* Stability problems and numerical integration on the Lie group SO ( 3 ) × R 3 × R

The optimal control problems on the Lie groups were studied very often in deep connection with mechanical systems. We can nd a large list of such examples, like the dynamics of an underwater vehicle, with SE(3,R) = SO(3) × R3 as space con guration (see [1]), the ball-plate problem, with R2 × SO(3) as space con guration [2], the rolling-penny dynamics having the Lie group SE(2,R) × SO(2) as space con guration [3], the control tower problem from air tra c, modeled on the Special Euclidean Group SE(3), the spacecraft dynamics modeled on the special orthogonal group SO(3), [4], the buoyancy’s dynamics on the Lie group SO(3) × R3 × R3, (see [5] for details), and the list may go on. Similar methods were used in [6-10]. Taking into consideration that in many cases the dynamics can be viewed as a left-invariant, drift-free control system on the considered Lie group, we became interested in the study of such systems. The problem of nding the optimal controls that minimize a quadratic cost function for the general left-invariant drift-free control system . X = X (A1u1 + A2u2 + A3u3 + A5u5 + A7u7) , (1)


Introduction
The optimal control problems on the Lie groups were studied very often in deep connection with mechanical systems. We can nd a large list of such examples, like the dynamics of an underwater vehicle, with SE( , R) = SO( ) × R as space con guration (see [1]), the ball-plate problem, with R × SO( ) as space con guration [2], the rolling-penny dynamics having the Lie group SE( , R) × SO( ) as space con guration [3], the control tower problem from air tra c, modeled on the Special Euclidean Group SE( ), the spacecraft dynamics modeled on the special orthogonal group SO( ), [4], the buoyancy's dynamics on the Lie group SO( ) × R × R , (see [5] for details), and the list may go on.
Taking into consideration that in many cases the dynamics can be viewed as a left-invariant, drift-free control system on the considered Lie group, we became interested in the study of such systems. The problem of nding the optimal controls that minimize a quadratic cost function for the general left-invariant drift-free control system on the Lie group G = SO( ) × R × R , where A i , i = , is the standard basis of the Lie algebra g ∶ Since the span of the set of Lie brackets generated by A , A , A , A , A coincides with g, the system (1) is controllable [11]. Considering now the cost function given by: the controls that minimize J and steer the system (1) from X = X at t = to X = X f at t = t f are given by: where x i 's are solutions of the following nonlinear system: ( The main goal of our paper is to establish some stability results of the equilibrium points were already obtained in [11], but the stability problem for the other equilibrium states remains unsolved. The paper is organized as follows: in the second paragraph we nd an appropriate control function in order to stabilize the equilibrium states e MNP . The third section brie y presents the Optimal Homotopy Asymptotic Method, developed in [12][13][14] and used in the last part in order to obtain the approximate analytic solutions of the controlled system.

Stabilization of e MNP by one linear control
Let us employ the control u ∈ C ∞ (R , R), for the system (2). The controlled system (2)−(3), explicitly given by has e MNP as an equilibrium state.
is the minus Lie-Poisson structure on the dual of the corresponding Lie algebra g * and the Hamiltonian function given by Proof. Indeed, one obtains immediately that and Π is a minus Lie-Poisson structure, see for details [11].

Remark 2.2 ([11]
). The functions C , C , C ∶ R → R given by are the Casimirs of our Poisson con guration.
The goal of this paragraph is to study the spectral and nonlinear stability of the equilibrium state e MNP of the controlled system ( ).
Let A be the matrix of linear part of our controlled system (4), that is At the equilibrium of interest its characteristic polynomial has the following expression: Hence we have ve zero eigenvalues and four purely imaginary eigenvalues. So we can conclude: Proof. For the proof we shall use Arnold's technique. Let us consider the following function The following conditions hold: which is positive de nite under the restriction λ > , and so is positive de nite. Therefore, via Arnold's technique, the equilibrium states e MNP , M, N, P ∈ R * are nonlinear stable, as required.

Basic ideas of the Optimal Homotopy Asymptotic Method
In order to compute analytical approximate solutions for the nonlinear di erential system given by the equations (4) with the boundary conditions we will use the Optimal Homotopy Asymptotic Method (OHAM) [12][13][14].
Let us start with a very short description of this method. The analytical approximate solutions can be obtained for equations of the general form: subject to the initial conditions of the type: x( ) = A, A ∈ R -given real number, where L is a linear operator (which is not unique), N is a nonlinear one and x(t) is the unknown smooth function of the Eq. (7). Following [12][13][14], we construct the homotopy given by: where p ∈ [ , ] is the embedding parameter, H(t, C i ), (H ≠ ) is an auxiliary convergence-control function, depending on the variable t and on the parameters C , C , ..., C s and the function X(t, p) has the expression: The following properties hold: and The governing equations of x (t) and x (t, C i ) can be obtained by equating the coe cients of p and p , respectively: The expression of x (t) can be found by solving the linear equation (13). Also, to compute x (t, C i ) we solve the equation (14), by taking into consideration that the nonlinear operator N presents the general form: where m is a positive integer and h i (t) and g i (t) are known functions depending both on x (t) and N.
Although the equation (14) is a nonhomogeneous linear one, in the most cases its solution can not be found.
In order to compute the function x (t, C i ) we will use the third modi ed version of OHAM (see [14] for details), consisting in the following steps: First we consider one of the following expressions for x (t, C i ): or These expressions of H i (t, h j (t), C j ) contain both linear combinations of the functions h j and the parameters C j , j = , s. The summation limit m is an arbitrary positive integer number. Next, by taking into account the equation (10), for p = , the rst-order analytical approximate solution of the equations (7) - (8) is: Finally, the convergence-control parameters C , C , ..., C s , which determine the rst-order approximate solution (18), can be optimally computed by means of various methods, such as: the least square method, the Galerkin method, the collocation method, the Kantorowich method or the weighted residual method.

De nition 3.1. [15]
We call an -approximate solution of the problem (7) on the domain ( , ∞) a smooth function x(t, C i ) of the form (18) which satis es the following condition: together with the initial condition from Eq. (8), where the residual function R(t, x(t, C i )) is obtained by substituting the Eq. (18) into Eq. (7), i.e.

De nition 3.2 ([15]). We call a week -approximate solution of the problem (7) on the domain ( , ∞) a smooth function x(t, C i ) of the form (18) which satis es the following condition:
together with the initial condition from Eq. (8).

Application of Optimal Homotopy Asymptotic Method for solving the nonlinear di erential system (4)
In order to solve the nonlinear di erential system given by the equations (4), each equation of the system (4) can be written in the form Eq. (7), where we can choose the linear operators as: with K > , K > the unknown parameters at this moment. The corresponding linear equations for initial approximations x i , i = , can be obtained by means of the Eqs. (13), (19) and (6): whose solutions are The corresponding nonlinear operators N [x i (t)], i = , are obtained from the equations (4): such that and therefore, substituting Eqs. (21) into Eqs. (22), we obtain On the other hand, the Eq. (14) becomes: where the linear operators L are given by Eq.
Using now the third-alternative of OHAM and the equations (18), the rst-order approximate solution can be put in the formx where x i (t) and x i (t, C i ) are given by (21)

Numerical examples and discussions
In this section, the accuracy and validity of the OHAM technique is proved using a comparison of our approximate solutions with numerical results obtained via the fourth-order Runge-Kutta method in the following case: we consider the initial value problem given by (4) with initial conditions (6) A i = . , i = , , M = and P = . One can show that these approximate solutions are week -approximate solutions by computing the numerical value of the integral of square residual function (to see the Table 4), i.e.

Conclusion
The paper presents the stabilization of a dynamical system using a linear control function. The Hamilton-Poisson formulation of the obtained system allows to use energy-methods in order to obtain stability results. In the last section the approximate analytic solutions of the considered controlled system (4) are established using the optimal homotopy asymptotic method (OHAM). Numerical simulations via Mathematica 9.0 software and the approximations deviations are presented. The accuracy of our results is pointed out by means of the approximate residual of the solutions. The next step we intend to do is a comparison between the Lie-Trotter integrator (which is a Poisson one, see [11]) and OHAM, regarding the numerical results.