâ€™s method (see [10]), the equilibrium states 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 (7)
Kahanâ€™s integrator (see [11]) for (7) can be written in the following form:
A long but straightforward computation or using eventually MATHEMATICA leads us to the following.
Proposition 9. Kahanâ€™s integrator (31) has the following properties: (i)It is not Poisson preserving.(ii)It does not preserve the Casimirs of our Poisson configuration .(iii)It does not preserve the Hamiltonian of our system (7).
We will discuss now the numerical integration of dynamics (7) via the LieTrotter integrator (see [12]). In the beginning, let us observe that the Hamiltonian vector field splits as follows: where
Their corresponding integral curves are, respectively, given by where
Then the LieTrotter integrator is given bythat is,
Now, using MATHEMATICA we obtain the following properties.
Proposition 10. The LieTrotter integrator (37) has the following properties: (i)It preserves the Poisson structure .(ii)It preserves the Casimirs of our Poisson configuration .(iii)It does not preserve the Hamiltonian of our system (7).(iv)Its restriction to the coadjoint orbit , where â€‰and is the KirilovKostantSouriau symplectic structure on gives rise to a symplectic integrator.
6. Conclusion
The paper presents a left invariant controllable system on the Lie group ; this arises naturally from the study of the buoyancyâ€™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 in [5], for in [14], for in [15], for in [16], for a specific Lie group 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 nonPoisson 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.
References
 N. E. Leonard, Averaging and motion control systems on Lie groups [Ph.D. thesis], University of Maryland, College Park, Md USA, 1994.
 V. Jurdjevic, â€śThe geometry of the plateball problem,â€ť Archive for Rational Mechanics and Analysis, vol. 124, no. 4, pp. 305â€“328, 1993. View at: Publisher Site  Google Scholar  MathSciNet
 C. Petrişor, â€śSome new remarks about the dynamics of an automobile with two trailers,â€ť Journal of Applied Mathematics, vol. 2014, Article ID 809408, 6 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 D. David and D. D. Holm, â€śMultiple LiePoisson structures, reductions, and geometric phases for the MaxwellBloch travelling wave equations,â€ť Journal of Nonlinear Science, vol. 2, no. 2, pp. 241â€“262, 1992. View at: Publisher Site  Google Scholar  MathSciNet
 A. Aron, C. Pop, and M. Puta, â€śAn optimal control problem on the lie group $SE(2,\mathbb{R})\times SO(2)$,â€ť Boletin de la Sociedad Mexicana, vol. 15, no. 2, pp. 129â€“140, 2009. View at: Google Scholar  MathSciNet
 N. E. Leonard, â€śStability of a bottomheavy underwater vehicle,â€ť Automatica, vol. 33, no. 3, pp. 331â€“346, 1997. View at: Publisher Site  Google Scholar  MathSciNet
 W.L. Chow, â€śUber systeme von linearen partiellen differentiagleichungen erster ordnung,â€ť Mathematische Annalen, vol. 117, no. 1, pp. 98â€“105, 1940. View at: Publisher Site  Google Scholar
 P. S. Krishnaprasad, â€śOptimal control and poisson reduction,â€ť Tech. Rep. 9387, Institute for System Research, University of Maryland, College Park, Md, USA, 1993. View at: Google Scholar
 B. HernandezBermejo and V. Fairen, â€śSimple evaluation of Casimir invariants in finitedimensional Poisson systems,â€ť Physics Letters. A, vol. 241, no. 3, pp. 148â€“154, 1998. View at: Publisher Site  Google Scholar  MathSciNet
 V. Arnold, â€śConditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid,â€ť Doklady Natsionalnoy Akademii Nauk, vol. 162, no. 5, pp. 773â€“777, 1965. View at: Google Scholar
 W. Kahan, Unconventional Numerical Methods for Trajectory Calculation, Lecture Notes, 1993.
 H. F. Trotter, â€śOn the product of semigroups of operators,â€ť Proceedings of the American Mathematical Society, vol. 10, pp. 545â€“551, 1959. View at: Publisher Site  Google Scholar  MathSciNet
 P. Birtea, M. Puta, R. Tudoran, and C. Voicu, â€śControllability problems in the charged dynamics,â€ť Systems and Control Letters, vol. 56, no. 78, pp. 512â€“515, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 R. Biggs and C. C. Remsing, â€śCostextended control systems on Lie groups,â€ť Mediterranean Journal of Mathematics, vol. 11, no. 1, pp. 193â€“215, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 A. Aron, I. Mos, A. Csaky, and M. Puta, â€śAn optimal control problem on the Lie group SO(4),â€ť International Journal of Geometric Methods in Modern Physics, vol. 5, no. 3, pp. 319â€“327, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 M. Craioveanu, C. Pop, A. Aron, and C. Petrisor, â€śAn optimal control problem on the special Euclidean group SE(3,R),â€ť in Proceedings of the International Conference of Differential Geometry and Dynamical Systems (DGDS '09), pp. 68â€“78, Bucharest, Romania, October 2009. View at: Google Scholar
 C. Lâzureanu and T. Bînzar, â€śOn a Hamiltonian version of controls dynamic for a driftfree left invariant control system G_{4},â€ť International Journal of Geometric Methods in Modern Physics, vol. 9, no. 8, Article ID 1250065, 2012. View at: Publisher Site  Google Scholar
 A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences: Control Theory and Optimization II, Springer, Berlin, Germany, 2004. View at: Publisher Site  MathSciNet
 M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, NY, USA, 1974. View at: MathSciNet
Copyright
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.