The role of the Schrödinger group in physics was always important and the literature on the subject is immense. The Schrödinger group and the Schrödinger algebra were first introduced by Niederer  and Hagen  as a nonrelativistic limit of the vector-field realization of the conformal algebra. The Schrödinger group could play the space configuration role in some concrete mechanical problems, like other important Lie groups do: SO(3) for aircraft dynamics, SE(3) for underwater dynamics, SO(2)× R2 for the rolling-penny dynamics, R2 × SO(3) for ball-plate problem, and so on. All these dynamics are modeled as invariant optimal control problems on the mentioned Lie groups. The study of the invariant optimal control problem on the Schrödinger Lie group naturally completes the short list from above. More details can be found in  or .
This article is concerned with some important comments on the behavior of a dynamical system arising from an optimal control problem on the Schrödinger Lie group presented in . More specifically, we are interested in improving the stability results, proving that some equilibrium states are nonlinearly stable and finding the periodic orbits around them, and finally, we propose an alternative method for finding analytic solution of the system.
The system, which can be put into the equivalent form: (1)
k ∈ ℝ, has the Hamilton-Poisson realization (see ):
is the minus Lie-Poisson strcture and
is the Hamiltonian.Also, there exists only one functionally independent Casimir of our Poisson configuration, given by C : ℝ5 → ℝ,
Following , the equilibrium states of the dynamics (1) are:
which are unstable for any P ∈ ℝ*;
which are spectrally stable for
which are spectrally stable for any P,Q ∈ ℝ*.
The paper is organized as follows: the second section is dedicated to the nonlinear stability results; also, the existence of the periodical orbits around the nonlinear stable states is proved. The existence of the Hamilton-Poisson realization offers the necessary geometric framework in order to obtain this results. Specific tools, such as the energy methods and the Moser Theorem, belong to this framework.
The Optimal Homotopy Asymptotic Method is presented in the third section. Using this method, the analytical approximate solution of the proposed dynamics is found in the fourth paragraph. A comparison with the Runge-Kutta 4th steps integrator obtained using MATHEMATICA 6.0 and numerical simulations are the subjects of the last part.
2 Nonlinear Stability Problems and Periodical Orbits
Let us present very briefly, the concept of nonlinear stable equilibrium point of a dynamical system. For more details, see .
An equilibrium state xe is said to be nonlinear stable if for each neighborhood U of xe in D there is a neighborhood V of xe in U such that trajectory x(t) initially in V never leaves U.
This definition supposes well-defined dynamics and a specified topology. In terms of a norm ‖ ‖. nonlinear stability means that for each ε > 0 there is δ > 0 such that if
It is clear that nonlinear stability implies spectral stability; the converse is not always true.
The equilibrium states are nonlinearly stable if
We will use Arnold’s technique. Consider the function
The following conditions hold:
then, for all v ∈ W, i.e. v = (0,a,0,b,c), a, b, c ∈ ℝ, we have
which is positive definite under the restriction and so
is also positive definite.
Hence, via Arnold’s technique, the equilibrium states are nonlinear stable, as required.
As a consequence, we can find the periodic orbits of the equilibrium points More exactly, we can prove:
the reduced dynamics to the coadjoint orbit (2)
has near the equilibrium point at least two periodic solution whose period is close to
We will use Moser-Weinstein theorem with zero eigenvalue, see  for details.
The restriction of our dynamics (1) to the coadjoint orbit (2) gives rise to a classical Hamiltonian system.
The matrix of the linear part of our reduced dynamics to (1) has purely imaginary roots at the equilibrium of interest:
then the reduced Hamiltonian has a local minimum at the equilibrium state
The equilibrium states are nonlinearly stable for any P,Q ∈ ℝ*.
We will use energy-Casimir method, see  for details. Let
be the energy-Casimir function, where φ : ℝ → ℝ is a smooth real valued function.
Now, the first variation of Fφ is given by
so we obtain
that is equals zero for any P,Q ∈ ℝ* if and only if (3)
The second variation of Fφ at the equilibrium of interest is given by
If we choose now φ such that the relation (3) is valid and
then the second variation of Fφ at the equilibrium of interest is positive definited and so our equilibrium states are nonlinearly stable.
3 The Optimal Homotopy Asymptotic Method
Let us pass now to find the analytical approximate solutions for the nonlinear differential system given by the Eq. (1) with the boundary conditions (4)
For equations of the general form: (5)
subject to the boundary/initial conditions of the type: (6)
where p ∈ [0, 1] is the embedding parameter, L is a linear operator and H(η, Ci), (H ≠ 0) is an auxiliary convergence-control function, depending on the variable η and on the parameters C1, C2, …, Cs.
One can prove that the following properties hold:
If the function F is giving by: (10)
then, by substituting (10) in (7), the following relation is obtained: (11)
Consider the operator 𝓗 of the specific expression: (12)
where the governing equations of F0(η) and F1(η, Ci) can be obtained by equating the coefficients of p0 and p1, respectively: (13)
The expression of F0(η) can be found solving the linear equation (13). Also, to compute F1(η, Ci) we solve the equation (14), by taking into consideration that the nonlinear operator N presents the general form: (15)
where m is a positive integer and hi(η) and gi(η) are known functions depending both on F0(η) and N.
Although equation (14) is a nonhomogeneous linear equation, in most cases its solution can not be found.
In order to compute the function F1(η, Ci) we will use the third modified version of OHAM (see  for details), consisting in the following steps:
– First we consider one of the following expressions for F1(η, Ci): (16)
These expressions of Hi(η, hj(η), Cj) contain both linear combinations of the functions hj and the parameters The summation limit m is an arbitrary positive integer number.
– Finally, the convergence-control parameters C1, C2, …,Cs, which determine the first-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 and so on. The first option should be minimizing the square residual error: (19)
where the residual R is given by (20)
The unknown parameters C1, C2, … , Cs can be identified from the conditions: (21)
With these parameters known (called optimal convergence-control parameters), the first-order approximate solution given by Eq.(18) is well-determined.
It should be emphasized that our procedure contains the auxiliary functions Hi(η, fi, Cj), i = 1, …, m, j = 1, …, s which provides us with a simple way to adjust and control the convergence of the approximate solutions. It is very important to properly choose these functions Hi(η, fi, Cj) which appear in the construction in the first-order approximation.
4 Application of Optimal Homotopy Asymptotic Method for Solving the Nonlinear Differential System (1)
Let us now apply the method described above to solve the nonlinear differential system given by the equations (1). We choose the linear operators: (22)
where K1 > 0 is unknown parameter.
The corresponding linear equations for the initial approximationsxi0, becomes (23)
whose solutions are (24)
The nonlinear operators are obtained from the equations (1) as follows: (25)
a) We have a large freedom to choice the linear operator L, such as:
and therefore, the nonlinear operator N becomes
where g1(t) is an arbitrary continuous function that convergence-control the solution.
b) The function g1(t) can be chosen as g1(t) = B1 cos ωt + C1 sin ωt or g1(t) = B1 cos ωt + C1 sin ωt + B2 cos ω2 t + C2 sin ω2 t or g1(t) = B1 cos ωt + C1 sin ωt + B2 cos 3ωt + C2 sin 3ωt and so on.
The auxiliary convergence-control functions Hi are chosen such that the product Hi · N[xi0(t) has the same form like N[xi0(t). Then, the first approximation is: (27)
Using now the third-alternative of OHAM and the equations (18), the first-order approximate solution can be put in the form (28)
where xi0(t) and xi1(t, Cj) are given by (24) and (27), respectively.
k ∈ ℝ.
5 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 Eq. (1) with initial conditions
The convergence-control parameters are optimally determined by means of the least-square method.
For all unknown functions
The first-order approximate solutions given by the Eq. (28) are respectively:
The Figures 1 – 5 present a comparison between the analytical approximate solutions obtained by OHAM method and numerical results obtained by Runge-Kutta 4th steps integrator. We can see that the analytical approximate solutions offer us the same numerical results as Runge-Kutta 4th integrator.
The main goal of this paper is to present some comments about the nonlinear stability and the periodic orbits for some equilibrium points of a dynamical system. Its solutions are the optimal controls that minimize a cost function which steer a left-invariant drift-free control system on the Schrödinger Lie group from an initial state to a final state (see ).
In the last section the approximate analytic solutions of the considered controlled system (1) are established using the optimal homotopy asymptotic method (OHAM). %Numerical simulations via Mathematica 6.0 software and the approximations deviations are presented. Both numerical simulations via Mathematica 6.0 software and approximations deviations were presented. The accuracy of our results is pointed out by means of the approximate residual of the solutions. We can see that the proposed method, even if it does not use a Poisson integrator, offers better results that the Lie-Trotter integrator used in .
We would like to thank the referees very much for their valuable comments and suggestions.
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About the article
Published Online: 2016-12-30
Published in Print: 2016-01-01
Conflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.
Citation Information: Open Physics, Volume 14, Issue 1, Pages 549–558, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2016-0060.
© 2016 Remus-Daniel Ene et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0