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

  • Camelia Pop and Remus-Daniel Ene EMAIL logo
Published/Copyright: March 20, 2018
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Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

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=000000−1001000000,A2=00100000−10000000,A3=0−100100000000000,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)=12∫0tfc1u12t+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=x4x5−x7x8x˙4=−x2x6+x3x5x˙5=x1x6−x3x4x˙6=−x1x5+x2x4x˙7=−x2x9+x3x8x˙8=x1x9−x3x7x˙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,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,Q∈R,e2MNP=(0,0,M,0,0,N,0,0,P),M,N,P∈R,e3MPQ=(0,0,0,0,M,0,0,P,Q),M,P,Q∈R,e4MNP=(0,M,0,0,N,0,0,P,0),M,N,P∈R,e5MNP=(M,N,P,0,0,0,0,0,0),M,N,P∈R,

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 u ∈ C∞ (R9, R),

u(x¯)=(0,−Mx3−2Px9,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=x7x9−Mx3−2Px9x˙3=x4x5−x7x8+Mx2+2Px8x˙4=−x2x6+x3x5x˙5=x1x6−x3x4−Mx6x˙6=−x1x5+x2x4+Mx5x˙7=−x2x9+x3x8x˙8=x1x9−x3x7−Mx9x˙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),

whereG = SO(3) × R3 × R3,

Π=0−x3x20−x6x50−x9x8x30−x1x60−x4x90−x7−x2x10−x5x40−x8x700−x6x5000000x60−x4000000−x5x400000000−x9x8000000x90−x7000000−x8x70000000(5)

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

H(x¯)=12(x12+x22+x32+x52+x72)−Mx1−2Px7.

Proof

Indeed, one obtains immediately that

Π⋅∇H=[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 functionsC1, C2, C3 : R9 → Rgiven 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=0000−x6−x500000−M000x90x7−2P0M0x5x40−x8−x7+2P00−x6x50x3−x2000x60−x4−x30x1−M000−x5x40x2−x1+M00000−x9x80000x3−x2x90−x7000−x30x1−M−x8x70000x2−x1+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 statese6MNPfor allM, N, P ∈ R∗.

Moreover we can prove:

Proposition 2.4

The controlled system(4)may be nonlinear stabilized about the equilibrium statese6MNPfor allM, N, P ∈ R∗.

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+x72−2Mx1−4Px7)++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 v ∈ W, i.e. v = (a, b, c, 0, d, e,0, f, g), a, b, c, d, e, f, g ∈ R we have

v⋅∇2Fλ,λ+N2P2,−NP(e6MNP)⋅vt=λa2+λb2+λc2+(1+λ)d2+e2+λ+N2P2f2+λ+N2P2g2−2NPfd−2NPeg

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, 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

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,A∈R - 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 anε-approximate solutionof the problem (7) on the domain (0, ∞) a smooth functionx(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 functionR(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 aweekε-approximate solutionof the problem (7) on the domain (0, ∞) a smooth functionx(t, Ci) of the form (18) which satisfies the following condition:

∫0∞R2(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)=A1⋅e−K1t,x20(t)=(A2cos⁡(Mt)−A3sin⁡(Mt))⋅e−K1t,x30(t)=(A3cos⁡(Mt)+A2sin⁡(Mt))⋅e−K1t,x40(t)=(A4cos⁡(Kt)−A7sin⁡(Kt))⋅e−K1t,x50(t)=(A5cos⁡(Mt)−A6sin⁡(Mt))⋅e−K1t,x60(t)=(A6cos⁡(Mt)+A5sin⁡(Mt))⋅e−K1t,x70(t)=(A7cos⁡(Kt)+A4sin⁡(Kt))⋅e−K1t,x80(t)=(A8cos⁡(Mt)−A9sin⁡(Mt))⋅e−K1t,x90(t)=(A9cos⁡(Mt)+A8sin⁡(Mt))⋅e−K1t,(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, 9are the linear combinations between the elementary functionse–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:

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)ωt⋅e−K1t,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 weekε-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 x̄3 given by Eq. (30) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx3 = |x3numerical – x̄3OHAM|)

tX3numericalx3oham given by Eq.(30)εx3
00.00010.00010
1/10–0.00026388686019–0.000263889207322.34712908 ·10–9
1/5–0.00098972163690–0.000989727431355.79444527 ·10–9
3/100.000801215911930.000801215167557.44372891 ·10–10
2/50.002051131771450.002051139539037.76758435 ·10–9
1/2–0.00105407568168–0.001054075784831.03151043 ·10–10
3/5–0.00322482848832–0.003224831701273.21294763 ·10–9
7/100.000995737819770.000995738978781.15901067 ·10–9
4/50.004446042758390.004446049089796.33140294 ·10–9
9/10–0.00061159885752–0.000611597462431.39509107 ·10–9
1–0.00564686848672–0.005646876859418.37269479 ·10–9

Table 2

The comparison between the approximate solutions x̄5 given by Eq. (32) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx5 = |x5numerical – x̄5OHAM|)

tX5numericalx5oham given by Eq.(32)εx5
00 0.00010.00010
1/10–0.00009267394569–0.000092673840411.05273724 ·10–10
1/5–0.00011309924594–0.000113099661294.15348823 ·10–10
3/100.000076658349930.000076658510281.60350492 ·10–10
2/50.000123915987330.000123916360933.73592167 ·10–10
1/2–0.00005910519311–0.000059105370991.77875786 ·10–10
3/5–0.00013222655382–0.000132226673161.19336773 ·10–10
7/100.000040374303820.000040374169071.34745601 ·10–10
4/50.000137862223590.000137862396901.73302447 ·10–10
9/10–0.00002084981273–0.000020849840222.74919318 ·10–11
1–0.00014071144636–0.000140711225442.20921854 ·10–10

Table 3

The comparison between the approximate solutions x̄8 given by Eq. (35) and the corresponding numerical solutions for M = 15 and P = 20 (relative errors: εx8 = |x8numerical – x̄8OHAM|)

tX8numericalx8oham given by Eq.(32)εx8
00.00010.00010
1/10–0.00009267394569–0.000092673840411.05273724 ·10–10
1/5–0.00011309924594–0.000113099661294.15348823 ·10–10
3/100.000076658349930.000076658510281.60350492 ·10–10
2/50.000123915987330.000123916360933.73592167 ·10–10
1/2–0.00005910519311–0.000059105370991.77875786 ·10–10
3/5–0.00013222655382–0.000132226673161.19336773 ·10–10
7/100.000040374303820.000040374169071.34745601 ·10–10
4/50.000137862223590.000137862396901.73302447 ·10–10
9/10–0.00002084981273–0.000020849840222.74919318 ·10–11
1–0.00014071144636–0.000140711225442.20921854 ·10–10

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

i∫01Ri2(t)dt
17.464737808519274 ·10–17
28.459118984995915 ·10–13
31.0225368942228159 ·10–13
42.6769875640036004 ·10–23
59.659432242298964 ·10–17
68.086505295586057 ·10–17
71.4525406989853368 ·10–23
89.659432355712418 ·10–17
98.086505295410683 ·10–17

∫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 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.

  1. for x̄1 : The convergence-control parameters are respectively:

    B1=0.00243871503187,B2=−0.00359273871118,B3=0.00052965854847,B4=0.00114795966364,B5=−0.00056092131600,B6=6.47335130⋅10−6,B7=0.00003434745346,B8=−3.49402157⋅10−6,C1=−0.00350555321652,C2=−0.00076136918414,C3=0.00251066466427,C4=−0.00087065292393,C5=−0.00027493879386,C6=0.00019640146954,C7=−0.00002061370631,C8=−1.94094963⋅10−6,K=2.57114627223466,K1=0.58656793719790,ω=1.01132823106464.

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

    x¯1(t)=e−0.58656793719790t(0.0001+0.00243871503187cos⁡(ωt)−−0.00359273871118cos⁡(3ωt)+0.00052965854847cos⁡(5ωt)++0.00114795966364cos⁡(7ωt)−0.00056092131600cos⁡(9ωt)++6.47335130511943⋅10−6cos⁡(11ωt)+0.00003434745346cos⁡(13ωt)−−3.49402157853902⋅10−6cos⁡(15ωt))+(−0.00350555321652sin⁡(ωt)−−0.00076136918414sin⁡(3ωt)+0.00251066466427sin⁡(5ωt)−−0.00087065292393sin⁡(7ωt)−0.00027493879386sin⁡(9ωt)++0.00019640146954sin⁡(11ωt)−0.00002061370631sin⁡(13ωt)−−1.94094963⋅10−6sin⁡(15ωt))e−0.58656793719790t.(28)

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

  2. for x̄2:

    x¯2(t)=e−0.58656793719790t(−0.46417071707583cos⁡(ωt)++0.55436638784881cos⁡(3ωt)+0.23179980739892cos⁡(5ωt)−−0.52457790349642cos⁡(7ωt)+0.20239755363119cos⁡(9ωt)++0.03102295410884cos⁡(11ωt)−0.04104869574268cos⁡(13ωt)++0.01021061332716cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)−−0.0001sin⁡(15t))+e−0.586567937197t(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)

  3. for x̄3:

    x¯3(t)=e−0.58656793719790t(1.62690195527876cos⁡(ωt)−−2.91905245722799cos⁡(3ωt)+1.41052417394739cos⁡(5ωt)++0.18894557284618cos⁡(7ωt)−0.45976960570592cos⁡(9ωt)++0.17490980799902cos⁡(11ωt)−0.02436756038517cos⁡(13ωt)++0.00190811324772cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)++0.0001sin⁡(15t))+e−0.58656793719790t(−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)

  4. for x̄4:

    x¯4(t)=e−0.58656793719790t(1.94037601⋅10−6cos⁡(ωt)++0.00001098412604cos⁡(3ωt)−0.00001943075801cos⁡(5ωt)++7.22410959926685⋅10−6cos⁡(7ωt)−4.23016798075448⋅10−7cos⁡(9ωt)−−3.67489997706953⋅10−7cos⁡(11ωt)+7.48336663534119⋅10−8cos⁡(13ωt)−−2.18052201200004⋅10−9cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(0.80363574138731t)−−0.0001sin⁡(0.80363574138731t))+e−0.58656793719790t(0.00024662613164sin⁡(ωt)−−0.00005660016305sin⁡(3ωt)+0.00001080462985sin⁡(5ωt)++3.31362820627460⋅10−6sin⁡(7ωt)−2.30237733420243⋅10−6sin⁡(9ωt)++3.66350337276902⋅10−7sin⁡(11ωt)+1.43826553061835⋅10−8sin⁡(13ωt)−−5.09710156758535⋅10−9sin⁡(15ωt)).(31)

  5. for x̄5:

    x¯5(t)=e−0.58656793719790t(0.01369917160029cos⁡(ωt)−−0.02516198187512cos⁡(3ωt)+0.01340403032961cos⁡(5ωt)++0.00013038596714cos⁡(7ωt)−0.00335904322615cos⁡(9ωt)++0.00158445050516cos⁡(11ωt)−0.00035976290254cos⁡(13ωt)++0.00006274960160cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)−−0.0001Sin(15t))+e−0.58656793719790t(−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)

  6. for x̄6:

    x¯6(t)=e−0.58656793719790t(0.01331723000622cos⁡(ωt)−−0.02136690535515cos⁡(3ωt)+0.00599018344061cos⁡(5ωt)++0.00533244090418cos⁡(7ωt)−0.00400304532847cos⁡(9ωt)++0.00056272420145cos⁡(11ωt)+0.00027367239189cos⁡(13ωt)−−0.00010630026076cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)++0.0001sin⁡(15t))+e−0.58656793719790t(−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)

  7. for x̄7:

    x¯7(t)=e−0.58656793719790t(0.00014174642928cos⁡(ωt)−−0.00012283316033cos⁡(3ωt)−0.00002041841606cos⁡(5ωt)++9.42449906⋅10−7cos⁡(7ωt)+8.49879531⋅10−7cos⁡(9ωt)−−3.23563916⋅10−7cos⁡(11ωt)+3.63337271⋅10−8cos⁡(13ωt)++4.78659301⋅10−11cos⁡(15ωt))+(0.0001cos⁡(3.72653706867179t)++0.0001sin⁡(3.72653706867179t))e−0.58656793719790t++(0.00013001568894sin⁡(ωt)−0.00011137613515sin⁡(3ωt)−−0.00002806958703sin⁡(5ωt)+6.45157686⋅10−6sin⁡(7ωt)−1.35713708⋅10−6sin⁡(9ωt)+6.39039531⋅10−8sin⁡(11ωt)+2.93113481⋅10−8sin⁡(13ωt)−3.32424397⋅10−9sin⁡(15ωt))e−0.58656793719790t.(34)

  8. for x̄8:

    x¯8(t)=e−0.58656793719790t(0.01369917160029cos⁡(ωt)−−0.02516198187512cos⁡(3ωt)+0.01340403032961cos⁡(5ωt)++0.00013038596714cos⁡(7ωt)−0.00335904322615cos⁡(9ωt)++0.00158445050516cos⁡(11ωt)−0.00035976290254cos⁡(13ωt)++0.00006274960160cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)−−0.0001sin⁡(15t))+e−0.58656793719790t(−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)

  9. for x̄9:

    x¯9(t)=e−0.58656793719790t(0.01331723000622cos⁡[ωt)−−0.02136690535515cos⁡(3ωt)+0.00599018344061cos⁡(5ωt)++0.00533244090418cos⁡(7ωt)−0.00400304532847cos⁡(9ωt)++0.00056272420145cos⁡(11ωt)+0.00027367239189cos⁡(13ωt)−−0.00010630026076cos⁡(15ωt))+e−0.58656793719790t(0.0001cos⁡(15t)++0.0001sin⁡(15t))+e−0.58656793719790t(−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 x̄3, x̄5 and x̄8 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Comparison between the approximate solutions x̄9 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.

  1. Conflict of interest

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

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Received: 2016-11-23
Accepted: 2018-01-19
Published Online: 2018-03-20

© 2018 Pop and Ene, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2018-0019
Keywords for this article
Nonlinear ordinary differential systems; Nonlinear stability; Lie group; Optimal homotopy asymptotic method
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  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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