Abstract
A controllable drift-free system on the Lie group is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on are studied, where is the minus Lie-Poisson 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 ball-plate problem (see [2]), for the dynamics of the car with two trailers (see [3]), for the Maxwell-Bloch 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 Lie-Poisson dynamics, considering and group multiplication in defined by for any , .
The underwater vehicle dynamics has Lie-Poisson 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 Lie-Poisson 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 drift-free 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, drift-free 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 Chow’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 ’s are solutions of
Proof. Let us apply Krishnaprasad’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 Lie-Poisson structure generated by the following matrix: Then the optimal controls are given by where ’s are solutions of the reduced Hamilton’s equations given by which are nothing else other than the required equations (7).
Proposition 3. Dynamics (7) has the following Hamilton-Poisson realization: where
Proof. Indeed, it is not hard to see that dynamics (7) can be put in the following equivalent form: as required.
Via Bermejo-Feiren’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 Arnold’s method (see [10]). Let given by Follwing Arnold’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 Arnold’s method (see [10]). Let given by Follwing Arnold’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 Arnold’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)
Kahan’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. Kahan’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 Lie-Trotter 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 Lie-Trotter integrator is given bythat is,
Now, using MATHEMATICA we obtain the following properties.
Proposition 10. The Lie-Trotter 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 and is the Kirilov-Kostant-Souriau 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 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 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 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.