On comparison between iterative methods for solving

Article Journal of Vibration and Control 1–7 ! The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546315590039 jvc.sagepub.com

On comparison between iterative methods for solving nonlinear optimal control problems Hossein Jafari1,2, Saber Ghasempour1 and Dumitru Baleanu3,4

Abstract Recently some semi-analytical methods have been introduced for solving a class of nonlinear optimal control problems such as the Adomian decomposition method, homotopy perturbation method and modified variational iteration method. In this manuscript we compare these methods for solving a type of nonlinear optimal control problem. We prove that these methods are equivalent, which means that they use the same iterative formula to obtain the approximate/analytical solution.

Keywords Nonlinear optimal control problem, Adomian decomposition methods, homotopy perturbation method, Pontryagin’s maximum principle, variational homotopy perturbation method

1. Introduction The theory of optimal control and its applications are now widely used in multi-disciplinary applications such as aircraft systems, biomedicine, robotics, etc. Optimal control of nonlinear systems is a challenging task which has been widely studied. It is known that a nonlinear optimal control problem (OCP) can be converted to a Hamilton–Jacobi–Bellman (HJB) partial differential equation or a nonlinear two-point boundary value problem (TPBVP). Many researchers have attempted to solve these two problems (see Diehl et al., 2005; Fakharian and Behshti, 2008; Fakharian et al., 2010; Fakharzadeh, 2012; Ha¨ma¨laïnen and Halme, 1976; Jajarmi et al., 2011; Manseur et al., 2005; Matinfar and Saeidy, 2014; Nik et al., 2012, and the references therein). Several computational methods have been used for solving nonlinear OCPs. Fakharian et al. used the modified Adomian decomposition method (ADM) (Fakharian and Behshti, 2008; Fakharian et al., 2010), Saberi Nik et al. applied the homotopy perturbation method (HPM) (Nik et al., 2012), Jajarmi et al. used the series method (Jajarmi et al., 2011), Manseur et al. applied the coupled Adomian/Alienor method (Manseur et al., 2005), Fakharzadeh used differential transformation method (DTM) (Fakharzadeh, 2012)

and so on (Diehl et al., 2005; Ha¨ma¨laïnen and Halme, 1976). Recently Matinfar and Saeidy presented a new technique for solving a class of nonlinear OCP (Matinfar and Saeidy, 2014), which they call the modified variational iteration method (MVIM). This method is similar to an existing method called the variational homotopy perturbation method (VHPM) (Matinfar and Ghasemi, 2010; Matinfar et al., 2010; Noor and Mohyud-Din, 2008). In Matinfar and Saeidy, (2014) the following nonlinear OCP was considered Z

tf

min J ¼

ðxT ðtÞQxðtÞ þ uT ðtÞRuðtÞÞdt

t0

1

Department of Mathematics, University of Mazandaran, Babolsar, Iran Department of Mathematical Sciences, University of South Africa, Pretoria, South Africa 3 Department of Mathematics and Computer Science, C ¸ ankaya University, Ankara, Turkey 4 Institute of Space Sciences, Magurele-Bucharest, Romania 2

Received: 16 November 2014; accepted: 6 May 2015 Corresponding author: Hossein Jafari, Department of Mathematics, University of Mazandaran, PO Box 47416-95447, Babolsar, Iran. Email: [email protected]

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Journal of Vibration and Control s:t:

x_ ¼ Fðt, xðtÞÞ þ Gðt, xðtÞÞuðtÞ; xðt0 Þ ¼ x0 ,

t 2 ½t0 , tf xðtf Þ ¼ xf

ð1Þ

where x 2 Rn and u 2 Rm are the state and control vectors respectively, Q 2 Rnn and R 2 Rmm are positive semi-definite and positive definite matrices respectively, and x0 and xf are the given initial and final states at t0 and tf respectively. Also, F and G are continuously differentiable functions in all arguments. According to Pontryagin’s maximum principle (Pinch, 1993), and following the method presented in Jajarmi et al., (2011), they converted equation (1) into the following form of nonlinear TPBVP

We want to obtain a solution u of equation (3) in Hilbert space. If equation (3) does not have a unique solution, then these methods give only a solution among other possible solutions.

2.1. The ADM for solving equation (3) For solving equation (3) using the ADM (Abbaoui and Cherruault, 1994; Adomian Ret al., 1996; Wazwaz, t n1 1 2011), we apply L1 ½: ¼ ðn1Þ! ½:d on both 0 ðt Þ side of equation (3). Thus uðtÞ ¼

m 1 X

ck

k¼0

x_ ¼ Fðt,xÞ þ Gðt,xÞ½R1 gT ðt,xÞy, @Fðt,xÞT y_ ¼ Qx þ y @x ! n X @G i T þ yi ½R1 GT ðt, xÞy , @x i¼1

tk þ L1 ð gðtÞÞ L1 ðR½uðtÞÞ k!

ð5Þ

1

L ðN½uðtÞÞ, t 4 0

ð2Þ

The ADM consists of the solution to equation (5) as an infinite series uðxÞ ¼

1 X

ui ðxÞ

ð6Þ

i¼0

xðt0 Þ ¼ x0 , yðt0 Þ ¼

and N(u(x)) is also decomposed as where 2 R is an unknown parameter which will be determined later by using boundary conditions. Finally, they introduced an MVIM for solving equation (2). In this paper, we prove that the ADM, the HPM and the VHPM are equivalent when used to solve nonlinear differential equations with initial or boundary conditions. Equivalent means these methods use the same iterative formula for solving a class of nonlinear OCP.

2. The methods In Section 1, we saw that the nonlinear OCP (1) can be converted to a nonlinear TPBVP (2). In this section, we briefly recall the ADM, the HPM and the VHPM for solving equation (2). To convey the basic idea of the above methods, we applied these methods to the following nonlinear equation L½uðtÞ þ R½uðtÞ þ N½uðtÞ ¼ gðtÞ,

t40

NðuðxÞÞ ¼

m

u ð0Þ ¼ ck ,

k ¼ 0, 1, 2, . . . , m 1

ð7Þ

where An, n ¼ 1,2,3,. . . are called the Adomian polynomials, which are calculated by Abbaoui and Cherruault (1994), Adomian et al. (1996), and Wazwaz (2011) as X n 1 dn i An ¼ u p N i n! dpn p¼0 i¼0

ð8Þ

Here p is a parameter introduced for convenience. Substituting equations (6) and (7) into equation (5) yields X 1 m 1 1 X X tk 1 1 ui ðtÞ ¼ ck þ L ð gðtÞÞ L ui ðtÞ R k! i¼0 i¼0 k¼0 X 1 1 L Ai ð9Þ i¼0

In view of the convergence of the series into equation (9), the components of the series in equation (6) are computed by following formula u0 ¼

m 1 X

ck

k¼0

tk þ L1 ð gðtÞÞ k!

1

1

unþ1 ¼ L ðR½un Þ L ðAn Þ, ðkÞ

Ai

i¼0

ð3Þ

where L ¼ dtd m , m 2 N is a linear operator with the highest order derivative, R[u(t)] (remainder linear term) is the linear differential operator of lower order than m, N[u(t)] is a nonlinear operator and g(t) is the source inhomogeneous term, subject to the initial conditions

1 X

ð10Þ n ¼ 0, 1, 2, . . .

ð4Þ When the independent variable (time) is unbounded, the series solution of equation (6) will diverge from


Jafari et al.

3

the true solution at larger values of time. This is where the discretization of the time axis makes itself indispensable. An estimate of local error P over a particular n time interval is given by Errorl ¼ 1 i¼nþ1 ui Oðh Þ. The global error order is one integral order fewer than the corresponding local error order. It is Errorl O(d hn1). So it can achieve a more accurate solution and get a higher rate of convergence by increasing the number of series terms (Az-Zo’bi, 2012; Ramana and Raghu Prasad, 2014). For convergence of this method we refer to Abbaoui and Cherruault (1994), Adomian et al. (1996), and Az-Zo’bi (2012).

which a few authors call He’s polynomials! It must be mentioned here that the authors of this paper have proved that He’s polynomials are only Adomian polynomials (Jafari et al., 2013). Substituting equations (13) and (15) into equation (11) or (12) and arranging it according to the powers of p, we have

2.2. The HPM for solving equation (3)

pn : Lðn Þ þ Rðn1 Þ þ Hn1 ¼ 0,

The HPM was developed by combining two techniques: standard homotopy and perturbation. For solving equation (3) according to He’s HPM (He, 1999), we first construct a homotopy as Hð; pÞ ¼ ð1 pÞ½LðÞ Lðu0 Þ þ p½LðÞ þ RðÞ þ NðÞ gðtÞ ¼ 0

ð12Þ

ð13Þ

is considered as the solution of equation (12). Substituting p ¼ 1 into equation (11) gives our original equation (3). Also, as p tends to 1 in equation (13) we have uðtÞ ¼ lim ¼ 0 þ 1 þ 2 þ p!1

p2 : Lð2 Þ þ Rð1 Þ þ H1 ¼ 0, 2 ðkÞ ð0Þ ¼ 0,

k ¼ 0, 1, 2, . . . , m 1

.. . n ðkÞ ð0Þ ¼ 0,

n ¼ 2, 3, . . .

ð17Þ

By solving the above equations, we obtain the components i, i ¼ 0,1,2, . . . of equation (13). For convergence of this method we refer to He (1999).

2.3. The VHPM for solving equation (3)

where p 2 [0,1] and u0 is an initial guess of equation (3), which satisfies equation (4). In the HPM, a power series of p ¼ 0 þ 1 p þ 2 p2 þ

p1 : Lð1 Þ þ Lðu0 Þ þ Rð0 Þ þ H0 gðtÞ ¼ 0

ð11Þ

or Hð; pÞ ¼ LðÞ Lðu0 Þ þ pLðu0 Þ þ p½RðÞ þ NðÞ gðtÞ ¼ 0

p0 : Lð0 Þ Lðu0 Þ ¼ 0

ð14Þ

Now we briefly describe an alternative approach of the variational iteration method (VIM) which is called MVIM (Matinfar and Saeidy, 2014) or VHPM (Matinfar and Ghasemi, 2010; Matinfar et al., 2010). This method is proposed as the coupling of the VIM and the HPM. For solving equation (3) using the VHPM, first according to He’s VIM (He, 2007; Inokuti et al., 1978), a correction function for equation (3) is constructed as Z unþ1 ðtÞ ¼ un ðtÞ þ þ

t

lðÞfLun ðÞ þ R un ðÞ

0 N un ðÞ

gðÞgd,

ð18Þ

n0

where l is a general Lagrangian multiplier, which can be identified optimally via variational theory. Here, restricted variation is applied for the nonlinear term Nu. Then we can determine l easily. In general (Wazwaz, 2010), we have

Like the ADM, N() is decomposed as NðÞ ¼

1 X

l¼ pi Hi ¼ H0 þ pH1 þ p2 H2 þ

ð15Þ

i¼0

X n 1 @n i , p N i n n! @p p¼0 i¼0

n ¼ 0, 1, 2, . . .

ð19Þ

After finding the value of l, unlike the VIM and similar to the HPM, we decompose the solution of equation (3) as the following series

where Hn is calculated as Hn ð0 , 1 , 2 , . . . , n Þ ¼

ð1Þm ð tÞðm1Þ ðm 1Þ!

¼ 0 þ 1 p þ 2 p2 þ ð16Þ

ð20Þ

Substituting p ¼ 1 into equation (20), yields the approximate solution of equation (18). Also, the


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Journal of Vibration and Control

nonlinear term is written as NðÞ ¼ similar to the HPM, we have 1 X

n

Z

t

n p ¼ u0 þ p

i¼0

Hi pi . Now,

X 1 n lðÞ R n p

0

n¼0

P1

Proof. See Jafari et al., (2013).

n¼0

# X 1 n þN n p gðÞ d

ð21Þ

n¼0

Finally, by sorting coefficients with respect to powers of p, we have 0

p :

p2 :

Theorem 2. The HPM for solving equation (3) is equivalent to the ADM when the homotopy H(;p) is considered as equation (11). Proof. Applying L1 to both sides of equation (17) we have 0 ¼

0 ¼ u0 Zt 1 ¼ lðÞ½Rð0 Þ þ H0 ð0 Þ gðÞd 0 Zt 2 ¼ lðÞ½Rð1 Þ þ H1 ð0 , 1 Þd

p1 :

Theorem 1. The He’s polynomials equation (16) is the Adomian’s polynomials equation (8).

m 1 X

ck k t , k! k¼0

1 ¼ L1 R½0 L1 H0 þ L1 gðtÞ, ð25Þ

2 ¼ L1 R½1 L1 H1 , .. .

0

n ¼ L1 R½n1 L1 Hn1

.. .

Z

pn :

t

lðÞ½Rðn1 Þ þ Hn1 ð0 , 1 , . . . , n1 Þd

n ¼ 0

According to Theorem 1 we have Hn ¼ An. In view of equations (9) and (13) we have

ð22Þ which is called the VHPM using He’s polynomials. For the selective zeroth approximation 0 we used the initial values from equation (4). In the VHPM the initial approximation 0 has been selected as 0 ðtÞ ¼

p!1

1 X

i pi ¼

i¼0

m 1 X

ck k t þ L1 ½ gðÞ k! k¼0

L1 Rð0 Þ L1 A0 1 X ¼ u0 þ u1 þ ¼ ui ¼ u i¼0

m 1 X

ck k t k! k¼0

ð23Þ

For convergence of this method we refer to Matinfar and Ghasemi (2010).

Hð; pÞ ¼

As we discussed in Section 2, those methods assumed the solution of equation (3) as an infinite series, computing the components of the series by using an iterative formula. Now, we want to prove analytically that the ADM, HPM and VHPM use the same iterative formula to obtain an approximate/analytical solution of equation (3). Definition 1. The well-known Cauchy formula for an nfold integral (Wazwaz, 2011) Z x1 Z xn1 1 dx1 dx2 . . . f ðxn Þdxn ¼ ðn 1Þ! a a Zxa ðx tÞn1 f ðtÞdt

Theorem 3. If we consider the homotopy H( ;p) as equation (21) for the VHPM. Then the VHPM is equivalent to the ADM. Proof. Substituting equations (13) and (15) into equation (21), we have

3. Comparison between the ADM, HPM, and VHPM for solving equation (3)

Z

limp!1 ¼ lim

x

ð24Þ

1 X

n pn u0 p

Z

t

X 1 lðÞ R n pn

0

n¼0

n¼0

X 1 þN n pn gðÞ d ¼ 0 n¼0

Zt lðÞ½Rð0 Þ ) 0 u0 þ p 1 0 i þ Hð0 Þ gðÞd Zt lðÞ½Rðn Þ þ Hn nþ1 pnþ1 d ¼ 0 0

By arranging the above equation according to the powers of p, we have

a


p0 :

0 u0 ¼ 0

Jafari et al.

5 Z

p1 :

t

L1 ½Rð1 Þ þ A1 þ

lðÞ½Rð0 Þ þ H0 ð0 Þ gðÞd ¼ 0

1 0

Z

2

p :

hence

t

lðÞ½Rð1 Þ þ H1 ð0 , 1 Þd ¼ 0

2 0

.. . p

nþ1

lim ¼

p!1

Z :

t

lðÞ½Rðn Þ þ Hn ð0 , 1 , . . . , n Þd ¼ 0,

nþ1 0

n ¼ 0, 1, 2, . . .

ð26Þ

X 1 ck k t L1 ½ gðÞ L1 Rði Þ k! i¼0 k¼0 X 1 L1 Ai ¼ u m1 X

i¼0

So, we prove that lim ¼ u. In a similar way we can p!1

prove that u ¼ lim v. In view of Theorems 2 and 3 we

From equation (26) we have

p!1

have the following result. 0 ¼ u0 Zt 1 ¼ lðÞ½Rð0 Þ þ H0 ð0 Þ gðÞd 0 Zt 2 ¼ lðÞ½Rð1 Þ þ H1 ð0 ,1 Þd

Theorem 4. Let l be equation (19) and the homotopy H(; p) be considered as equation (21). Then the VHPM for solving equation (3) is equivalent to the HPM.

0

.. .

Z

t

lðÞ½Rðn Þ þ Hn ð0 ,1 ,...,n Þd, n ¼ 0, 1, 2, .. .

nþ1 ¼ 0

ð27Þ According to Theorem 1 we have Hn ¼ An. In view of equations (19) and (23), substituting equation (27) into equation (20) leads us to

Proof. From equation (30) we have Rt 1 lðÞ½:d ¼ L ½: and substituting it into equation 0 (21) we have 1 X

X 1 n n p ¼ u0 pL R n p n

1

n¼0

þN

X 1

n¼0

n p

n

ð31Þ

gðÞ

n¼0

¼ 0 þ 1 p þ 2 p2 þ Z t m 1 X ck k t þ lðÞ½Rð0 Þ þ A0 gðÞd p ¼ ð28Þ k! 0 k¼0 Z t þ lðÞ½Rð1 Þ þ A1 d p2 þ 0

so Zt ck k t lðÞ gðÞd lim ¼ p!1 k! 0 k¼0 Z t þ lðÞ½Rð0 Þ þ A0 dÞ 0 Z t þ lðÞ½Rð1 Þ þ A1 d þ m 1 X

ð29Þ

ð30Þ

Substituting equation (30) into equation (29) we have

p!1

m 1 X

ck k t L1 ½ gðÞ L1 ½Rð0 Þ þ A0 k! k¼0

Equation (32) is equivalent to equation (5). That means the VHPM for equation (3) is same as the HPM for equation (5). Example. Consider the following type of nonlinear OCP (Fakharzadeh, 2012; Matinfar and Saeidy, 2014; Nik et al., 2012) Zt min J ¼ u2 ðtÞdt such that

0

lim ¼

u ¼ u0 þ L1 ð gðtÞÞ L1 ðR½uðtÞÞ L1 ðN½uðtÞÞ ð32Þ

0

0

In Jafari (2014), the first author proves that Zt lðÞ½:d ¼ L1 ½:

We take the limit of equation (31) as p ! 1 so that we have

1 x_ ¼ x2 ðtÞ sin xðtÞ þ uðtÞ; 2 t 2 ½0, 1, xð0Þ ¼ 0, xð1Þ ¼ 0:5

ð33Þ

Solution. It can be transformed to the following nonlinear TPBVP according to Pontryagin’s maximum principle (Pinch, 1993) 1 x_ ¼ x2 ðtÞ sin xðtÞ þ u ðtÞ, 2


t 2 ½0, 1

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Journal of Vibration and Control

1 y_ ¼ yðtÞxðtÞ sin xðtÞ yðtÞx2 ðtÞ cos xðtÞ, 2 xð0Þ ¼ 0, yð0Þ ¼

t 2 ½0, 1 ð34Þ

where the optimal control law is given by 1 u ðtÞ ¼ yðtÞ 2 To solve equation (34) we apply L1 ½: ¼ both sides of equation (34), and obtain

Rt 0

½:d on

1 2 1 x ðtÞ sin xðtÞ yðtÞ d xðtÞ ¼ 2 2 0 Z t 1 2 yðtÞ ¼ þ yðtÞxðtÞ sin xðtÞ yðtÞx ðtÞ cos xðtÞ d 2 0 ð35Þ Z t

If we apply the ADM (Fakharian et al., 2010) or the HPM (Nik et al., 2012) to equation (35), we find same iterative formula that is given in Matinfar and Saeidy (2014) by using the VHPM.

4. Conclusions In this paper, it has been shown that the VHPM provides exactly the same iterative formula as the ADM and HPM for solving nonlinear OCPs. It has been proved analytically that those methods, the ADM, the HPM and the VHPM, are equivalent for solving nonlinear differential equations. However the volume of calculation for the VHPM is more than for the other two methods. Specifically we show that the modified VIM or the VHPM which was recently proposed by Matinfar and Saeidy (2014) (see equation (20)) is the same as the HPM for solving integral equations or the ADM for solving differential equations. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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