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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Stabilization and Analytic Approximate Solutions of an Optimal Control Problem

Camelia Pop / Camelia Petrişor / Remus-Daniel Ene
Published Online: 2018-08-20 | DOI: https://doi.org/10.1515/phys-2018-0064

Abstract

This paper analyses a dynamical system derived from a left-invariant, drift-free optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 in deep connection with the important role of the Lie groups in tackling the various problems occurring in physics, mathematics, engineering and economic areas [1, 2, 3, 4, 5]. The stability results for the initial dynamics were inconclusive for a lot of equilibrium points (see [6]), so a linear control has been considered in order to stabilize the dynamics. The analytic approximate solutions of the resulting nonlinear system are established and a comparison with the numerical results obtained via the fourth-order Runge-Kutta method is achieved.

Keywords: ordinary differential equations; solution of equations; optimal control; nonlinear stability; optimal homotopy asymptotic method

PACS: 02.60.Cb; 02.60.Lj

1 Introduction

In the last of years, the optimal control problems on the Lie groups has drawn a lot of attention. The opportunity to study a mechanical system using geometric tools have had a big success in many areas of physics (see for instance [7, 8, 9, 10, 11]), engineering [12, 13, 14, 15]) and biology [16]. The left-invariant, drift-free control systems on the Lie groups belong to this class; having a Lie group as space configuration, the existence of a Hamilton-Poisson formulation is assured and therefore several powerful techniques can be employed such as the reduction of the symmetric system, the determination of the relative equilibria from a global point of view, the energy methods for studying the stability, specific methods for finding the periodical orbits, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems. Using the Lie group framework of the system, the determination of the equilibria and the energy method for determining the nonlinear stability, were performed in [6]. However, the nonlinear stability of some equilibria has remained unsolved, both methods - the energy-Casimir and Arnold’s - being unsuccessful.

The Hamilton-Poisson systems play an important role in mathematics, geometrical mechanics and physics (classical, quantum, or relativistic) due to their deep connections to metriplectic systems and, being useful to achieve their syncronization in information security and cryptography [17, 18, 19, 20].

In this paper, a linear control function is proposed in order to stabilize an equilibria and the analytic approximate solutions via the Optimal Homotopy Asymptotic Method (OHAM) of the controlled system are presented.

2 The Geometrical Framework of the Problem and the Stabilization by one Linear Control

To begin with let us introduce very briefly the left-invariant, drift-free control system on the Lie group G = SO(3) × ℝ3 × ℝ3 considered in [6]

X.=XA1u1+A2u2+A3u3+A5u5+A7u7,(1)

where XG, ui, i = 1, 9 are the controls and {Ai}i = 1,9 is the standard basis of the corresponding Lie algebra g. Then, the following results hold:

Proposition 2.1

[6]

  • The system (1) is controllable.

  • The controls that minimize the function

    J(u1,u2,u3,u5,u7)=

    =120tfu12t+u22t+u32t+u52t+u72tdt

    and steer the system (1) from X = X0 at t = 0 to X = Xf at t = tf are given by the solutions of

    x˙1=x5x6,x˙2=x7x9,x˙3=x4x5x7x8,x˙4=x2x6+x3x5,x˙5=x1x6x3x4,x˙6=x1x5+x2x4,x˙7=x2x9+x3x8,x˙8=x1x9x3x7,x˙9=x1x8+x2x7.(2)

The main goal of this section is to establish some stability results of the equilibrium points e8MNP = (M, 0, N, P, 0, NPM, 0, 0, 0), M, N, P ∈ ℝ*, of the above system. Some stability results regarding the equilibrium states

e1MNPQ=(0,0,0,M,0,N,0,P,Q),M,N,P,QR,e2MNP=(0,0,M,0,0,N,0,0,P),M,N,PR,e3MNP=(0,0,0,0,M,0,0,N,P),M,N,PR,e4MNP=(0,M,0,0,N,0,0,P,0),M,N,PR,e5MNP=(M,N,P,0,0,0,0,0,0),M,N,PR

were already obtained in [6], but the stability problem for the other equilibrium states remains unsolved.

In order to stabilize the equilibrium states e8MNP of the system (2) we employ the linear control uC(ℝ9, ℝ) given by

u(x¯)=(Nx2+Nx5,Mx3+Nx1+Mx6Nx4,Mx2Mx5,Nx5,Nx4Mx6,Mx5,Nx8,Mx9+Nx7,Mx8),(3)

so the controlled system (2)-(3) can be explicitly written:

x˙1=x5x6Nx2+Nx5x˙2=x7x9Mx3+Nx1+Mx6Nx4x˙3=x4x5x7x8+Mx2Mx5x˙4=x2x6+x3x5Nx5x˙5=x1x6x3x4+Nx4Mx6x˙6=x1x5+x2x4+Mx5x˙7=x2x9+x3x8Nx8x˙8=x1x9x3x7Mx9+Nx7x˙9=x1x8+x2x7+Mx8.(4)

Proposition 2.2

The controlled system (4) has the Hamilton-Poisson realization

(G,Π,H),

where Π is the minus Lie-Poisson structure on the dual g* of the Lie algebra, given by

Π=0x3x20x6x30x1x60x2x10x5x40x6x500x60x400x5x40000x9x800x90x700x8x7000x50x9x8x4x90x70x8x70000000000000000000000000

and the Hamiltonian function

H(x¯)=12(x12+x22+x32+x52+x72)Mx1Nx3+Mx4+Nx6.

Proof

Indeed, it is not hard to see that the dynamics (4) can be put into the equivalent form:

x˙=ΠH,

as required. More details about Poisson structures can be found in [21]. □

Proposition 2.3

[6] The functions C1, C2 and C3 given by:

C1(x¯)=12(x42+x52+x62),C2(x¯)=12(x72+x82+x92),C3(x¯)=x1x4+x2x5+x3x6

are Casimirs of our configuration.

Let B be the matrix of linear part of our controlled system (4), that is

B=0N00x6+NN0MN00M0x5x4M0x6x50x3Nx60x4x3+N0x5x40x2x1+M0x9x800x90x700x8x7000x5000Mx90x70x8x70x2000x1M000000000x3Nx20x3+N0x1M0x2x1+M0.

At the equilibrium of interest, its characteristic polynomial has the following expression

PA(e8MNP)(λ)=λ5λ4+1PM2(M2+N2)λ2++P2M2(M2+N2)21PM.

If MP < 0, we have five zero eigenvalues and four purely imaginary eigenvalues, so we can conclude:

Proposition 2.4

If MP < 0, M, N, P ∈ ℝ*, then the controlled system (4) may be spectral stabilized about the equilibrium states e8MNP

Moreover we can prove:

Proposition 2.5

If MP < 0, M, N, P ∈ ℝ*, then the controlled system (4) may be nonlinear stabilized about the equilibrium states e8MNP.

Proof

Let us consider the function (see [22] for details):

Fλ,μ(x¯)=C2(x¯)+λH(x¯)+μC1(x¯)=12(x72+x82+x92)+λ2(x12+x22+x32+x52+x722Mx12Nx3+2Mx4+2Nx6)+μ2(x42+x52+x62).

Then we have successively:

  1. Fλ,μ( e8MNP) = 0 iff μ = MPλ;

  2. Considering now

    W=ker[dH(e8MNP)]ker[dC1(e8MNP)]=Span100000000,010000000,001000000,000010000,000000100,000000010,000000001,

    then, for all vW, i.e. v = (a, b, c, 0, d, 0, e, f, g), a, b, c, d, e, f, g ∈ ℝ, we have

    v2Fλ,MPλ(e8MNP)vt=λa2+λb2+λc2+λMPλd2+1+λe2+f2+g2

    which is positive definite under the restriction λ > 0 and MP < 0, and so

    2Fλ,MPλ(e8MNP)|W×W

    is positive definite.

Therefore, if MP < 0, then the equilibrium states e8MNP are nonlinear stable, as required. □

3 Basic Ideas of the Optimal Homotopy Asymptotic Method

In order to find the approximate analytic solutions of the controlled system (2.4), we shall start with a brief description of this method. For an equation of the form:

L(x(t))+N(x(t))=0,(5)

subject to the initial conditions of the type:

x(0)=A,AR - given real number,(6)

where 𝓛 is a linear operator, 𝓝 is a nonlinear one and x(t) is the unknown smooth function of the Eq. (5), the approximate analytic solutions can be obtained using the third modified version of Optimal Homotopy Asymptotic Method described hereinafter (see [23] for details).

Following [23, 24] we construct the homotopy given by:

H[L(X(t,p)),H(t,Cj),N(X(t,p))]L(x0(t))++p[L(x1(t,Cj))H(t,Cj)N(x0(t))]=0,(7)

where p ∈ [0, 1] is the embedding parameter, H(t, Cj), (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:

X(t,p)=x0(t)+px1(t,Cj).(8)

The following properties hold:

H[L(X(t,0)),H(t,Cj),N(X(t,0))]L(x0(t))=0(9)

and

H[L(X(t,1)),H(t,Cj),N(X(t,1))]L(x1(t,Cj))H(t,Cj)N(x0(t))=0.(10)

The governing equations of x0(t) and x1(t, Cj) can be obtained by equating the coefficients of p0 and p1, respectively:

L(x0(t))=0,x(0)=A,(11)

L(x1(t,Cj))=H(t,Cj)N(x0(t)),x1(0,Cj)=0,j=1,s¯.(12)

The expression of x0(t) can be found by solving the linear Eq. (11). Also, to compute x1(t, Cj) we solve the Eq. (12), by taking into consideration that the nonlinear operator N presents the general form:

Nx0(t)=k=1mhk(t)gk(t),(13)

where m is a positive integer and hk(t) and gk(t) are known functions depending both on x0(t) and N.

Although the Eq. (12) is a nonhomogeneous linear one in the most cases its solution can not be found.

In order to compute the function x1(t, Cj), we do not solve Eq. (12) but from the theory of differential equation, taking into consideration the method of variation of parameters, the method of influence function, Cauchy method, the operator method [25], it is more convenient to consider the unknown function x1(t, Cj) using the following steps:

  • First we consider one of the following expressions for x1(t, Cj)

    x1(t,Cj)=k=1mHk(t,hj(t),Cj)gk(t),j=1,s¯,(14)

    or

    x1(t,Cj)=k=1mHk(y,gj(t),Cj)hk(t),x1(0,Cj)=0,j=1,s¯.(15)

    These expressions of Hk(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.

  • Next, by taking into account the Eq. (8), for p = 1, the first-order analytical approximate solution of the Eqs. (5) - (6) is:

    x¯(t,Cj)=X(t,1)=x0(t)+x1(t,Cj).(16)

  • Finally, the convergence-control parameters C1, C2, …, Cs, which determine the first-order approximate solution (16), 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. ▪

In what follows, we will show that the approximate analytic solution x(t, Cj) given by Eq. (16) is a weak ϵ-approximate solution of the problem Eq. (5) with the initial condition Eq. (6).

Definition 3.1

[26] We call an ϵ-approximate solution of the problem (5) on the domain (0, ∞) a smooth function x(t, Cj) of the form (16) which satisfies the following condition:

|R(t,x¯(t,Cj))|<ϵ,

together with the initial condition from Eq. (6), where the residual function R(t, x(t, Cj)) is obtained by substituting the Eq. (16) into Eq. (5), i.e.

R(t,x¯(t,Cj))=L(x¯(t,Cj))+N(x¯(t,Cj)).

Definition 3.2

[26] We call an weak ϵ-approximate solution of the problem (5) on the domain (0, ∞) a smooth function x(t, Cj) of the form (16) which satisfies the following condition:

0R2(t,x¯(t,Cj))dη<ϵ,

together with the initial condition from Eq. (6).

4 Analytic Approximate Solutions of the System (4) using Optimal Homotopy Asymptotic Method

We now apply the procedure described above to the nonlinear differential system given by the Eqs. (4) with the initial conditions

xi(0)=Ai,,AiR - given real number,i=1,9¯.(17)

Considering the linear operators 𝓛 [xi(t)]:

Lx1(t)=x˙1(t)+K1x1(t)+Nx2(t)Nx5(t),Lx2(t)=x˙2(t)+K1x2(t)+Mx3(t)Nx1(t),Lx3(t)=x˙3(t)+K1x3(t)Mx2(t)+Mx5(t),Lx4(t)=x˙4(t)+K1x4(t)+Kx7(t),Lx5(t)=x˙5(t)+K1x5(t)+Mx6(t),Lx6(t)=x˙6(t)+K1x6(t)Mx5(t),Lx7(t)=x˙7(t)+K1x7(t)Kx4(t)+Nx8(t),Lx8(t)=x˙8(t)+K1x8(t)+Mx9(t),Lx9(t)=x˙9(t)+K1x9(t)Mx8(t)(18)

and 𝓝[xi(t)], i = 1, 9:

Nx1(t)=K1x1(t)+x5(t)x6(t),Nx2(t)=K1x2(t)x7(t)x9(t)Mx6(t)+Nx4(t),Nx3(t)=K1x3(t)x4(t)x5(t)+x7(t)x8(t),Nx4(t)=K1x4(t)Kx7(t)+x2(t)x6(t)x3(t)x5(t)+Nx5(t),Nx5(t)=K1x5(t)x1(t)x6(t)+x3(t)x4(t)Nx4(t),Nx6(t)=K1x6(t)+x1(t)x5(t)x2(t)x4(t),Nx7(t)=K1x7(t)+x2(t)x9(t)x3(t)x8(t),Nx8(t)=K1x8(t)x1(t)x9(t)+x3(t)x7(t)Nx7(t),Nx9(t)=K1x9(t)+x1(t)x8(t)x2(t)x7(t),(19)

where K > 0 and K1 > 0 are unknown parameters, then the solutions of the Eq. (11) for the initial approximations xi0, i = 1, 9 are given by:

x10(t)=A1M2+N2(M2+N2cos(M2+N2t))A2N22A5(M2+N2)NM2+N2sin(M2+N2t)A32(M2+N2)(1+cos((M2+N2)t))+A5N(Msin(Mt)+M2+N2sin(M2+N2t))A6MN(cos(Mt)cos(M2+N2t))eK1t,x20(t)=A1NM2+N2sin(M2+N2t)+(A22A5(M2+N2)N2cos(M2+N2t)+A3M2M2+N2sin(M2+N2t)+A5(M2+N2)N2(cos(Mt)+cos(M2+N2t))+A6N2((M2+N2)sin(Mt)+MM2+N2sin(M2+N2t))eK1t,x30(t)=A1MNM2+N2(1+cos((M2+N2)t))+A2N22A5(M2+N2)MN2M2+N2sin(M2+N2t)+A32(1+cos((M2+N2)t))A5N2(Msin(Mt)+M2+N2sin(M2+N2t))+A6M2N2(1cos(M2+N2t))eK1t,x40(t)=A4cos(Kt)A7sin(Kt)KNA8K2M2(cos(Kt)cos(Mt))NA9K2M2(Msin(Kt)+Ksin(Mt))eK1t,x50(t)=(A5cos(Mt)A6sin(Mt))eK1t,x60(t)=(A6cos(Mt)+A5sin(Mt))eK1t,x70(t)=A7cos(Kt)+A4sin(Kt)+MNA9K2+M2(cos(Kt)cos(Mt))NA8K2M2(Ksin(Kt)Msin(Mt))eK1t,x80(t)=(A8cos(Mt)A9sin(Mt))eK1t,x90(t)=(A9cos(Mt)+A8sin(Mt))eK1t.(20)

By substituting Eqs. (20) into Eqs. (19), we reach the following results:

Nx10(t)=K1x10(t)+x50(t)x60(t),Nx20(t)=K1x20(t)x70(t)x90(t)Mx60(t)+Nx40(t),Nx30(t)=K1x30(t)x40(t)x50(t)+x70(t)x80(t),Nx40(t)=K1x40(t)Kx70(t)+x20(t)x60(t)x30(t)x50(t)+Nx50(t),Nx50(t)=K1x50(t)x10(t)x60(t)+x30(t)x40(t)Nx40(t),Nx60(t)=K1x60(t)+x10(t)x50(t)x20(t)x40(t),Nx70(t)=K1x70(t)+x20(t)x90(t)x30(t)x80(t),Nx80(t)=K1x80(t)x10(t)x90(t)+x30(t)x70(t)Nx70(t),Nx90(t)=K1x90(t)+x10(t)x80(t)x20(t)x70(t).(21)

Remark 4.1

It is easy to see that the nonlinear operators 𝓝[xi0(t)], i = 1, 9 can be written as linear combinations of the functions

eK1tcos(Mt),eK1tsin(Mt),e2K1tcos2(Mt),e2K1tsin2(Mt),e2K1tcos(Mt)sin(Mt),eK1tcos(Kt),eK1tsin(Kt),e2K1tcos(Kt)cos(Mt),e2K1tsin(Kt)sin(Mt),e2K1tcos(Kt)sin(Mt),e2K1tsin(Kt)cos(Mt),eK1tcos(M2+N2t),eK1tsin(M2+N2t),e2K1tcos2(M2+N2t),e2K1tsin2(M2+N2t),e2K1tcos(M2+N2t)sin(M2+N2t),e2K1tcos(Mt)sin(M2+N2t),e2K1tcos(M2+N2t)sin(Mt),e2K1tcos(Kt)sin(M2+N2t),e2K1tcos(M2+N2t)sin(Kt).

Although the Eq. (12) is a nonhomogeneous linear one, finding its solution could be a difficult problem.

In order to compute the functions xi1(t, Bs, Cs), i = 1, 9, by the Remark 4.1 and the previous Section, the first approximation xi1, i = 1, 9 can be written as:

xi1(t,Bs,Cs)==n=18Bncos(2n+1)ωt+Cnsin(2n+1)ωteK1t,i=1,9¯,s=1,8¯.(22)

Now, using the Eqs. (20) and Eq. (22), the first-order analytic approximate solution given by Eq. (16) of the Eqs. (5) - (17) is well-determined and its becomes:

x¯i(t,Bs,Cs)=xi0(t)+xi1(t,Bs,Cs),i=1,9¯,s=1,8¯.(23)

5 Numerical Examples and Discussions

The goal of this section is to prove the accuracy and the validity of the OHAM technique. We make use of the comparison of our approximate solutions with numerical results obtained via the fourth-order Runge-Kutta method in the following case, as follows: we consider the initial value problem given by the system (4) with initial conditions Ai = 0.001, i = 1, 9, M = 15 and N = 20; then, the convergence-control parameters K, K1, ω, Bi, Ci, i = 1, 8 are optimally determined by means of the least-square method using the Wolfram Mathematica 9.0 software.

One can show that these approximate solutions given by Eq. (23) are weak ϵ-approximate solutions by computing the numerical values of the integral of square residual function (to see the Table 4 from Appendix 6), i.e. 01Ri2(t)dt,i=1,,9, where

R1(t)=x¯˙1(t)+x¯5(t)x¯6(t)+Nx¯2(t)Nx¯5(t),R2(t)=x¯˙2(t)x¯7(t)x¯9(t)+Mx¯3(t)Nx¯1(t)Mx¯6(t)+Nx¯4(t),R3(t)=x¯˙3(t)x¯4(t)x¯5(t)+x¯7(t)x¯8(t)Mx¯2(t)+Mx¯5(t),R4(t)=x¯˙4(t)+x¯2(t)x¯6(t)x¯3(t)x¯5(t)+Nx¯5(t),R5(t)=x¯˙5(t)x¯1(t)x¯6(t)+x¯3(t)x¯4(t)Nx¯4(t)+Mx¯6(t),R6(t)=x¯˙6(t)+x¯1(t)x¯5(t)x¯2(t)x¯4(t)Mx¯5(t),R7(t)=x¯˙7(t)+x¯2(t)x¯9(t)x¯3(t)x¯8(t)+Nx¯8(t),R8(t)=x¯˙8(t)x¯1(t)x¯9(t)+x¯3(t)x¯7(t)+Mx¯9(t)Nx¯7(t),R9(t)=x¯˙9(t)+x¯1(t)x¯8(t)x¯2(t)x¯7(t)Mx¯8(t),(24)

with xi(t), i = 1, … 9, given by Eq. (23).

  • For x1 the convergence-control parameters are respectively:

B1=0.6870910041,B2=0.4489602178,B3=0.5460639209,B4=0.1352692445,B5=0.2275764517,B6=0.0091553612,B7=0.0517476470,B8=0.0059904466,C1=0.6043513189,C2=0.6632485829,C3=0.2759472577,C4=0.3824870710,C5=0.0477875461,C6=0.1139833895,C7=0.0026759104,C8=0.0144167229,K=2.4390311791,K1=7.00700137106,ω=1.6655500501.

The first-order approximation of the solution given by the Eq. (16) becomes:

x¯1(t)=eK1t(0.6870910041cos(ωt)0.4489602178cos(3ωt)0.5460639209cos(5ωt)+0.1352692445cos(7ωt)+0.2275764517cos(9ωt)0.0091553612cos(11ωt)0.0517476470cos(13ωt)+0.0059904466cos(15ωt))+(0.6043513189sin(ωt)0.6632485829sin(3ωt)+0.2759472577sin(5ωt)+0.3824870710sin(7ωt)0.0477875461sin(9ωt)0.1139833895sin(11ωt)0.0026759104sin(13ωt)+0.0144167229sin(15ωt))eK1t+0.0017eK1tsin(25t)+0.00005eK1t(15sin(15t)+25sin(25t))0.00075eK1t(cos(15t)cos(25t))8107eK1t(1+cos(25t))+1.6106eK1t(225+400cos(25t)).(25)

Finally, Table 1 and Tables 2-3 (see Appendix 6), respectively, emphasizes the accuracy of the OHAM technique by comparing the approximate analytic solutions x3, x5 and x8 respectively presented above with the corresponding numerical integration values.

Table 1

The comparison between the approximate solutions x3 given by Eq. (A.2) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx3 = |x3numericalx3OHAM|)

Table 2

The comparison between the approximate solutions x5 given by Eq. (A.4) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx5 = |x5numericalx5OHAM|)

Table 3

The comparison between the approximate solutions x8 given by Eq. (A.7) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx8 = |x8numericalx8OHAM|)

Table 4

The numerical values of the integral of square residual function given by Eq. (24) corresponding to the approximate solutions given by Eqs. (25), (A.1)-(A.8) for M = 15 and N = 20.

On the other hand, the approximate analytic solutions x2 ÷ x9 can be found in Appendix 6.

Remark 5.1

The Figures 1-9 present numerical comparisons between the analytical approximate solutions obtained by OHAM Method and numerical results obtained by Runge-Kutta 4-steps integrator. We can see that the analytical approximate solutions offer us the same numerical results as Runge-Kutta 4-steps integrator.

Comparison between the approximate solutions x1 given by Eq. (25) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 1

Comparison between the approximate solutions x1 given by Eq. (25) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x2 given by Eq. (A.1) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 2

Comparison between the approximate solutions x2 given by Eq. (A.1) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x3 given by Eq. (A.2) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 3

Comparison between the approximate solutions x3 given by Eq. (A.2) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x4 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 4

Comparison between the approximate solutions x4 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x5 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 5

Comparison between the approximate solutions x5 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x6 given by Eq. (A.5) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 6

Comparison between the approximate solutions x6 given by Eq. (A.5) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x7 given by Eq. (A.6) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 7

Comparison between the approximate solutions x7 given by Eq. (A.6) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x8 given by Eq. (A.7) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 8

Comparison between the approximate solutions x8 given by Eq. (A.7) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

Comparison between the approximate solutions x9 given by Eq. (A.8) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Figure 9

Comparison between the approximate solutions x9 given by Eq. (A.8) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution

6 Conclusion

In this paper, the stabilization of a dynamical system arising from an optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 is investigated. The conditions of the spectral and nonlinear stability of a specific equilibria were obtained using a linear control added to the initial system. Due to the existence of the Poisson formulation, the results were obtained using specific tools, such as Arnold’s method.

Finally, the analytic approximate solutions of the controlled dynamics are obtained via the Optimal Homotopy Asymptotic Method. Numerical simulations and a comparison with Runge-Kutta 4-steps integrator are also presented.

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Appendix

The following conditions K1 = 7.00700137 ⋅ 10–6, ω = 1.6655500501 hold for all unknown functions xi, i = 1, 9, so:

  • for x2:

    x¯2(t)=eK1t(0.4389432372cos(ωt)+0.5020290567cos(3ωt)+0.2237312287cos(5ωt)0.3256680044cos(7ωt)0.0683191602cos(9ωt)+0.1147324797cos(11ωt)+0.0126641080cos(13ωt)0.0202264714cos(15ωt))0.002125eK1tcos(25t)+0.0015625eK1t(cos(15t)+cos(25t))+eK1t(0.5027080265sin(ωt)+0.3350256550sin(3ωt)0.4352899954sin(5ωt)0.1306954331sin(7ωt)+0.2103586786sin(9ωt)+0.0300134573sin(11ωt)0.0576550229sin(13ωt)+0.0061020354sin(15ωt))+0.0011eK1tsin(25t)+2.5106eK1t(625sin(15t)+375sin(25t));(A.1)

  • for x3:

    x¯3(t)=eK1t(0.5160419993cos(ωt)+0.3372418512cos(3ωt)+0.4100224611cos(5ωt)0.1016916740cos(7ωt)0.1702969101cos(9ωt)+0.0069229986cos(11ωt)+0.0388494598cos(13ωt)0.0050061873cos(15ωt))+(0.4538511906sin(ωt)+0.4980536525sin(3ωt)0.2073422773sin(5ωt)0.2871797742sin(7ωt)+0.0364900541sin(9ωt)+0.0855759933sin(11ωt)+0.0019833945sin(13ωt)0.0116955965sin(15ωt))eK1t0.001275eK1tsin(25t)2.5106eK1t(15sin(15t)+25sin(25t))+0.0005625000eK1t(1cos(25t))0.00048eK1t(1+cos(25t))+0.0005eK1t(1+cos(25t));(A.2)

  • for x4:

    x¯4(t)=eK1t(0.0014047411cos(ωt)0.0021067093cos(3ωt)0.0001754437cos(5ωt)+0.0003714480cos(7ωt)+0.0004300377cos(9ωt)0.0000814697cos(11ωt)8.497125106cos(13ωt)+0.0001658929cos(15ωt))+(0.0014137145sin(ωt)+0.0002567584sin(3ωt)+0.0007210711sin(5ωt)+0.0001595064sin(7ωt)0.0005212098sin(9ωt)0.0000413490sin(11ωt)+0.0000357107sin(13ωt)0.0008061102sin(15ωt))eK1t+0.001eK1tcos(3.5665110571t)+0.0003360195eK1t(cos(3.5665110571t)cos(15t))0.001eK1tsin(3.5665110571t)+0.0000942151eK1t(15sin(3.5665110571t)+3.5665110571sin(15t));(A.3)

  • for x5:

    x¯5(t)=eK1t(0.0003950998cos(ωt)0.0001844516cos(3ωt)0.0003597136cos(5ωt)+8.918677107cos(7ωt)0.0008339975cos(9ωt)+0.0000152088cos(11ωt)0.0000418091cos(13ωt)+0.0010087715cos(15t))+(0.0003071207sin(ωt)0.0004137084sin(3ωt)+0.0000724864sin(5ωt)+0.0002604951sin(7ωt)+0.0010261920sin(9ωt)0.0000865232sin(11ωt)1.773634106sin(13ωt)+0.0002071073sin(15ωt))eK1t+eK1t(0.001cos(15t)0.001sin(15t));(A.4)

  • for x6:

    x¯6(t)=eK1t(0.0005541313cos(ωt)+0.0006533041cos(3ωt)+0.0001646139cos(5ωt)0.0002330474cos(7ωt)0.0010805238cos(9ωt)+0.0000595280cos(11ωt)+6.366934106cos(13ωt)0.0001243732cos(15t))+(0.0014141404sin(ωt)+0.0001011217sin(3ωt)0.0004019421sin(5ωt)0.0001317507sin(7ωt)0.0008720894sin(9ωt)+0.0000307159sin(11ωt)0.0000263820sin(13ωt)+0.0006045821sin(15ωt))eK1t+eK1t(0.001cos(15t)+0.001sin(15t));(A.5)

  • for x7:

    x¯7(t)=eK1t(0.0007187903cos(ωt)0.0008984587cos(3ωt)0.0002156138cos(5ωt)+0.0003149641cos(7ωt)+0.0014403433cos(9ωt)0.0000798255cos(11ωt)8.460837106cos(13ωt)+0.0001658418cos(15ωt))+(0.0020538216sin(ωt)0.0001277630sin(3ωt)+0.0005463646sin(5ωt)+0.0001740129sin(7ωt)+0.0011614636sin(9ωt)0.0000408001sin(11ωt)+0.0000352719sin(13ωt)0.0008061122sin(15ωt))eK1t+0.001eK1tcos(0.1923247828t)+0.0013335525eK1t(cos(0.1923247828t)cos(15t))+0.001eK1tsin(0.1923247828t)+0.0000889035eK1t(0.1923247828sin(0.1923247828t)15sin(15t));(A.6)

  • for x8:

    x¯8(t)=eK1t(0.0003950998cos(1ωt)0.0001844516cos(3ωt)0.0003597136cos(5ωt)+8.918677107cos(7ωt)0.0008339975cos(9ωt)+0.0000152088cos(11ωt)0.0000418091cos(13ωt)+0.0010087715cos(15ωt))+(0.0003071207sin(ωt)0.0004137084sin(3ωt)+0.0000724864sin(5ωt)+0.0002604951sin(7ωt)+0.0010261920sin(9ωt)0.0000865232sin(11ωt)1.773634106sin(13ωt)+0.0002071073sin(15ωt))eK1t+0.001eK1tcos(15t)0.001eK1tsin(15t);(A.7)

  • for x9:

    x¯9(t)=eK1t(0.0005541313cos(ωt)+0.0006533041cos(3ωt)+0.0001646139cos(5ωt)0.0002330474cos(7ωt)0.0010805238cos(9ωt)+0.0000595280cos(11ωt)+6.366933106cos(15ωt)0.0001243732cos(13ωt))+(0.0014141404sin(1ωt)+0.0001011217sin(3ωt)0.0004019421sin(5ωt)0.0001317507sin(7ωt)0.0008720894sin(9ωt)+0.0000307159sin(11ωt)0.0000263820sin(13ωt)+0.0006045821sin(15ωt))eK1t+0.001eK1tcos(15t)+0.001eK1tsin(15t);(A.8)

About the article

Received: 2017-03-14

Accepted: 2018-06-22

Published Online: 2018-08-20


Conflict of InterestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.


Citation Information: Open Physics, Volume 16, Issue 1, Pages 476–487, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0064.

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© 2018 C. Pop et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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