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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Stability problems and numerical integration on the Lie group SO(3) × R3 × R3

Camelia Pop / Remus-Daniel Ene
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/math-2018-0019

Abstract

The paper is dealing with stability problems for a nonlinear system on the Lie group SO(3) × R3 × R3. The approximate analytic solutions of the considered system via Optimal Homotopy Asymptotic Method are presented, too.

Keywords: Nonlinear ordinary differential systems; Nonlinear stability; Lie group; Optimal homotopy asymptotic method

MSC 2010: 34H15; 65Nxx; 65P40; 70H14; 74G10; 74H10

1 Introduction

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 [1]), the ball-plate problem, with R2 × SO(3) as space configuration [2], the rolling-penny dynamics having the Lie group SE(2, R) × SO(2) as space configuration [3], 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), [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 [6,7,8,9,10].

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 finding the optimal controls that minimize a quadratic cost function for the general left-invariant drift-free control system

X˙=XA1u1+A2u2+A3u3+A5u5+A7u7,(1)

on the Lie group G = SO(3) × R3 × R3, where Ai, i = 1, 9 is the standard basis of the Lie algebra g:

A1=0000001001000000,A2=0010000010000000,A3=0100100000000000,A4=0001000000000000,A5=0000000100000000,A7=0000000000010000.

Since the span of the set of Lie brackets generated by A1, A2, A3, A5, A7 coincides with g, the system (1) is controllable [11].

Considering now the cost function given by:

J(u1,u2,u3,u5,u7)=120tfc1u12t+c2u22t+c3u32t+c5u52t+c7u72tdtc1>0,c2>0,c3>0,c5>0,c7>0,

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:

u1=1c1x1,u2=1c2x2,u3=1c3x3,u5=1c5x5,u7=1c7x7,

where xi’s are solutions of the following nonlinear system:

x˙1=x5x6x˙2=x7x9x˙3=x4x5x7x8x˙4=x2x6+x3x5x˙5=x1x6x3x4x˙6=x1x5+x2x4x˙7=x2x9+x3x8x˙8=x1x9x3x7x˙9=x1x8+x2x7.(2)

The main goal of our paper is to establish some stability results of the equilibrium points

e6MNP=(M,0,0,N,0,0,P,0,0),M,N,PR,

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,e3MPQ=(0,0,0,0,M,0,0,P,Q),M,P,QR,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 [11], 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 e6MNP . 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 e6MNP by one linear control

Let us employ the control uC (R9, R),

u(x¯)=(0,Mx32Px9,Mx2+2Px8,0,Mx6,Mx5,0,Mx9,Mx8),(3)

for the system (2). The controlled system (2)(3), explicitly given by

x˙1=x5x6x˙2=x7x9Mx32Px9x˙3=x4x5x7x8+Mx2+2Px8x˙4=x2x6+x3x5x˙5=x1x6x3x4Mx6x˙6=x1x5+x2x4+Mx5x˙7=x2x9+x3x8x˙8=x1x9x3x7Mx9x˙9=x1x8+x2x7+Mx8,(4)

has e6MNP as an equilibrium state.

Proposition 2.1

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

(G,Π,H),

where G = SO(3) × R3 × R3,

Π=0x3x20x6x50x9x8x30x1x60x4x90x7x2x10x5x40x8x700x6x5000000x60x4000000x5x400000000x9x8000000x90x7000000x8x70000000(5)

is the minus Lie-Poisson structure on the dual of the corresponding Lie algebra g and the Hamiltonian function given by

H(x¯)=12(x12+x22+x32+x52+x72)Mx12Px7.

Proof

Indeed, one obtains immediately that

ΠH=[x˙1x˙2x˙3x˙4x˙5x˙6x˙7x˙8x˙9]t,

and Π is a minus Lie-Poisson structure, see for details [11].  □

Remark 2.2

([11]). The functions C1, C2, C3 : R9R given by

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

and

C3(x¯)=x4x7+x5x8+x6x9

are the Casimirs of our Poisson configuration.

The goal of this paragraph is to study the spectral and nonlinear stability of the equilibrium state e6MNP of the controlled system (4).

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

A=0000x6x500000M000x90x72P0M0x5x40x8x7+2P00x6x50x3x2000x60x4x30x1M000x5x40x2x1+M00000x9x80000x3x2x90x7000x30x1Mx8x70000x2x1+M0.

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

pA(e2MNP)(λ)=λ5[λ4+(M2+N2+2P2)λ2+P2(N2+P2)].

Hence we have five zero eigenvalues and four purely imaginary eigenvalues. So we can conclude:

Proposition 2.3

The controlled system (4) may be spectral stabilized about the equilibrium states e6MNP for all M, N, PR.

Moreover we can prove:

Proposition 2.4

The controlled system (4) may be nonlinear stabilized about the equilibrium states e6MNP for all M, N, PR.

Proof

For the proof we shall use Arnold’s technique. Let us consider the following function

Fλ,μ,ν=C1+λH+μC2+νC3==12(x42+x52+x62)+λ2(x12+x22+x32+x52+x722Mx14Px7)++μ2(x72+x82+x92)+ν(x4x7+x5x8+x6x9).

The following conditions hold:

  1. Fλ,μ,ν(e6MNP)=0 iff μ=λ+N2P2,ν=NP;

  2. Considering now

    W=ker[dH(e6MNP)]ker[dC2(e6MNP)]ker[dC3(e6MNP)]==Span100000000,010000000,001000000,000010000,000001000,000000010,000000001,

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

v2Fλ,λ+N2P2,NP(e6MNP)vt=λa2+λb2+λc2+(1+λ)d2+e2+λ+N2P2f2+λ+N2P2g22NPfd2NPeg

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

2Fλ,λ+N2P2,NP(e6MNP)|W×W

is positive definite.

Therefore, via Arnold’s technique, the equilibrium states e6MNP , M, N, PR 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

xi(0)=Ai,i=1,9¯,(6)

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:

L(x(t))+N(x(t))=0,(7)

subject to the initial conditions of the type:

x(0)=A,AR - given real number,(8)

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:

H[L(X(t,p)),H(t,Ci),N(X(t,p))]L(x0(t))++p[L(x1(t,Ci))H(t,Ci)N(x0(t))]=0,(9)

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:

X(t,p)=x0(t)+px1(t,Ci).(10)

The following properties hold:

H[L(X(t,0)),H(t,Ci),N(X(t,0))]=L(x0(t))(11)

and

H[L(X(t,1)),H(t,Ci),N(X(t,1))]=L(x1(t,Ci))H(t,Ci)N(x0(t)).(12)

The governing equations of X0(t) and x1(t, Ci) can be obtained by equating the coefficients of p0 and p1, respectively:

L(x0(t))=0,x(0)=A,(13)

L(x1(t,Ci))=H(t,Ci)N(x0(t)),x1(0,Ci)=0,i=1,s¯.(14)

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:

Nx0(t)=i=1mhi(t)gi(t),(15)

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 [14] for details), consisting in the following steps:

First we consider one of the following expressions for x1(t, Ci):

x1(t,Ci)=i=1mHi(t,hj(t),Cj)gi(t),j=1,s¯,(16)

or

x1(t,Ci)=i=1mHi(y,gj(t),Cj)hi(t),j=1,s¯,x1(0,Ci)=0.(17)

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.

Next, by taking into account the equation (10), for p = 1, the first-order analytical approximate solution of the equations (7) - (8) is:

x¯(t,Ci)=X(t,1)=x0(t)+x1(t,Ci).(18)

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.

Definition 3.1

[15] We call an ε-approximate solution of the problem (7) on the domain (0, ∞) a smooth function x(t, Ci) of the form (18) which satisfies the following condition:

|R(t,x¯(t,Ci))|<ϵ,

together with the initial condition from Eq. (8), where the residual function R(t, x(t, Ci)) is obtained by substituting the Eq. (18) into Eq. (7), i.e.

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

Definition 3.2

([15]). We call a week ε-approximate solution of the problem (7) on the domain (0, ∞) a smooth function x(t, Ci) of the form (18) which satisfies the following condition:

0R2(t,x¯(t,Ci))dη<ϵ,

together with the initial condition from Eq. (8).

4 Application of Optimal Homotopy Asymptotic Method for solving the nonlinear differential system (4)

In order to solve the nonlinear differential 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:

Lx1(t)=x˙1(t)+K1x1(t),Lx2(t)=x˙2(t)+K1x2(t)+Mx3(t),Lx3(t)=x˙3(t)+K1x3(t)Mx2(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),Lx8(t)=x˙8(t)+K1x8(t)+Mx9(t),Lx9(t)=x˙9(t)+K1x9(t)Mx8(t),(19)

with K > 0, K1 > 0 the unknown parameters at this moment.

The corresponding linear equations for initial approximations xi0, i = 1, 9 can be obtained by means of the Eqs. (13), (19) and (6):

Lxi0(t)=0,xi0(0)=Ai,i=1,9¯,(20)

whose solutions are

x10(t)=A1eK1t,x20(t)=(A2cos(Mt)A3sin(Mt))eK1t,x30(t)=(A3cos(Mt)+A2sin(Mt))eK1t,x40(t)=(A4cos(Kt)A7sin(Kt))eK1t,x50(t)=(A5cos(Mt)A6sin(Mt))eK1t,x60(t)=(A6cos(Mt)+A5sin(Mt))eK1t,x70(t)=(A7cos(Kt)+A4sin(Kt))eK1t,x80(t)=(A8cos(Mt)A9sin(Mt))eK1t,x90(t)=(A9cos(Mt)+A8sin(Mt))eK1t,(21)

The corresponding nonlinear operators N[xi(t)], i = 1, 9 are obtained from the equations (4):

Nx1(t)=K1x1(t)+x5(t)x6(t),Nx2(t)=K1x2(t)x7(t)x9(t)+2Px9(t),Nx3(t)=K1x3(t)x4(t)x5(t)+x7(t)x8(t)2Px8(t),Nx4(t)=K1x4(t)Kx7(t)+x2(t)x6(t)x3(t)x5(t),Nx5(t)=K1x5(t)x1(t)x6(t)+x3(t)x4(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),Nx9(t)=K1x9(t)+x1(t)x8(t)x2(t)x7(t),(22)

such that

Lx1(t)+Nx1(t)=x˙1(t)+x5(t)x6(t),Lx2(t)+Nx2(t)=x˙2(t)x7(t)x9(t)+Mx3(t)+2Px9(t),Lx3(t)+Nx3(t)=x˙3(t)x4(t)x5(t)+x7(t)x8(t)Mx2(t)2Px8(t),Lx4(t)+Nx4(t)=x˙4(t)+x2(t)x6(t)x3(t)x5(t)Lx5(t)+Nx5(t)=x˙5(t)x1(t)x6(t)+x3(t)x4(t)+Mx6(t),Lx6(t)+Nx6(t)=x˙6(t)+x1(t)x5(t)x2(t)x4(t)Mx5(t),Lx7(t)+Nx7(t)=x˙7(t)+x2(t)x9(t)x3(t)x8(t),Lx8(t)+Nx8(t)=x˙8(t)x1(t)x9(t)+x3(t)x7(t)+Mx9(t)Lx9(t)+Nx9(t)=x˙9(t)+x1(t)x8(t)x2(t)x7(t)Mx8(t),

and therefore, substituting Eqs. (21) into Eqs. (22), we obtain

Nx10(t)=K1x10(t)+x50(t)x60(t),Nx20(t)=K1x20(t)x70(t)x90(t)+2Px90(t),Nx30(t)=K1x30(t)x40(t)x50(t)+x70(t)x80(t)2Px80(t),Nx40(t)=K1x40(t)Kx70(t)+x20(t)x60(t)x30(t)x50(t),Nx50(t)=K1x50(t)x10(t)x60(t)+x30(t)x40(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),Nx90(t)=K1x90(t)+x10(t)x80(t)x20(t)x70(t).(23)

Remark 4.1

Now, we observe that the nonlinear operators N[xi0(t)], i = 1, 9 are the linear combinations between the elementary functions eK1t · cos(Mt), eK1t · sin(Mt), e–2K1t · cos2(Mt), e–2K1t · sin2(Mt), e–2K1t · cos(Mt) sin(Mt), eK1t · cos(Kt), eK1t · 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:

L(xi1(t,Cj))=H(t,Cj)N(xi0(t)),xi1(0,Cj)=0,i=1,9¯,j=1,s¯,(24)

where the linear operators L are given by Eq. (19) and the expressions N(xi0(t)), i = 1, 9 are given by Eq. (23).

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:

xi1=n=17Bncos(2n+1)ωt+Cnsin(2n+1)ωteK1t,i=1,9¯.(25)

Using now the third-alternative of OHAM and the equations (18), the first-order approximate solution can be put in the form

x¯i(t,Ci)=xi0(t)+xi1(t,Ci),i=1,9¯(26)

where xi0(t) and xi1(t, Ci) are given by (21) and (25), respectively.

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),

Table 1

The comparison between the approximate solutions 3 given by Eq. (30) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx3 = |x3numerical3OHAM|)

Table 2

The comparison between the approximate solutions 5 given by Eq. (32) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx5 = |x5numerical5OHAM|)

Table 3

The comparison between the approximate solutions 8 given by Eq. (35) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx8 = |x8numerical8OHAM|)

Table 4

The numerical values of the integral of square residual function given by Eq. (27) corresponding to the approximate solutions given by Eqs. (28)-(36) for M = 15 and P = 20

01Ri2(t)dt , i = 1, …, 9,

where

R1(t)=x¯˙1(t)+x¯5(t)x¯6(t),R2(t)=x¯˙2(t)x¯7(t)x¯9(t)+Mx¯3(t)+2Px¯9(t),R3(t)=x¯˙3(t)x¯4(t)x¯5(t)+x¯7(t)x¯8(t)Mx¯2(t)2Px¯8(t),R4(t)=x¯˙4(t)+x¯2(t)x¯6(t)x¯3(t)x¯5(t),R5(t)=x¯˙5(t)x¯1(t)x¯6(t)+x¯3(t)x¯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),R8(t)=x¯˙8(t)x¯1(t)x¯9(t)+x¯3(t)x¯7(t)+Mx¯9(t),R9(t)=x¯˙9(t)+x¯1(t)x¯8(t)x¯2(t)x¯7(t)Mx¯8(t),(27)

with i(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 1 : The convergence-control parameters are respectively:

    B1=0.00243871503187,B2=0.00359273871118,B3=0.00052965854847,B4=0.00114795966364,B5=0.00056092131600,B6=6.47335130106,B7=0.00003434745346,B8=3.49402157106,C1=0.00350555321652,C2=0.00076136918414,C3=0.00251066466427,C4=0.00087065292393,C5=0.00027493879386,C6=0.00019640146954,C7=0.00002061370631,C8=1.94094963106,K=2.57114627223466,K1=0.58656793719790,ω=1.01132823106464.

    The first-order approximate solutions given by the Eq. (26) are respectively:

    x¯1(t)=e0.58656793719790t(0.0001+0.00243871503187cos(ωt)0.00359273871118cos(3ωt)+0.00052965854847cos(5ωt)++0.00114795966364cos(7ωt)0.00056092131600cos(9ωt)++6.47335130511943106cos(11ωt)+0.00003434745346cos(13ωt)3.49402157853902106cos(15ωt))+(0.00350555321652sin(ωt)0.00076136918414sin(3ωt)+0.00251066466427sin(5ωt)0.00087065292393sin(7ωt)0.00027493879386sin(9ωt)++0.00019640146954sin(11ωt)0.00002061370631sin(13ωt)1.94094963106sin(15ωt))e0.58656793719790t.(28)

    For all unknown functions i, i = 1, 9, we have K1 = 0.58656793719790 and ω = 1.01132823106464.

  • for 2:

    x¯2(t)=e0.58656793719790t(0.46417071707583cos(ωt)++0.55436638784881cos(3ωt)+0.23179980739892cos(5ωt)0.52457790349642cos(7ωt)+0.20239755363119cos(9ωt)++0.03102295410884cos(11ωt)0.04104869574268cos(13ωt)++0.01021061332716cos(15ωt))+e0.58656793719790t(0.0001cos(15t)0.0001sin(15t))+e0.586567937197t(0.42041532210572sin(ωt)++0.47464526230797sin(3ωt)0.66980627335479sin(5ωt)++0.08449154306675sin(7ωt)+0.23139718247432sin(9ωt)0.13429737097902sin(11ωt)+0.02641377996654sin(13ωt)0.00262253548558sin(15ωt)).(29)

  • for 3:

    x¯3(t)=e0.58656793719790t(1.62690195527876cos(ωt)2.91905245722799cos(3ωt)+1.41052417394739cos(5ωt)++0.18894557284618cos(7ωt)0.45976960570592cos(9ωt)++0.17490980799902cos(11ωt)0.02436756038517cos(13ωt)++0.00190811324772cos(15ωt))+e0.58656793719790t(0.0001cos(15t)++0.0001sin(15t))+e0.58656793719790t(3.16149837021824sin(ωt)++0.42959981904576sin(3ωt)+1.45120190775009sin(5ωt)1.12246042334992sin(7ωt)+0.24077988316763sin(9ωt)++0.07360271218550sin(11ωt)0.05047592981365sin(13ωt)++0.01049752842422sin(15ωt)).(30)

  • for 4:

    x¯4(t)=e0.58656793719790t(1.94037601106cos(ωt)++0.00001098412604cos(3ωt)0.00001943075801cos(5ωt)++7.22410959926685106cos(7ωt)4.23016798075448107cos(9ωt)3.67489997706953107cos(11ωt)+7.48336663534119108cos(13ωt)2.18052201200004109cos(15ωt))+e0.58656793719790t(0.0001cos(0.80363574138731t)0.0001sin(0.80363574138731t))+e0.58656793719790t(0.00024662613164sin(ωt)0.00005660016305sin(3ωt)+0.00001080462985sin(5ωt)++3.31362820627460106sin(7ωt)2.30237733420243106sin(9ωt)++3.66350337276902107sin(11ωt)+1.43826553061835108sin(13ωt)5.09710156758535109sin(15ωt)).(31)

  • for 5:

    x¯5(t)=e0.58656793719790t(0.01369917160029cos(ωt)0.02516198187512cos(3ωt)+0.01340403032961cos(5ωt)++0.00013038596714cos(7ωt)0.00335904322615cos(9ωt)++0.00158445050516cos(11ωt)0.00035976290254cos(13ωt)++0.00006274960160cos(15ωt))+e0.58656793719790t(0.0001cos(15t)0.0001Sin(15t))+e0.58656793719790t(0.02795989598259sin(ωt)++0.00526407948570sin(3ωt)+0.01119295776584sin(5ωt)0.00969533499261sin(7ωt)+0.00279014392623sin(9ωt)++0.00019457655388sin(11ωt)0.00035980785989sin(13ωt)++0.00010360505021sin(15ωt)).(32)

  • for 6:

    x¯6(t)=e0.58656793719790t(0.01331723000622cos(ωt)0.02136690535515cos(3ωt)+0.00599018344061cos(5ωt)++0.00533244090418cos(7ωt)0.00400304532847cos(9ωt)++0.00056272420145cos(11ωt)+0.00027367239189cos(13ωt)0.00010630026076cos(15ωt))+e0.58656793719790t(0.0001cos(15t)++0.0001sin(15t))+e0.58656793719790t(0.02188514843412sin(ωt)0.00146001826477sin(3ωt)+0.01365871149717sin(5ωt)0.00673797188735sin(7ωt)0.00067903495719sin(9ωt)++0.00146666027085sin(11ωt)0.00045540606550sin(13ωt)++0.00007291228376sin(15ωt)).(33)

  • for 7:

    x¯7(t)=e0.58656793719790t(0.00014174642928cos(ωt)0.00012283316033cos(3ωt)0.00002041841606cos(5ωt)++9.42449906107cos(7ωt)+8.49879531107cos(9ωt)3.23563916107cos(11ωt)+3.63337271108cos(13ωt)++4.786593011011cos(15ωt))+(0.0001cos(3.72653706867179t)++0.0001sin(3.72653706867179t))e0.58656793719790t++(0.00013001568894sin(ωt)0.00011137613515sin(3ωt)0.00002806958703sin(5ωt)+6.45157686106sin(7ωt)1.35713708106sin(9ωt)+6.39039531108sin(11ωt)+2.93113481108sin(13ωt)3.32424397109sin(15ωt))e0.58656793719790t.(34)

  • for 8:

    x¯8(t)=e0.58656793719790t(0.01369917160029cos(ωt)0.02516198187512cos(3ωt)+0.01340403032961cos(5ωt)++0.00013038596714cos(7ωt)0.00335904322615cos(9ωt)++0.00158445050516cos(11ωt)0.00035976290254cos(13ωt)++0.00006274960160cos(15ωt))+e0.58656793719790t(0.0001cos(15t)0.0001sin(15t))+e0.58656793719790t(0.02795989598259sin(ωt)++0.00526407948570sin(3ωt)+0.01119295776584sin(5ωt)0.00969533499261sin(7ωt)+0.00279014392623sin(9ωt)++0.00019457655388sin(11ωt])0.00035980785989sin(13ωt)++0.00010360505021sin(15ωt).(35)

  • for 9:

    x¯9(t)=e0.58656793719790t(0.01331723000622cos[ωt)0.02136690535515cos(3ωt)+0.00599018344061cos(5ωt)++0.00533244090418cos(7ωt)0.00400304532847cos(9ωt)++0.00056272420145cos(11ωt)+0.00027367239189cos(13ωt)0.00010630026076cos(15ωt))+e0.58656793719790t(0.0001cos(15t)++0.0001sin(15t))+e0.58656793719790t(0.02188514843412sin(ωt)0.00146001826477sin(3ωt)+0.01365871149717sin(5ωt)0.00673797188735sin(7ωt)0.00067903495719sin(9ωt)++0.00146666027085sin(11ωt)0.00045540606550sin(13ωt)++0.00007291228376sin(15ωt)).(36)

Finally, Tables 1, 2 and 3 emphasizes the accuracy of the OHAM technique by comparing the approximate analytic solutions 3, 5 and 8 respectively presented above with the corresponding numerical integration values.

The Figs. 1-9 depicted a comparison between the obtained approximate solutions given by Eqs. (28)-(36) with corresponding numerical integration.

Comparison between the approximate solutions x̄1 given by Eq. (28) and the corresponding numerical solutions
Fig. 1

Comparison between the approximate solutions 1 given by Eq. (28) and the corresponding numerical solutions

Comparison between the approximate solutions x̄2 given by Eq. (29) and the corresponding numerical solutions
Fig. 2

Comparison between the approximate solutions 2 given by Eq. (29) and the corresponding numerical solutions

Comparison between the approximate solutions x̄3 given by Eq. (30) and the corresponding numerical solutions
Fig. 3

Comparison between the approximate solutions 3 given by Eq. (30) and the corresponding numerical solutions

Comparison between the approximate solutions x̄4 given by Eq. (31) and the corresponding numerical solutions
Fig. 4

Comparison between the approximate solutions 4 given by Eq. (31) and the corresponding numerical solutions

Comparison between the approximate solutions x̄5 given by Eq. (32) and the corresponding numerical solutions
Fig. 5

Comparison between the approximate solutions 5 given by Eq. (32) and the corresponding numerical solutions

Comparison between the approximate solutions x̄6 given by Eq. (33) and the corresponding numerical solutions
Fig. 6

Comparison between the approximate solutions 6 given by Eq. (33) and the corresponding numerical solutions

Comparison between the approximate solutions x̄7 given by Eq. (34) and the corresponding numerical solutions
Fig. 7

Comparison between the approximate solutions 7 given by Eq. (34) and the corresponding numerical solutions

Comparison between the approximate solutions x̄8 given by Eq. (35) and the corresponding numerical solutions
Fig. 8

Comparison between the approximate solutions 8 given by Eq. (35) and the corresponding numerical solutions

Comparison between the approximate solutions x̄9 given by Eq. (36) and the corresponding numerical solutions
Fig. 9

Comparison between the approximate solutions 9 given by Eq. (36) and the corresponding numerical solutions

6 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.

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About the article

Received: 2016-11-23

Accepted: 2018-01-19

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.

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© 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

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