The optimal control problems on the Lie groups were studied very often in deep connection with mechanical systems. We can find a large list of such examples, like the dynamics of an underwater vehicle, with SE(3, R) = SO(3) × R3 as space configuration (see ), the ball-plate problem, with R2 × SO(3) as space configuration , the rolling-penny dynamics having the Lie group SE(2, R) × SO(2) as space configuration , the control tower problem from air traffic, modeled on the Special Euclidean Group SE(3), the spacecraft dynamics modeled on the special orthogonal group SO(3), , the buoyancy’s dynamics on the Lie group SO(3) × R3 × R3, (see  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 finding the optimal controls that minimize a quadratic cost function for the general left-invariant drift-free control system
on the Lie group G = SO(3) × R3 × R3, where Ai, i = 1, 9 is the standard basis of the Lie algebra g:
Considering now the cost function given by:
the controls that minimize J and steer the system (1) from X = X0 at t = 0 to X = Xf at t = tf are given by:
where xi’s are solutions of the following nonlinear system:
The main goal of our paper is to establish some stability results of the equilibrium points
of the above system. Some stability results regarding the equilibrium states
were already obtained in , but the stability problem for the other equilibrium states remains unsolved.
The paper is organized as follows: in the second paragraph we find an appropriate control function in order to stabilize the equilibrium states . The third section briefly 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.
2 Stabilization of by one linear control
Let us employ the control u ∈ C∞ (R9, R),
has as an equilibrium state.
The controlled system (4) has the Hamilton-Poisson realization
where G = SO(3) × R3 × R3,
is the minus Lie-Poisson structure on the dual of the corresponding Lie algebra g∗ and the Hamiltonian function given by
Indeed, one obtains immediately that
and Π is a minus Lie-Poisson structure, see for details . □
(). The functions C1, C2, C3 : R9 → R given by
are the Casimirs of our Poisson configuration.
The goal of this paragraph is to study the spectral and nonlinear stability of the equilibrium state of the controlled system (4).
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 five zero eigenvalues and four purely imaginary eigenvalues. So we can conclude:
The controlled system (4) may be spectral stabilized about the equilibrium states for all M, N, P ∈ R∗.
Moreover we can prove:
The controlled system (4) may be nonlinear stabilized about the equilibrium states for all M, N, P ∈ R∗.
For the proof we shall use Arnold’s technique. Let us consider the following function
The following conditions hold:
then, for all v ∈ W, i.e. v = (a, b, c, 0, d, e,0, f, g), a, b, c, d, e, f, g ∈ R we have
which is positive definite under the restriction λ > 0, and so
is positive definite.
Therefore, via Arnold’s technique, the equilibrium states , M, N, P ∈ R∗ are nonlinear stable, as required. □
3 Basic ideas of the Optimal Homotopy Asymptotic Method
In order to compute analytical approximate solutions for the nonlinear differential system given by the equations (4) with the boundary conditions
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:
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).
where p ∈ [0, 1] is the embedding parameter, H(t, Ci ), (H ≠ 0) is an auxiliary convergence-control function, depending on the variable t and on the parameters C1, C2, …, Cs and the function x(t, p) has the expression:
The following properties hold:
The governing equations of X0(t) and x1(t, Ci) can be obtained by equating the coefficients of p0 and p1, respectively:
The expression of x0(t) can be found by solving the linear equation (13). Also, to compute x1(t, Ci) 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 hi(t) and gi(t) are known functions depending both on x0(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 x1(t, 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 x1(t, Ci):
These expressions of Hi(t, hj(t), Cj) contain both linear combinations of the functions hj and the parameters Cj, j = 1, s. 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.
4 Application of Optimal Homotopy Asymptotic Method for solving the nonlinear differential system (4)
with K > 0, K1 > 0 the unknown parameters at this moment.
whose solutions are
The corresponding nonlinear operators N[xi(t)], i = 1, 9 are obtained from the equations (4):
Now, we observe that the nonlinear operators N[xi0(t)], i = 1, 9 are the linear combinations between the elementary functions e–K1t · cos(Mt), e–K1t · sin(Mt), e–2K1t · cos2(Mt), e–2K1t · sin2(Mt), e–2K1t · cos(Mt) sin(Mt), e–K1t · cos(Kt), e–K1t · sin(Kt), e–2K1t · cos(Kt) cos(Mt), e–2K1t · sin(Kt) sin(Mt), e–2K1t · cos(Kt) sin(Mt), e–2K1t · sin(Kt) cos(Mt).
On the other hand, the Eq. (14) becomes:
The auxiliary convergence-control functions Hi are chosen such that the product between hi · N[xi0(t)] has the same form of the N[xi0(t)]. Then, the first approximation becomes:
Using now the third-alternative of OHAM and the equations (18), the first-order approximate solution can be put in the form
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 (4) with initial conditions (6) Ai = 0.0001, i = 1, 9, M = 15 and P = 20.
One can show that these approximate solutions are week ε-approximate solutions by computing the numerical value of the integral of square residual function (to see the Table 4),
, i = 1, …, 9,
with x̄i(t), i = 1, …, 9 given by Eq. (26).
The convergence-control parameters K, K1, ω, Bi, Ci, i = 1, 9 are optimally determined by means of the least-square method.
for x̄1 : The convergence-control parameters are respectively:
The first-order approximate solutions given by the Eq. (26) are respectively:
For all unknown functions x̄i, i = 1, 9, we have K1 = 0.58656793719790 and ω = 1.01132823106464.
Finally, Tables 1, 2 and 3 emphasizes the accuracy of the OHAM technique by comparing the approximate analytic solutions x̄3, x̄5 and x̄8 respectively presented above with the corresponding numerical integration values.
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 ) and OHAM, regarding the numerical results.
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About the article
Published Online: 2018-03-20
Conflict of interestConflict of interests: The authors declare that there is no conflict of interests regarding the publication of this paper.
Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 219–234, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0019.
© 2018 Pop and Ene, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0