The significant advances in nonlinear stochastic dynamics and control in Hamiltonian formulation during the past decade are reviewed. The exact stationary solutions and equivalent nonlinear system method of Gaussian-white -noises excited and dissipated Hamiltonian systems, the stochastic averaging method for quasi Hamiltonian systems, the stochastic stability, stochastic bifurcation, first-passage time and nonlinear stochastic optimal control of quasi Hamiltonian systems are summarized. Possible extension and applications of the theory are pointed out. This review article cites 158 references.
Issue Section:
Review Articles
1.
Bolotin
, V. V.
Random Vibration of Elastic Systems
, Matinus Nijhoff Publishers
, The Hague
.2.
Ibrahim
, R. A.
Parametric Random Vibration
, Research Studies Press LTD
, Taunton, England
.3.
Dimentberg
, M. F.
Statistical Dynamics of Nonlinear and Time-Varying Systems
, Research Studies Press LtD
, Taunton, England
.4.
Roberts
, J. B.
Spanos
, P. D.
Random Vibration and Stctistical Linearization
, Wiley
, New York
.5.
Zhu
, W. Q.
Random Vibration
, Science Press
, Beijing (In Chinese)
.6.
Soong
, T. T.
Grigoriu
, M.
Random Vibration of Mechanical and Structural Systems
, PTR Prentice-Hall
, Englewood Cliffs, New Jersey
.7.
Soize
, C.
The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solution
, World Scientific
, Singapore
.8.
Lin
, Y. K.
Cai
, G. Q.
Probabilistic Structural Dynamics, Advanced Theory and Applications
, McGraw-Hill
, New York
.9.
Grigoriu
, M.
Applied Non-Gaussian Processes; Examples, Theory, Simulations, Linear Random Vibration and MATLAB Solutions
, Prentice Hall
, Englewood Cliffs, NJ, USA
.10.
Khasminskii
, R. Z.
Stochastic Stability of Differential Equations
, Sijthoff and Noorchoff, Alphen aan den Rijn
, The Netherlands
.11.
Mao
, X.
Exponential Stability of Stochastic Differential Equations
, Marcel Dekker Inc.
, New York
.12.
Arnold
, L.
Random Dynamical Systems
, Springer
, Berlin
.13.
Stengel
, H.
Stochastic Optimal Control
, Wiley
, New York
.14.
Fleming
, W. H.
Soner
, H. M.
Controlled Markov Processes and Viscosity Solutions
, Springer-Verlag
, New York
.15.
Yong
, J. M.
Zhou
, X. Y.
Stochastic Control, Hamiltonian Systems and HJB Equations
, Springer-Verlag
, New York
.16.
Ziegler
, F.
Sehuëller
, G. I.
Nonlinear Stochastic Dynamic Engineering Systems
, Proc. IUTAM Symp.
, Springer-Verlag
, Berlin
.17.
Bellomo
, N.
Casciati
, F.
Nonlinear Stochastic Mechanics
, Proc. IUTAM Symp.
, Springer-Verlag
, Berlin
.18.
Naess
, A.
Krenk
, S.
Advances in Nonlinear Stochastic Mechanics
, Proc. IUTAM Symp.
, Kluwer Academic Publishers
, Dordrecht
.19.
Narayanan
, S.
Iyengar
, R. N.
Nonlinearity and Stochastic Structural Dynamics
, Proc. IUTAM Symp.
, Kluwer Academic Publishers
, Dordrecht
.20.
Sri Namachchivaya
, N.
Lin
, Y. K.
Nonlinear Stochastic Dynamics
, Proc. IUTAM Symp.
, Kluwer Academic Publishers
, Dordrecht
.21.
Crandall
, S. H.
Zhu
, W. Q.
Random Vibration: A Survey of Recent Developments
,” ASME J. Appl. Mech.
0021-8936, 50
, 50th Anniversary Issue, pp. 953
–962
.22.
Sehuëller
, G. I.
A State-of-the-Art Report on Computational Stochastic Mechanics
,” Probab. Eng. Mech.
0266-8920, 12
, pp. 197
–321
.23.
Zhu
, W. Q.
Cai
, G. Q.
Nonlinear Stochastic Dynamics: A Survey of Recent Developments
,” Acta Mech.
0001-5970, 18
, pp. 551
–566
.24.
Zhu
, W. Q.
Nonlinear Stochastic Dynamics and Control—Hamiltonian Theoretical Framework
, Science Press
, Beijing
(In Chinese).25.
Tabor
, M.
Chaos and Integrability in Nonlinear Dynamics, An Introduction
, Wiley
, New York
.26.
Lichtenberg
, A. J.
Lieberman
, M. A.
Regular and Stochastic Motion
, Springer-Verlag
, Berlin
.27.
Das
, A.
Integrable Models
, World Scientific
, Singapore
.28.
Yoshida
, H.
A New Necessary Condition for the Integrability of Hamiltonian Systems With Two-Dimensional Homogeneous Potential
,” Physica D
0167-2789, 128
, pp. 53
–87
.29.
Arnold
, V. I.
Mathmatical Methods of Classical Mechanics
, 2nd ed.
, Springer-Verlag
, New York
.30.
Arnold
, V. I.
Kozlov
, V. V.
Neishtadt
, A. I.
Mathematical Aspects of Classical and Celestial Mechanics, In Dynamical Systems III
, Viarnold
Springer-Verlag
, New York
.31.
Boundtis
, T.
Segar
, H.
Vivaldi
, F.
Integrable Hamiltonian Systems and the Painleve Property
, Phys. Rev. A
1050-2947, Third Series, 25
, pp. 1257
–1264
.32.
Whittaker
, E. T.
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
, Cambridge University Press
, Cambridge, UK
.33.
Hénon
, M.
Heiles
, C.
The Applicability of the Third Integral of Motion; Some Numerical Experiments
,” Astron. J.
0004-6256, 69
, pp. 73
–79
.34.
Fuller
, A. T.
Analysis of Nonlinear Stochastic Systems by Means of the Fokker-Planck Equation
,” Int. J. Control
0020-7179, 9
, pp. 603
–655
.35.
Zhu
, W. Q.
Cai
, G. Q.
Lin
, Y. K.
On Exact Stationary Solutions of Stochastically Perturbed Hamiltonian Systems
,” Probab. Eng. Mech.
0266-8920, 5
, pp. 84
–87
.36.
Zhu
, W. Q.
Cai
, G. Q.
Lin
, Y. K.
Stochastically Perturbed Hamiltonian Systems
,” Nonlinear Stochastic Mechanics
, N.
Bellomo
F.
Casciaki
Springer-Verlag
, Berlin
, pp. 543
–552
.37.
Caughey
, T. K.
Nonlinear Theory of Random Vibration
,” Advances in Applied Mechanics 11
, Academic Press
, New York
.38.
Caughey
, T. K.
Ma
, F.
The Exact Steady-State Solution of a Class of Nonlinear Stochastic Systems
,” Int. J. Non-Linear Mech.
0020-7462, 17
, pp. 137
–142
.39.
Caughey
, T. K.
Ma
, F.
The Steady-State Response of a Class of Dynamical Systems to Stochastic Excitation
,” ASME J. Appl. Mech.
0021-8936, 49
, pp. 629
–632
.40.
Dimentberg
, M. F.
An Exact Solution to a Certain Nonlinear Random Vibration Problem
,” Int. J. Non-Linear Mech.
0020-7462, 17
, pp. 231
–236
.41.
Lin
, Y. K.
Cai
, G. Q.
Exact Stationary-Response Solution for Second Order Nonlinear Systems Under Parametric and External Excitations, Part II
,” ASME J. Appl. Mech.
0021-8936, 55
, pp. 702
–705
.42.
Zhu
, W. Q.
Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems Under Parametric and External White Noise Excitations
,” J. Appl. Math. Mech.
0021-8928, 11
, pp. 165
–175
.43.
Zhu
, W. Q.
Yang
, Y. Q.
Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 63
, pp. 493
–500
.44.
Huang
, Z. L.
Liu
, Z. H.
Zhu
, W. Q.
Stationary Response of Multi-Degree-of-Freedom Vibro-Impact Systems Under White Noise Excitations
,” J. Sound Vib.
0022-460X, 275
, pp. 223
–240
.45.
Cai
, G. Q.
Lin
, Y. K.
Exact and Approximate Solutions for Randomly Excited MDOF Nonlinear Systems
,” Int. J. Non-Linear Mech.
0020-7462, 31
, pp. 623
–647
.46.
Zhu
, W. Q.
Huang
, Z. L.
Exact Stationary Solutions of Stochastically Excited and Dissipated Partially Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 36
, pp. 39
–48
.47.
Ying
, Z. G.
Zhu
, W. Q.
Exact Stationary Solutions of Stochastically Excited and Dissipated Gyroscopic Systems
,” Int. J. Non-Linear Mech.
0020-7462, 35
, pp. 837
–848
.48.
Huang
, Z. L.
Zhu
, W. Q.
Exact Stationary Solutions of Stochastically and Harmonically Excited and Dissipated Integrable Hamiltonian Systems
,” J. Sound Vib.
0022-460X, 230
, pp. 709
–720
.49.
Caughey
, T. K.
On Response of Nonlinear Oscillators to Stochastic Excitation
,” Probab. Eng. Mech.
0266-8920, 1
, pp. 2
–4
.50.
Lutes
, L. D.
Approximate Technique for Treating Random Vibration of Hysteretic Systems
,” J. Acoust. Soc. Am.
0001-4966, 48
, pp. 299
–306
.51.
Cai
, G. Q.
Lin
, Y. K.
A New Approximate Solution Technique for Randomly Excited Nonlinear Oscillators
,” Int. J. Non-Linear Mech.
0020-7462, 23
, pp. 409
–420
.52.
Zhu
, W. Q.
Yu
, J. S.
The Equivalent Nonlinear System Method
,” J. Sound Vib.
0022-460X, 129
, pp. 385
–395
.53.
To
, C. W. S.
Li
, D. M.
Equivalent Nonlinearization of Nonlinear Systems to Random Excitation
,” Probab. Eng. Mech.
0266-8920, 6
, pp. 184
–192
.54.
Lei
, Z.
Qiu
, C.
A New Equivalent Nonlinearization Method for Random Vibration of Nonlinear Systems
,” Mech. Res. Commun.
0093-6413, 23
, pp. 131
–136
.55.
Zhu
, W. Q.
Soong
, T. T.
Lei
, Y.
Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 61
, pp. 618
–623
.56.
Zhu
, W. Q.
Lei
, Y.
Equivalent Nonlinear System Method for Stochastically Excited and Dissipated Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 64
, pp. 209
–216
.57.
Zhu
, W. Q.
Deng
, M. L.
Equivalent Nonlinear Systems Method for Stochastically Excited and Dissipated Integrable Hamiltonian Systems-Resonant Case
,” J. Sound Vib.
0022-460X, 274
, pp. 1110
–1122
.58.
Zhu
, W. Q.
Huang
, Z. L.
Suzuki
, Y.
Equivalent Nonlinear System Method for Stochastically Excited and Dissipated Partially Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 36
, pp. 773
–786
.59.
Stratonovich
, R. L.
Topics in the Theory of Random Noise
, Vols. 1
and 2
, Gordon Breach
, New York
.60.
Khasminskii
, R. Z.
A Limit Theorem for Solution of Differential Equations With Random Right-Hand Side
,” Theor. Probab. Appl.
0040-585X, 11
, pp. 390
–406
.61.
Papanicolaou
, G. C.
Kohler
, W.
Asymptotic Theory of Mixing Stochastic Ordinary Differential Equations
,” Commun. Pure Appl. Math.
0010-3640, 27
, pp. 641
–668
.62.
Blankenship
, G. L.
Papanicolaou
, G. C.
Stability and Control of Stochastic Systems With Wide-Band Noise Disturbances I
,” SIAM J. Appl. Math.
0036-1399, 34
, pp. 437
–476
.63.
Landa
, P. S.
Stratonovich
, R. L.
Theory of Stochastic Transition of Various Systems Between States
,” Vestn. Mosk. Univ., Ser. 3: Fiz., Astron.
, pp. 33
–45
(in Russian).64.
Khasminskii
, R. Z.
Behavior of a Conservative System With Small Friction and Small Random Noise
,” Prikl. Mat. Mekh.
0032-8235, 28
, pp. 1126
–1130
(in Russian).65.
Khasminskii
, R. Z.
On the Averaging Principle for Itô Stochastic Differential Equations
,” Kybernetika
0023-5954, 3
, pp. 260
–279
(in Russian).66.
Papanicolaou
, G.
Stroock
, D.
Varadhan
, S. R. S.
Martingale Approach to Some Limit Theorems
,” Duke Univ. Math. Ser. III
.67.
Roberts
, J. B.
Energy Method for Nonlinear Systems With Nonwhite Excitation
,” Random Vibrations and Reliabiity, Proc. IUTAM Symp.
, K.
Hennig
Academic-Verlag
, Berlin
, pp. 285
–294
.68.
Red-Horse
, J. R.
Spanos
, P. D.
A Generalization to Stochastic Averaging in Random Vibration
,” Int. J. Non-Linear Mech.
0020-7462, 27
, pp. 85
–101
.69.
Cai
, G. Q.
Lin
, Y. K.
Random Vibration of Strongly Nonlinear Systems
,” Nonlinear Dyn.
0924-090X, 24
, pp. 3
–15
.70.
Zhu
, W. Q.
Huang
, Z. L.
Suzuke
, Y.
Response and Stability of Strongly Nonlinear Oscillators Under Wide-Band Random Excitation
,” Int. J. Non-Linear Mech.
0020-7462, 36
, pp. 1235
–1250
.71.
Huang
, Z. G. L.
Zhu
, W. Q.
Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and White Noise Excitations
,” J. Sound Vib.
0022-460X, 238
, pp. 233
–256
.72.
Huang
, Z. L.
Zhu
, W. Q.
Ni
, Y. Q.
Ko
, J. M.
Stochastic Averaging of Strongly Nonlinear Oscillator Under Bounded Noise Excitation
, J. Sound Vib.
0022-460X, 254
, pp. 245
–267
.73.
Sri Namachchivaya
, N.
Sowers
, R. B.
Rigorous Stochastic Averaging at a Center With Additive Noise
,” Meccanica
0025-6455, 37
, pp. 85
–114
.74.
Freidlin
, M. I.
Wentzell
, A. D.
Random Perturbation of Dynamical Systems
, 2nd ed
, Springer-Verlag
, Berlin
.75.
Sowers
, R. B.
Stochastic Averaging Near Homoclinic Robets Via Singular Perturbations
,” in Ref. (20).76.
Roberts
, J. B.
Spanos
, P. D.
Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,” Int. J. Non-Linear Mech.
0020-7462, 21
, pp. 111
–134
.77.
Zhu
, W. Q.
Stochastic Averaging Methods in Random Vibration
,” Appl. Mech. Rev.
0003-6900, 41
(5
), pp. 189
–199
.78.
Zhu
, W. Q.
Recent Developments and Applications of the Stochastic Averaging Method in Random Vibration
,” Appl. Mech. Rev.
0003-6900, 49
(10
), pp. s72
–s82
.79.
Zhu
, W. Q.
Yang
, Y. Q.
Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 64
, pp. 157
–164
.80.
Zhu
, W. Q.
Huang
, Z. L.
Yang
, Y. Q.
Stochastic Averaging of Quasi Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 64
, pp. 975
–984
.81.
Zhu
, W. Q.
Huang
, Z. L.
Suzuki
, Y.
Stochastic Averaging and Lyapunov Exponent of Quasi Partially Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 37
, pp. 419
–437
.82.
Deng
, M. L.
Zhu
, W. Q.
Stationary Motion of Active Brownian Particle
,” Phys. Rev. E
1063-651X, 69
, p. 046105
.83.
Zhu
, W. Q.
Deng
, M. L.
Huang
, Z. L.
Optimal Bounded Control of First-Passage Failure of Quasi Integrable Hamiltonian Systems With Wide-Band Random Excitation
,” Nonlinear Dyn.
0924-090X, 33
, pp. 189
–207
.84.
Huang
, Z. L.
Zhu
, W. Q.
Exact Stationary Solutions of Averaged Equations of Stochastically and Harmonically Excited MDOF Quasi-Linear Systems With Internal and / Or External Resonance
,” J. Sound Vib.
0022-460X, 204
, pp. 249
–258
.85.
Huang
, Z. L.
Zhu
, W. Q.
Stochastic Averaging of Quasi Integrable Hamiltonian Systems Under Combined Harmonic and White Noise Excitations
,” Int. J. Non-Linear Mech.
0020-7462, 39
, pp. 1421
–1434
.86.
Huang
, Z. L.
Zhu
, W. Q.
Stochastic Averaging of Quasi Integrable Hamiltonian Systems Under Bounded Noise Excitation
,” Probab. Eng. Mech.
0266-8920, 19
, pp. 219
–228
.87.
Huang
, Z. L.
Zhu
, W. Q.
Averaging Method for Quasi-Integrable Hamiltonian Systems
,” J. Sound Vib.
0022-460X, 284
, pp. 325
–341
.88.
Kozin
, F.
A Survey of Stability of Stochastic Systems
,” Automatica
0005-1098, 5
(1
), pp. 95
–112
.89.
Naprestek
, J.
Stochastic Exponential and Asymptotic Stability of Simple Nonlinear Systems
,” Int. J. Non-Linear Mech.
0020-7462, 31
(5
), pp. 693
–705
.90.
Oseledec
, V. I.
A Multiplicative Ergodic Theorem, Lyapunov Characteristic Numbers for Dynamical Systems
,” Trans. Mosc. Math. Soc.
0077-1554, 19
, pp. 197
–231
.91.
Khasminskii
, R. Z.
Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems
,” Theor. Probab. Appl.
0040-585X, 11
, pp. 144
–147
.92.
Kozin
, F.
Zhang
, Z. Y.
On Almost Sure Sample Stability of Nonlinear Itô, Differential Equations
,” Probab. Eng. Mech.
0266-8920, 6
(2
), pp. 92
–95
.93.
Talay
, D.
Lyapunov Exponent of the Euler Scheme for Stochastic Differential Equation
,” in Stochastic Dynamics
, H.
Crauel
M.
Gundlach
Springer-Verlag
, New York
, pp. 241
–258
.94.
Wihstutz
, V.
Perturbation Methods for Lyapunov Exponents
,” in Stochastic Dynamics
, H.
Crauel
M.
Gundlach
Springer-Verlag
, New York
, pp. 209
–239
.95.
Ariaratnam
, S. T.
Xie
, W.-C.
Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation
,” ASME J. Appl. Mech.
0021-8936, 59
, pp. 664
–673
.96.
Ariaratnam
, S. T.
Abdelrahman
, N. M.
Stochastic Stability of Coupled Oscillators in Internal Resonance
,” in Ref. (20).97.
Arnold
, L.
A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems
.” in Lyapunov Exponents (Lecture Notes in Mathematics 1186)
, L.
Arnold
V.
Wihstutz
Springer-Verlag
, Berlin
, pp. 793
–802
.98.
Arnold
, L.
Doyle
, M. M.
Sri Namachchivaya
, N.
Small Noise Expansion of Moment Lyapunov Exponents for Two-Dimensional Systems
,” Dyn. Stab. Syst.
0268-1110, 12
(3
), pp. 187
–211
.99.
Khasminskii
, R. Z.
Moshchuk
, N.
Moment Lyapunov Exponent and Stability Index for Linear Conservative System With Small Random Perturbation
,” SIAM J. Appl. Math.
0036-1399, 58
(1
), pp. 245
–256
.100.
Xie
, W. C.
Moment Lyapunov Exponent of a Two-Dimensional System Under Real Noise Excitation
,” J. Sound Vib.
0022-460X, 239
(1
), pp. 139
–155
.101.
Horsthemke
, W.
Lefever
, R.
Noise-Induced Transition
, Springer-Verlag
, Berlin
.102.
Sri Namachchivaya
, N.
Stochastic Bifurcation
,” Appl. Math. Comput.
0096-3003, 38
(1
), 101
–159
.103.
Arnold
, L.
Recent Progress in Stochastic Bifurcation Theory
,” In Ref. (19), pp. 15
–27
.104.
Zhu
, W. Q.
Lyapunov Exponent and Stochastic Stability of Quasi Non-Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 39
, pp. 645
–655
.105.
Pardoux
, E.
Wihstutz
, V.
Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems With Small Diffusion
,” SIAM J. Appl. Math.
0036-1399, 48
, pp. 442
–457
.106.
Zhu
, W. Q.
Huang
, Z. L.
Lyapunov Exponent and Stochastic Stability of Quasi Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 66
, pp. 211
–217
.107.
Huang
, Z. L.
Zhu
, W. Q.
Lyapunov Exponent and Almost Sure Asymptotic Stability of Quasi-Linear Gyroscopic Systems
,” Int. J. Non-Linear Mech.
0020-7462, 35
, pp. 645
–655
.108.
Huang
, Z. L.
Zhu
, W. Q.
A New Approach to Almost-Sure Asymptotic Stability of Stochastic Systems of Higher Dimension
,” Int. J. Non-Linear Mech.
0020-7462, 38
, pp. 239
–247
.109.
Zhu
, W. Q.
Huang
, Z. L.
Stochastic Stability of Quasi-Non-Integrable-Hamiltonian Systems
,” J. Sound Vib.
0022-460X, 218
, pp. 769
–789
.110.
Zhu
, W. Q.
Huang
, Z. L.
Stochastic Hopf Bifurcation of Quasi Non-Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 34
, pp. 437
–447
.111.
Lyon
, R. H.
Heckl
, M.
Hazelgrove
, C. B.
Response of Hard-Spring Oscillator to Narrow-Band Excitation
,” J. Acoust. Soc. Am.
0001-4966, 33
, pp. 1404
–1411
.112.
Zhu
, W. Q.
Lu
, M. Q.
Wu
, Q. T.
Stochastic Jump and Bifurcation of a Duffing Oscillator Under Narrow-Band Excitation
,” J. Sound Vib.
0022-460X, 165
, pp. 285
–304
.113.
Kapitaniak
, T.
Chaotic Distribution of Nonlinear Systems Perturbed by Random Noise
,” Phys. Lett. A
0375-9601, 116
, pp. 251
–254
.114.
Bulsara
, A. R.
Schieve
, W. C.
Jacobs
, E. W.
Homoclinic Chaos in Systems Perturbed by Weak Langevin Noise
,” Phys. Rev. A
1050-2947, 41
, pp. 668
–681
.115.
Frey
, M.
Simiu
, E.
Noise Induced Chaos and Phase Space Flux
,” Physica D
0167-2789, 63
, pp. 321
–340
.116.
Liu
, W. Y.
Zhu
, W. Q.
Huang
, Z. L.
Effect of Bounded Noise on Chaotic Motion of Duffing Oscillator Under Parametric Excitation
,” Chaos, Solitons Fractals
0960-0779, 12
, pp. 527
–537
.117.
Liu
, Z. H.
Zhu
, W. Q.
Homoclinic Bifurcation and Chaos in Simple Pendulum Under Bounded Noise Excitation
,” Chaos, Solitons Fractals
0960-0779, 20
, pp. 593
–607
.118.
Zhu
, W. Q.
Liu
, Z. H.
Homoclinic Bifurcation and Chaos in Coupled Simple Pendulum and Harmonic Oscillator Under Bounded Noise Excitation
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 15
(1
), pp. 234
–243
.119.
Roberts
, J. B.
First-Passage Probabilities for Randomly Excited Systems: Diffusion Methods
,” Probab. Eng. Mech.
0266-8920, 1
, pp. 66
–81
.120.
Cai
, G. Q.
Lin
, Y. K.
On Statistics of First-Passage Failure
,” ASME J. Appl. Mech.
0021-8936, 61
(1
), pp. 93
–99
.121.
Gan
, C. B.
Zhu
, W. Q.
First-Passage Failure of Quasi-Non-Integrable-Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 36
, pp. 209
–220
.122.
Zhu
, W. Q.
Deng
, M. L.
Huang
, Z. L.
First-Passage Failure of Quasi-Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 69
, pp. 274
–282
.123.
Zhu
, W. Q.
Huang
, Z. L.
Deng
, M. L.
First-Passage Failure and its Feedback Minimization of Quasi-Partially Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 38
, pp. 1133
–1148
.124.
Zhu
, W. Q.
Lei
, Y.
First Passage Time for State Transition of Randomly Excited Systems
,” Proc. of 47th Session of Int. Statistical Inst. Vol. LIII
(Invited papers), Book3, pp. 517
–531
.125.
Zhu
, W. Q.
Wu
, Y. J.
First-Passage Time of Duffing Oscillator Under Combined Harmonic and White-Noise Excitations
,” Nonlinear Dyn.
0924-090X, 32
, pp. 291
–305
.126.
Housner
, G. W.
et al.
Structural Control: Past, Present and Future
,” J. Eng. Mech.
0733-9399, 123
, pp. 897
–971
.127.
Yoshida
, K.
A Method of Optimal Control of Nonlinear Stochastic Systems With Nonquadratic Criteria
,” Int. J. Control
0020-7179, 39
(2
), pp. 279
–291
.128.
Chang
, R. J.
Optimal Linear Feedback Control for a Class of Nonlinear Nonquadratic Non-Gaussian Problem
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 113
(4
), 569
–574
.129.
Liberzon
, D.
Brockett
, R. W.
Nonlinear Feedback Systems Perturbed by Noise: Steady-State Probability Distribution and Optimal Control
,” IEEE Autom. Contr.
, 45
(6
), pp. 1116
–1130
.130.
Bratus
, A.
et al.
Hybrid Solution Method for Dynamic Programming Equations for MDOF Stochastic Systems
,” Dyn. Control
0925-4668, 10
(1
), pp. 107
–116
.131.
Crespo
, L. G.
Sun
, J. Q.
Nolinear Stochastic Control via Stationary Response Design
,” Probab. Eng. Mech.
0266-8920, 18
, pp. 79
–86
.132.
Kushner
, H. J.
Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems
, Birkhauser
, Boston
.133.
Kushner
, H. J.
Rungguldier
, W. J.
Nearly Optimal State Feedback Control for Stochastic Systems With Wide-Band Noise Disturbances
,” SIAM J. Control Optim.
0363-0129, 25
, pp. 298
–315
.134.
Zhu
, W. Q.
Ying
, Z. G.
Optimal Nonlinear Feedback Control of Quasi-Hamiltonian Systems
,” Sci. China, Ser. A: Math., Phys., Astron.
1006-9283, 42
, pp. 1213
–1219
.135.
Zhu
, W. Q.
Ying
, Z. G.
Soong
, T. T.
An Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems
,” Nonlinear Dyn.
0924-090X, 24
, pp. 31
–51
.136.
Zhu
, W. Q.
Deng
, M. L.
Optimal Bounded Control for Minimizing the Response of Quasi Nonintegrable Hamiltonian Systems
,” Nonlinear Dyn.
0924-090X, 35
, pp. 81
–100
.137.
Zhu
, W. Q.
Deng
, M. L.
Optimal Bounded Control for Minimizing the Response of Quasi Integrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 39
, pp. 1535
–1546
.138.
Zhu
, W. Q.
Wu
, Y. J.
Optimal Bounded Control of Strongly Nonlinear Oscillator Under Combined Harmonic and White-Noise Excitations
,” Probab. Eng. Mech.
0266-8920, 20
(1
), pp. 1
–9
.139.
Zhu
, W. Q.
Ying
, Z. G.
Ni
, Y. Q.
KOJM
, 2000, “Optimal Nonlinear Stochastic Control of Hysteretic Structures
,” J. Eng. Mech.
0733-9399, 126
, pp. 1027
–1032
.140.
Wonham
, W. M.
On the Separation Theorem of Stochastic Control
,” SIAM J. Control
0036-1402, 6
, pp. 312
–326
.141.
Bensoussan
, A.
Stochastic Control of Partially Observable Systems
, Cambridge University Press
, Cambridge, UK
.142.
Zhu
, W. Q.
Ying
, Z. G.
Nonlinear Stochastic Optimal Control of Partially Observable Linear Structures
,” Eng. Struct.
0141-0296, 24
, pp. 333
–342
.143.
Zhu
, W. Q.
Luo
, M.
Ying
, Z. G.
Nonlinear Stochastic Optimal Control of Tall Buildings Under Wind Loading
,” Eng. Struct.
0141-0296, 26
, pp. 1561
–1572
.144.
Charalambous
, C. D.
Elliott
, R. J.
Classes of Nonlinear Partially Observable Stochastic Optimal Control Problems With Explicit Optimal Control Law
,” SIAM J. Control Optim.
0363-0129, 36
, pp. 542
–578
.145.
Zhu
, W. Q.
Ying
, Z. G.
On Stochastic Optimal Control of Partially Observable Nonlinear Quasi Hamiltonian Systems
,” J. Zhejiang Univ. Sci.
, 5
(11
), pp. 1313
–1317
.146.
Jansen
, L. M.
Dyke
, S. J.
Semi-Active Control Strategy for MR Dampers: Comparative Study
,” J. Eng. Mech.
0733-9399, 126
, pp. 795
–803
.147.
Ying
, Z. G.
Zhu
, W. Q.
Soong
, T. T.
A Stochastic Optimal Semi-Active Control Strategy for ER/MR Dampers
,” J. Sound Vib.
0022-460X, 259
, pp. 45
–62
.148.
Dong
, L.
Ying
, Z. G.
Zhu
, W. Q.
Stochastic Optimal Semi-Active Control of Nonlinear Systems Using MR Damper
,” Adv. Struct. Eng.
1369-2768, 7
, pp. 485
–494
.149.
Zhu
, W. Q.
Luo
, M.
Dong
, L.
Semi-Active Control of Wind Excited Building Structures Using MR/ER Dampers
,” Probab. Eng. Mech.
0266-8920, 19
, pp. 279
–285
.150.
Florchinger
, P.
Feedback Stabilization of Affine in the Control Stochastic Differential Systems by the Control Lyapunov Function Method
,” SIAM J. Control Optim.
0363-0129, 35
, pp. 500
–511
.151.
Zhu
, W. Q.
Feedback Stabilization of Quasi Nonintegrable Hamiltonian Systems by Using Lyapunov Exponent
,” Nonlinear Dyn.
0924-090X, 36
, pp. 455
–470
.152.
Zhu
, W. Q.
Huang
, Z. L.
Feedback Stabilization of Quasi-Integrable Hamiltonian Systems
,” ASME J. Appl. Mech.
0021-8936, 70
, pp. 129
–136
.153.
Zhu
, W. Q.
Huang
, Z. L.
Stochastic Stabilization of Quasi Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent
,” Nonlinear Dyn.
0924-090X, 33
, pp. 209
–224
.154.
Zhu
, W. Q.
Huang
, Z. L.
Ko
, J. M.
Ni
, Y. Q.
Optimal Feedback Control of Strongly Nonlinear Oscillator Excited by Bounded Noise
,” J. Sound Vib.
0022-460X, 274
, pp. 701
–724
.155.
Zhu
, W. Q.
Huang
, Z. L.
Stochastic Stabilization of Quasi Noniintegrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 39
, pp. 879
–895
.156.
Zhu
, W. Q.
Huang
, Z. L.
Deng
, M. L.
Feedback Minimization of First-Passage Failure of Quasi Nonintegrable Hamiltonian Systems
,” Int. J. Non-Linear Mech.
0020-7462, 37
, pp. 1057
–1071
.157.
Zhu
, W. Q.
Wu
, Y. J.
Optimal Bounded Control of First-Passage Failure of Strongly Nonlinear Oscillators Under Combined Harmonic and White Noise Excitations
,” J. Sound Vib.
0022-460X, 271
(1
), pp. 83
–101
.158.
Cheng
, M.
Zhu
, W. Q.
Ying
, Z. G.
Stochastic Optimal Semi-Active Control of Hysteretic Systems by Using a Magnetorheological Damper
,” Smart Mater. Struct.
0964-1726, 15
, pp. 711
–718
.Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
