Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

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Volume 2011, Article ID 635823,21pages doi:10.1155/2011/635823

Research Article

Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

Yongjun Wu,

¹

Changshui Feng,

²

and Ronghua Huan

³

1Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

2Institute of Mechatronic Engineering, Hangzhou Dianzi University, Hangzhou 310018, China

3Department of Mechanics, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Yongjun Wu,yongjun.wu@yahoo.com Received 29 March 2011; Revised 8 July 2011; Accepted 25 July 2011

Academic Editor: Angelo Luongo

Copyrightq2011 Yongjun Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the stochastic optimal bounded control for minimizing the stationary response of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The stochastic averaging method and the dynamical programming principle are combined to obtain the fully averaged It ô stochastic differential equations which describe the original controlled strongly nonlinear system approximately. The stationary joint probability density of the amplitude and phase difference of the optimally controlled systems is obtained from solving the corresponding reduced Fokker-Planck-Kolmogorov FPK equation. An example is given to illustrate the proposed procedure, and the theoretical results are verified by Monte Carlo simulation.

1. Introduction

The well-known tool to solve the problem of stochastic optimal control is the dynamical programming principle, which was proposed by Bellman 1. According to this principle, a stochastic optimal control problem may be transformed into the problem of finding a solution to the so-called Hamilton-Jacobi-BellmanHJBequation. However, in most cases, the HJB equation cannot be solved analytically2, especially in the multidimensional case.

A powerful technique to solve stochastic optimal control problems is to combine the so- called stochastic averaging method3with Bellman’s dynamic programming principle. The idea is to replace the original stochastic system by the averaged one and then to apply the dynamic programming principle to the averaged system. It has been justified that the optimal control for the averaged system is nearly optimal for the original system4. This combination strategy has two notable advantages. Firstly, the dimension of the averaged

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system can be reduced remarkably, and the corresponding HJB equation is low-dimensional.

Secondly, the diffusion matrix of the original stochastic system is usually singular, while the averaged one is usually nonsingular. This unique characteristic enables the HJB equation for the averaged system to have classical rather than viscous solution. In recent years, a notable nonlinear stochastic optimal control strategy has been proposed for the control of quasi- Hamiltonian systems under random excitations by Zhu based on the stochastic averaging method for quasi-Hamiltonian systems and the stochastic dynamical programming principle 3. Examples have shown that this strategy is very effective and efficient.

In practice, the excitations of dynamical systems can be classified as either deterministic or random. The random excitation is often modeled as Gaussian white noise, wide- band colored noise, or bounded noise. On the other hand, the magnitudes of the control forces are usually limited due to the saturation in actuators; that is, the control forces are bounded. The optimal bounded control of linear or nonlinear systems under random excitations has been studied by many researchers5–8. In all these studies, the excitations of the systems are random excitations alone. However, many physical or mechanical systems are subjected to both random and deterministic harmonic excitations. For example, such combined excitations arise in the study of stochastic resonance9or uncoupled flapping motion of rotor blades of a helicopter in forward flight under the eﬀect of atmospheric turbulence10. Due to the existence of the deterministic harmonic excitation, the response of the stochastic system is not time homogeneous and the stationary behavior is not easy to capture. Stochastic averaging method is such a method that can approximate the original system by time-homogeneous stochastic processes. By using stochastic averaging method, many researchers have studied the linear or strongly nonlinear systems under combined harmonic and wide-band random excitations11–16. On the other hand, optimal bounded control of a linear or nonlinear oscillator subject to combined harmonic and Gaussian white noise excitations has been studied17–19. However, Gaussian white noise is an ideal model.

In most cases, the random excitation should be modeled as wide-band colored noise. So far, little work has been done on the optimal bounded control of a strongly nonlinear oscillator subject to combined harmonic and wide-band noise excitations20.

In the present paper, a procedure for designing optimal bounded control to minimize the stationary response of a strongly nonlinear oscillator under combined harmonic and wide-band colored noise excitations is proposed. In Section 2, based on the stochastic averaging method, the equation of motion of a weakly controlled strongly nonlinear oscillator under combined harmonic and wide-band noise excitations is reduced to partially averaged It ô stochastic differential equations. In Section 3, a dynamical programming equation for the control problem of minimizing response of the system is formulated from the partially averaged It ô stochastic differential equations by applying the dynamical programming principle. The optimal control law is determined by the dynamical programming equation and the control constraint. InSection 4, the reduced FPK equation governing the stationary joint probability density of the amplitude and phase difference of the optimally controlled system is established. InSection 5, a nonlinearly damped Duffing oscillator under combined harmonic and wide-band noise excitations is taken as an example to illustrate the application of the proposed procedure. All theoretical results are verified by Monte Carlo simulation.

2. Partially Averaged It ˆo Stochastic Differential Equations

Consider a weakly controlled strongly nonlinear oscillator subject to weak linear or nonlinear damping and weak external or parametric excitation by a combination of harmonic functions

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and wide-band colored noises. The equation of motion of the system has the following form

X¨ gX εf

X,X,˙ Ωt εu

X,X˙

ε^1/2h_k X,X˙

ξ_kt, k 1,2, . . . , r, 2.1 where the termgXis stiﬀness, which is assumed as an arbitrary nonlinear function ofX.ε is a small parameter;εfX,X,˙ Ωtaccounts for light damping and weak harmonic excitation with frequencyΩ;ε^1/2h_kξ_ktrepresent weak external and/or parametric random excitations;

ξktare wide-band stationary and ergodic noises with zero mean and correlation functions R_klτor spectral densitiesS_klω;εudenotes a weakly feedback control force. The repeated subscripts indicate summation.

Whenu 0,2.1describes a large class of physical or structural systems. Under the conditions specified by Xu and Cheung21, the solutions of system2.1can be expressed as the following form:

Xt AcosΦt BA, Xt ˙ −AνA,ΦsinΦt, 2.2 where

Φt μt Θt, 2.3

νA,Φ dμ dt

2PAB−PAcosΦ B

A²sin²Φ , 2.4

Px _x

0

g y

dy. 2.5

A, Φ, μ, Θ, and ν are all stochastic processes. Px is the potential energy of the system2.1. The functions cosΦtand sinΦtare called generalized harmonic functions.

ObviouslyνA,Φis the instantaneous frequency of the system2.1.

Whenε 0,2.1degenerates to the following nonlinear conservative oscillator:

X¨ gX 0. 2.6

The average frequency of the conservative oscillator2.6can be obtained by the following formula:

ωA 2π

_2π

0 dΦ/νA,Φ. 2.7

Then the following approximate relation exists:

Φt≈ωAt Θt. 2.8

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Treating2.2as generalized Van der Pol transformation fromX,X˙ toA, Θ , one can obtain the following equations forAandΘ:

dA

dt εF₁¹A,Φ,Ωt εF₁²A,Φ, u ε^1/2U_1kA,Φξ_kt, dΘ

dt εF₂¹A,Φ,Ωt εF₂²A,Φ, u ε^1/2U_2kA,Φξ_kt,

2.9

where

F₁¹ −A

gAB1hfAcosΦ B,−AνA,ΦsinΦ,ΩtνA,ΦsinΦ, F₁² −Au

gAB1hνA,ΦsinΦ, F₂¹ −1

gAB1hfAcosΦ B,−AνA,ΦsinΦ,ΩtνA,ΦcosΦ h, F₂² −u

gAB1hνA,ΦcosΦ h,

U_1k −A

gAB1hh_kAcosΦ B,−AνA,ΦsinΦνA,ΦsinΦ,

U_2k −1

gAB1hh_kAcosΦ B,−AνA,ΦsinΦνA,ΦcosΦ h, h dB

dA

g−AB gAB g−AB−gAB.

2.10

According to the Stratonovich-Khasminskii limit theorem22,23,AandΘconverge weakly to 2-dimensional diffusive Markov processes in a time interval ofε⁻¹order asε → 0, which can be represented by the following It ô stochastic differential equations:

dA ε

S₁A,Φ,Ωt F₁²A,Φ, u

dtε^1/2G_1kA,ΦdB_kt, dΘ ε

S2A,Φ,Ωt F₂²A,Φ, u

dtε^1/2G2kA,ΦdBkt,

2.11

whereBktare independent unit Wiener processes,

Si F_i¹ ₀

−∞

∂U_ik

∂A

t

·U1l|_tτRklτ ∂U_ik

∂Φ

t

·U2l|_tτRklτ

dτ,

b_ij εG_ikG_jk _∞

−∞

U_ik|_t·U_jl

tτR_klτ

dτ, i, j 1,2, k, l 1, . . . , r.

2.12

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System2.1has harmonic excitation and two cases can be classified: resonant case and nonresonant case. In the nonresonant case, the harmonic excitation has no eﬀect on the first approximation of the response. Thus, we are interested in the resonant case, namely,

Ω ωA

m

n εσ, 2.13

where m and n are relatively prime positive small integers and εσ is a small detuning parameter. In this case, multiplying2.13bytand utilizing the approximate relation2.8 yield

Ωt m

nΦ εσμ− m

nΘ. 2.14

Introduce a new angle variableΓsuch that Γ εσμ− m

n Θ 2.15

which is a measure of the phase diﬀerence between the response and the harmonic excitation.

Then2.14can be rewritten as

Ωt m

nΦ Γ. 2.16 Using the It ô differential formula, one can obtain the following It ô stochastic differential equations forA,Γ, andΦ:

dA ε

S₁A,Φ,Γ F₁²A,Φ, u

dtε^1/2G_1kA,ΦdB_kt, dΓ ε

σωA− m n

S2A,Φ,Γ F₂²A,Φ, u

dt−ε^1/2m

nG2kA,ΦdBkt, dΦ

ωA ε

S₂A,Φ,Γ F₂²A,Φ, u

dtε^1/2G_2kA,ΦdB_kt.

2.17

Obviously,AandΓare slowly varying processes, whileΦis a rapidly varying process.

Averaging the drift and diffusion coefficients in2.17with respect toΦyields the following partially averaged It ô stochastic differential equations:

dA ε

m¹₁ A,Γ

F₁²A,Φ, u

Φ

dtε^1/2σ1rAdBrt, dΓ ε m¹₂ A,Γ−m

n

F₂²A,Φ, u

Φ

dtε^1/2σ_2rAdBrt,

2.18

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where

m¹₁ S1A,Φ,Γ_Φ, m¹₂ σωA− m

nS2A,Φ,Γ_Φ, b11A εσ1rσ1r b11A,Φ_Φ, b₂₂A εσ_2rσ_2r m²

n²b22A,Φ_Φ, b₁₂A b₂₁A εσ_1rσ_2r m

nb12A,Φ_Φ,

•_Φ 1 2π

_2π

0

•dΦ.

2.19

Herein,•_Φ denotes the averaging with respect toΦfrom 0 to 2π ·b_ij are diﬀusion coeﬃcients.

Note that there are two procedures of averaging in this section. One is stochastic averaging, and the other is deterministic time averaging. The procedures for obtainingS_i and bij in 2.12 are called stochastic averaging. The procedures for obtaining m¹_i and b_ij are called deterministic time averaging. To complete averaging and obtain the explicit expressions ofSiandbij, the functionsF_i¹andUikin2.12are expanded into Fourier series with respect toΦand the approximate relation2.8is used.

3. Dynamical Programming Equation and Optimal Control Law

For mechanical or structural systems,Atis the response amplitude andΓtis the phase diﬀerence between the response and the harmonic excitation. They represent the averaged state of system2.1. Usually, only the response amplitude is concerned, so it is meaningful to controlAtin a semi-infinite or finite time interval. Consider the ergodic control problem of system2.18in a semi-infinite time interval with the following performance index:

J lim

T→ ∞

1 T

_T

0

fAtdt. 3.1

Herein,f is a cost function. Based on the stochastic dynamical programming principle24, the following simplified dynamical programming equation can be established from the first equation of2.18:

η min

u

fA

m¹₁ F₁²

Φ

dV dA1

2b11d²V dA²

, 3.2

where VA is called value function; min_u∈U•denotes the minimum value of • with respect tou;ηis optimal performance index;u∈Uindicates the control constraint.

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The optimal control law can be determined from minimizing the right-hand side of 3.2with respect touunder control constraint. Suppose that the control constraint is of the form

|uA,Φ| ≤u0, 3.3

whereu0is a positive constant representing the maximum control force. The controluenters the performance index3.2only through the term

F²₁

Φ

dV dA

− u

gAB1hAνA,ΦsinΦ

Φ

dV

dA. 3.4

This expression attains its minimum value for u ±u0, depending on the sign of the coeﬃcient ofuin3.4. So the optimal control is

u^∗ u0sgn

AνA,ΦsinΦ gAB1h·dV

dA

, 3.5

where sgn denotes sign function.

It is reasonable to assume that the cost function f is a monotonously increasing function ofAt because the controller must do more work to suppress larger amplitudes At. ThendVA/dAis positive. Under the specified conditions21,gABand1h are both positive. Equation3.5is reduced to

u^∗ −u0sgn−AνA,ΦsinΦ −u0sgnXt˙

. 3.6

Equation3.6implies that the optimal control is a bang-bang control.u^∗has a constant magnitudeu0. It is in the opposite direction of ˙Xtand changes its direction at ˙Xt 0.

For the optimal bounded control of system2.18in a finite time interval, the following performance index is taken:

J E _t_f

0

fAtdtχ A

tf

, 3.7

wheret_fis the final control time andχis the final cost function.E•denotes the expectation operator. The following simplified dynamical programming equation can be established from the first equation of2.18:

∂V

∂t −min

u

fA

m¹₁ F₁²

Φ

∂V

∂A1 2b₁₁∂²V

∂A²

, 3.8

whereV VA, tis the value function. Equation3.8is subject to the following final time condition:

V A, t_f

χ A

t_f

. 3.9

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It is obvious that the same optimal control law as that in3.6can be derived if the control constraint is of the form of3.3.

Note that the optimal control for the averaged system2.18is nearly optimal for the original system 2.1. For simplicity, here it is called optimal control for both original and averaged systems.

4. Stationary Response of Optimally Controlled System

Inserting u^∗ from 3.6 into 2.18 to replace u and averagingF²_i , the following fully averaged It ˆo stochastic diﬀerential equations forAandΓcan be obtained:

dA εm₁A,Γdtε^1/2σ_1kAdBkt,

dΓ εm2A,Γdtε^1/2σ2kAdBkt, 4.1

where

m1A,Γ m¹₁

−u^∗AνA,ΦsinΦ gAB1h

Φ

,

m2A,Γ m¹₂ m n

u^∗νA,ΦcosΦ h gAB1h

Φ

.

4.2

The reduced FPK equation for the optimally controlled system is of the following form

0 − ∂

∂a

m1p

− ∂

∂γ

m2p 1

2

∂²

∂a²

b11p ∂²

∂a∂γ

b12p 1

2

∂²

∂γ²

b22p

, 4.3

wherep pa, γis the stationary joint probability density of the amplitudeAand the phase diﬀerenceΓ. Sincep is a periodic function ofγ, it satisfies the following periodic boundary condition with respect toγ:

p a, γ

p

a, γ2π

. 4.4

The boundary condition with respect toa 0 is

p finite ata 0 4.5

which implies thata 0 is a reflecting boundary. The other boundary condition is

p, ∂p

∂a−→0 asa−→ ∞. 4.6

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In addition to the boundary conditions, the stationary joint probability densitypa, γ satisfies the following normalization condition:

_2π

0

_∞

0

p a, γ

da dγ 1. 4.7

Usually, the partial diﬀerential equation4.3can be solved only numerically.

5. Example

To illustrate the proposed strategy in the previous sections, take the following controlled nonlinearly damped Duﬃng oscillator as an example. Duﬃng oscillator is a typical model in nonlinear analysis. The equation of motion of the system is of the form

X¨

β₁β₂X²

X˙ ω²₀XαX³ EcosΩtξ₁t Xξ₂t u, 5.1

where β1, β2, ω0, α, E, and Ω are positive constants; u is the feedback control with the constraint defined by3.3;ξ_kt k 1, 2are independent stationary and ergodic wide- band noises with zero mean and rational spectral densities

Siω Di

π 1

ω²ω²_i, i 1,2. 5.2

ξ_itcan be regarded as the output of the following first-order linear filter:

ξ˙_iω_iξ_i W_it, i 1,2, 5.3

whereWitare Gaussian white noises in the sense of Stratonovich with intensities 2Di. Note that the maximum value ofS_iωis D_i/πω_i². It is assumed thatβ₁,D_i/πω²_i, andEare all small.

For the system5.1, the instantaneous frequency defined by2.4has the following form:

νA,Φ

ω²₀3αA² 4

1λcos 2Φ _1/2

,

λ αA²

4

ω²₀3αA² /4.

5.4

νA,Φcan be approximated by the following finite sum with a relative error less than 0.03%:

νA,Φ b₀A b₂Acos 2Φ b₄Acos 4Φ b₆Acos 6Φ, 5.5

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where

b₀A

ω²₀3αA² 4

1/2 1− λ²

16

,

b2A

ω²₀3αA² 4

1/2 λ 2 3λ³

64

,

b4A

ω²₀3αA² 4

_1/2

−λ² 16

,

b6A

ω²₀3αA² 4

_1/2 λ³ 64

.

5.6

Then the averaged frequencyωAof the system5.1can be approximated byb0A.

In the case of primary external resonance, Ω

ωA 1εσ. 5.7

Introduce the new angel variable defined by 2.15, and complete the procedures shown in Sections 2–4 we obtain the following fully averaged It ˆo stochastic diﬀerential equations forAandΓ:

dA m1A,Γdt

b11AdB1t, dΓ m2A,Γdt

b22AdB2t,

5.8

wheremiandbii i 1,2are drift and diﬀusion coeﬃcients, respectively.miandbiiare given in the appendix.

The stationary joint probability densitypa, γof the optimally controlled system5.1 is governed by the following reduced FPK equation:

0 − ∂

∂a m1p

− ∂

∂γ m2p

1 2

∂²

∂a²

b11p 1

2

∂²

∂γ²

b22p

. 5.9

Solving the FPK equation5.9by finite diﬀerence method under the conditions4.4–

4.7, the stationary joint probability density of the amplitude and the phase diﬀerence of the optimally controlled system 5.1 can be obtained. Furthermore, the stationary mean amplitude of the optimally controlled system5.1can be obtained as follows:

Ea _∞

0

_2π

0

ap a, γ

da dγ. 5.10

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0 50 100 150 200 250 300 350 400

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

ξ1(t

t

)

Figure 1: Sample function of wide-band noiseξ1t·D1 10,ω1 30.

0 20 40 60 80 100 120 140 160 180 200

−0.15

−0.1

−0.05 0 0.05 0.1

u

t

Figure 2: Sample function of the control forceu,u0 0.1.

To check the accuracy of the proposed method, Monte Carlo digital simulation of the original system 5.1 is performed. The sample functions of independent wide-band noiseξ_itwere generated by inputting Gaussian white noises to the linear filter5.3. The response of system5.1was obtained numerically by using the fourth-order Runge-Kutta method with time step 0.02. The long-time solution after 1500,000 steps was regarded as the stationary ergodic response. 100 samples are used. For every sample, the amplitudeAand angle variable Γ are calculated from step 1500,001 to step 2000000 to obtain the statistical probability density of pa, γ. Figures 1 and 2 show the typical sample function of wide- band noise ξ₁t and control force u, respectively. Figure 3 shows pa, γ of the optimal controlled system5.1. It is seen that the theoretical result agrees very well with that from digital simulation. Figures 4a and 4b show the sample functions of displacement and velocity of the system5.1, respectively. It is obvious that the bounded control can reduce the displacement and velocity. Also, the amplitude of the original system5.1is reduced by control, which is verified byFigure 5.

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0 0 1

2 3 4 56 0.5

1.5 1 2

p(a,γ)

a γ

0 0.4

0.8 1.2 1.6 2

0 0.4 0.8 1.2 1.6 2

a

0 0 1

2 3 45 6 0.5

1 1.5 2

p(a,γ)

a γ

0 0.4

0.8 1.2 1.6 2

0 0.4 0.8 1.2 1.6 2

b 2.5

2

1.5

1

0.5

00 0.5 1 1.5 2 2.5

p(a)

a c

p(γ)

1 2 3γ 4 5 6

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00

d

Figure 3: Stationary joint probability densitypa, γof the optimally controlled system5.1in primary external resonance case.ω0 1.0,Ω 1.1,α 1.0, E 0.3, βi 0.05, ωi 30, D1 10, D2 5, u0 0.1 i 1,2.aTheoretical result;bMonte Carlo simulation of the original system5.1;cstationary marginal probability densitypa;dstationary marginal probability densitypγ. — theoretical results;

•results from Monte Carlo simulation.

Note that the system5.1is strongly nonlinear. Increasing the nonlinearity coeﬃcients β₁ or α in 5.1, one can see that the agreement between theoretical results and Monte Carlo digital simulation is still acceptablesee Figures6and7. This demonstrates that the proposed method is powerful to deal with strongly nonlinear problems, even though the nonlinearity is extremely strong.

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−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

X

30,000 30,100 30,200 30,300 30,400 t

a

X˙

−2

−3

−4 4 3 2 1 0

−1

30,000 30,100 30,200 30,300 30,400

t b

Figure 4: Sample functions of displacement and velocity of the system5.1,u0 0.4. Other parameters are the same as those inFigure 3. Red line: with control; blue line: without control.

0.05 0.1 0.15 0.2

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E[a]

α=1

α=3

Figure 5: Mean amplitude of the response of the optimally controlled system5.1in primary external resonance. The parameters are the same as those inFigure 3except thatu0is a variable. — theoretical results;•results from Monte Carlo simulation.

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0 0.5 1 1.5 2 2.5 0

1 2 3 4 5 6 7

β2=100

β2=6

β2=0.1

p(a)

a

Figure 6: Stationary marginal probability densitypaof the optimally controlled system5.1in primary external resonance case. Other parameters are the same as those inFigure 3. — theoretical results;• results from Monte Carlo simulation.

0 0.2 0.4 0.6 0.8 1

α=100

α=15

α=5

0 1 2 3 4 5 6 7

a

p(a)

Figure 7: Stationary marginal probability densitypaof the optimally controlled system5.1in primary external resonance case. Other parameters are the same as those inFigure 3. — theoretical results;• results from Monte Carlo simulation.

6. Conclusions

In the present paper, a combination procedure of the stochastic averaging method and Bellman’s dynamic programming for designing the optimal bounded control to minimize the response of strongly nonlinear systems under combined harmonic and wide-band noise excitations has been proposed. The procedure consists of applying the stochastic averaging method for weakly controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations, establishing the dynamical programming equation for the control problem of minimizing the response based on the partially averaged It ˆo stochastic diﬀerential equations and the dynamical programming principle, determining the optimal control from the dynamical programming equation and the control constraint without solving the dynamical programming equation. Then the stationary joint probability density

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and mean amplitude of the optimally controlled averaged system are obtained from solving the reduced FPK equation associated with the fully averaged It ô stochastic differential equations. A nonlinearly damped Duffing oscillator with hardening stiffness has been taken as an example to illustrate the application of the proposed procedure. The comparison between the theoretical results and those from Monte Carlo simulation shows that the proposed procedure works quite well even though the nonlinearity is extremely strong. The results show that the response amplitude of the system can be reduced remarkably by the feedback control.

The advantages of the proposed method are obvious. Note that; in the example, the wide-band noise is generated by a first-order linear filter. In principle, one could apply the stochastic dynamical programming method to the extended systemX,X, ξ˙ ₁, ξ₂^T to study the optimal control problem. However, the corresponding HJB equation is 5-dimensional or 4 dimensional. The corresponding reduced FPK equation is 4 dimensional, which is very difficult to solve. After stochastic averaging, the original system is represented by two-dimensional time-homogeneous diffusion Markov processes of amplitude and phase difference with nondegenerate diffusion matrix. The dynamical programming equation derived from the averaged equations is two dimensional or one dimensional. The corresponding reduced FPK equation is two dimensional, which is easy to solve. The other advantage of the proposed procedure is that it is not necessary to solve the dynamical programming equation for obtaining the optimal control law. Furthermore, the proposed method can be extended to multi-degrees-of-freedomMDOFsystems easily. This will be our future work.

Appendix

The drift coeﬃcients in5.8are as follows:

m1 H1A F10A,Γ 2u0−105b0A 35b2A 7b4A 3b6A 105π

ω²₀αA² , m₂ H₂A Ω−b₀A EcosΓ2b₀A b₂A

4A

αA²ω²₀ , F10A,Γ EsinΓ2b0A−b2A

4

αA²ω²₀

−A β₁

16ω²₀10αA²

A²β₂

4ω²₀3αA² 32

αA²ω²₀ , H1A m11m12m13m14,

H2A m21m22m23m24,

m₁₁ m₁₁₁S₁ωA m₁₁₃S₁3ωA m₁₁₅S₁5ωA m₁₁₇S₁7ωA, m13 m131S1ωA m133S13ωA m135S15ωA m137S17ωA, m₁₂ m₁₂₂S₂2ωA m₁₂₄S₂4ωA m₁₂₆S₂6ωA m₁₂₈S₂8ωA,

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m14 m142S22ωA m144S24ωA m146S26ωA m148S28ωA, m21 m211I1ωA m213I13ωA m215I15ωA m217I17ωA, m₂₃ m₂₃₁I₁ωA m₂₃₃I₁3ωA m₂₃₅I₁5ωA m₂₃₇I₁7ωA, m₂₂ m₂₂₂I₂2ωA m₂₂₄I₂4ωA m₂₂₆I₂6ωA m₂₂₈I₂8ωA, m₂₄ m₂₄₂I₂2ωA m₂₄₄I₂4ωA m₂₄₆I₂6ωA m₂₄₈I₂8ωA, Iiω Di

ω_i ω

ω²ω_i² i 1,2, m111 πb2A−2b0A

×

2αA2b0A−b₂A

A²αω²₀

db2A/dA−2db0A/dA

8

A²αω²₀3 ,

m113 πb2A−b4A

×

2αAb4A−b₂A

A²αω₀²

db2A/dA−db4A/dA

8

A²αω₀²3 ,

m₁₁₅ πb4A−b₆A

×

2αAb6A−b₄A

A²αω₀²

db4A/dA−db6A/dA

8

A²αω₀²3 ,

m117

πb₆A

−2αAb6A

A²αω²₀

db6A/dA

8

A²αω₀²3 , m122 πA2b0A−b4A

×

2b0A−b4A

A²α−ω²₀

−A

A²αω²₀

2db0A/dA−db4A/dA

32

A²αω²₀3 ,

m124 πAb2A−b6A

×

b6A−b2A

A²α−ω²₀ A

A²αω₀²

db2A/dA−db6A/dA

32

A²αω₀²3 ,

m₁₂₆ πAb₄A b₄A

A²α−ω²₀ A

A²αω²₀

db4A/dA

32

A²αω²₀3 ,

m128

πAb6A b6A

A²α−ω²₀ A

A²αω²₀

db6A/dA

32

A²αω²₀3 ,

(17)

m₁₃₁ −π

b²₂A−4b²₀A

8A

A²αω₀²2 ,

m133

−3π

b²₄A−b²₂A 8A

A²αω₀²2 ,

m₁₃₅ −5π

b²₆A−b²₄A

8A

A²αω₀²2 ,

m₁₃₇ 7πb₆²A

8A

A²αω²₀2,

m₁₄₂ πA2b0A−b₄A2b0A 2b₂A b₄A 16

A²αω₀²2 , m144 πAb2A−b6Ab2A 2b4A b6A

8

A²αω²₀2 , m146 3πAb4Ab4A 2b6A

16

A²αω₀²2 ,

m₁₄₈ πAb²₆A

4

A²αω²₀2,

m₂₁₁ 2b0A b2A² 8A²

αA²ω²₀2,

m213 3b2A b4A² 8A²

αA²ω²₀2,

m215 5b4A b6A² 8A²

αA²ω²₀2, m₂₁₇ 7b6A²

8A²

αA²ω²₀₂,

m222 2b0A 2b2A b4A² 16

αA²ω²₀2 , m₂₂₄ b2A 2b₄A b₆A²

8

αA²ω²₀₂ ,

(18)

m₂₂₆ 3b 4A 2b6A² 16

αA²ω²₀2 , m₂₂₈ b6A²

4

αA²ω²₀2, m231 2b0A−b2A

×

−b2A2b0A

3A²αω²₀ A

αA²ω₀²

2db0A/dA db2A/dA

8A²

αA²ω²₀3 ,

m233 b2A−b4A

×

−b2A b4A

3A²αω²₀ A

αA²ω²₀

db2A/dA db4A/dA

8A²

αA²ω²₀3 ,

m235 b4A−b6A 8A²

αA²ω²₀3

×

−b4A b₆A

3A²αω²₀ A

αA²ω²₀

db4A/dA db6A/dA

8A²

αA²ω²₀3 ,

m₂₃₇ b6A

−b6A

3A²αω²₀ A

αA²ω₀²

db6A/dA

8A²

αA²ω²₀3 , m242

−2Aα2b0A 2b2A b4A

A²αω²₀ 2db0A

dA 2db2A

dA db4A dA

×A2b 0A−b₄A 32

αA²ω²₀3 , m244

−2Aαb2A 2b4A b6A

A²αω²₀ db2A

dA 2db4A

dA db6A dA

×Ab 2A−b6A 32

αA²ω²₀3 , m₂₄₆ Ab₄A

×

−2Aαb4A 2b₆A

A²αω²₀

db4A/dA 2db6A/dA

32

αA²ω²₀3 ,

m₂₄₈ Ab₆A

−2Aαb6A

A²αω²₀

db6A/dA

32

αA²ω₀²₃ .

A.1

(19)

The diﬀusion coeﬃcients in5.8are as follows:

b11 b111b112, b22 b221b222, bij 0, i /j,

b111 b1111S1ωA b1113S13ωA b1115S15ωA b1117S17ωA, b221 b2211S1ωA b2213S13ωA b2215S15ωA b2217S17ωA, b₁₁₂ b₁₁₂₂S₂2ωA b₁₁₂₄S₂4ωA b₁₁₂₆S₂6ωA b₁₁₂₈S₂8ωA,

b₂₂₂ b₂₂₂₀S₂0 b₂₂₂₂S₂2ωA b₂₂₂₄S₂4ωA b₂₂₂₆S₂6ωA b₂₂₂₈S₂8ωA, b₁₁₁₁ πb2A−2b₀A²

4

αA²ω₀²₂ , b1113 πb 2A−b4A²

4

αA²ω²₀2 , b₁₁₁₅ πb 4A−b₆A²

4

αA²ω²₀₂ , b₁₁₁₇ πb²₆A

4

αA²ω²₀2, b1122

πA²b4A−2b₀A² 16

αA²ω²₀2 , b₁₁₂₄ πA²b2A−b6A²

16

αA²ω₀²2 , b1126

πA²b²₄A

16

αA²ω²₀2, b1128

πA²b²₆A 16

αA²ω²₀2, b₂₂₁₁ π2b 0A b₂A²

4A²

αA²ω₀²2 , b2213 πb 2A b4A²

4A²

αA²ω²₀2,

(20)

b₂₂₁₅ πb 4A b6A² 4A²

αA²ω²₀2,

b₂₂₁₇ πb²₆A

4A²

αA²ω²₀2,

b₂₂₂₀ π2b 0A b₂A² 8

αA²ω₀²₂ ,

b₂₂₂₂ π2b0A 2b₂A b₄A² 16

αA²ω₀²₂ ,

b₂₂₂₄ πb2A 2b₄A b₆A² 16

αA²ω²₀2 ,

b₂₂₂₆ πb 4A 2b₆A² 16

αA²ω²₀2 ,

b₂₂₂₈ πb²₆A

16

αA²ω²₀2.

A.2

Acknowledgments

The work reported in this paper was supported by the National Natural Science Foundation of China under Grant nos. 10802030, 10902096 and Specialized Research Fund for Doctoral Program of Higher Education of China under Grant no. 200802511005.

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