A Variational Method for Approximating the Response of Nonlinear Stochastic Systems

1974 ◽  
Vol 96 (3) ◽  
pp. 353-357
Author(s):  
L. D. Zirkle ◽  
L. G. Clark
Keyword(s):  
Approximate Solution ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Variational Method ◽  
Variational Methods ◽  
Stochastic Systems ◽  
Random Variables ◽  
Nonlinear Stochastic Systems ◽  
Non Gaussian

A method is introduced for determining approximate properties of the response of nonlinear stochastic systems. The method is based in concept on the variational methods of mechanics and allows the consideration of classes of systems not readily subject to analysis by existing techniques. Three examples are presented illustrating the application to nonlinear systems with non-stationary inputs, non-Gaussian inputs and with time delay. The main limitation of the technique is the necessity for assuming a meaningful form for the approximate solution in terms of arbitrary random variables.

scholarly journals Asymptotic Stability of Delay-Controlled Nonlinear Stochastic Systems with Actuator Failures

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
N. Zhou ◽  
R. H. Huan
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
Control Force ◽  
Nonlinear Stochastic Systems ◽  
Actuator Failures

The problem of asymptotic stability of delay-controlled nonlinear stochastic systems with actuator failures is investigated in this paper. Such a system is formulated as a continuous-discrete hybrid system based on the random switch model of failure-prone actuator. Time delay control force is converted into delay-free one by randomly periodic characteristic of the system. Using limit theorem and stochastic averaging, an approximate formula for the largest Lyapunov exponent of the original system is then derived, from which necessary and sufficient conditions for asymptotic stability are obtained. The validity and utility of the proposed procedure are demonstrated by using a stochastically driven nonlinear two-degree system with time delay feedback and actuator failure.

scholarly journals Forecasting COVID-19 with Importance-Sampling and Path-Integrals

2021 ◽  
Author(s):  
Lester Ingber
Keyword(s):  
Simulated Annealing ◽  
Nonlinear Systems ◽  
Financial Markets ◽  
Path Integral ◽  
Stochastic Systems ◽  
Path Integrals ◽  
Strong Support ◽  
Stochastic Nonlinear Systems ◽  
Nonlinear Stochastic Systems ◽  
Adaptive Simulated Annealing

Background: Forecasting nonlinear stochastic systems most often is quite difficult, without giving in to temptations to simply simplify models for the sake of permitting simple computations. Objective: Here, two basic algorithms, Adaptive Simulated Annealing (ASA) and path-integral codes PATHINT/PATHTREE (and their quantum generalizations qPATHINT/qPATHTREE) are offered to detail such systems. Method: ASA and PATHINT/PATHTREE have been effective to forecast properties in three disparate disciplines in neuroscience, financial markets, and combat analysis. Applications are described for COVID-19. Results: Results of detailed calculations have led to new results and insights not previously obtained. Conclusion: These 3 applications give strong support to a quite generic application of these tools to stochastic nonlinear systems.

scholarly journals H∞Control for Nonlinear Stochastic Systems with Time-Delay and Multiplicative Noise

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Ming Gao ◽  
Weihai Zhang ◽  
Zhengmao Zhu
Keyword(s):  
Time Delay ◽  
General Class ◽  
Multiplicative Noise ◽  
Stochastic Systems ◽  
Design Method ◽  
Infinite Horizon ◽  
Control Design ◽  
Mean Square ◽  
Nonlinear Stochastic Systems ◽  
Numerical Examples

This paper studies the infinite horizonH∞control problem for a general class of nonlinear stochastic systems with time-delay and multiplicative noise. The exponential/asymptotic mean squareH∞control design of delayed nonlinear stochastic systems is presented by solving Hamilton-Jacobi inequalities. Two numerical examples are provided to show the effectiveness of the proposed design method.

scholarly journals Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

Entropy ◽  
2018 ◽  
Vol 20 (7) ◽  
pp. 509 ◽  
Author(s):  
Nan Chen ◽  
Andrew Majda
Keyword(s):  
Large Scale ◽  
Stochastic Systems ◽  
Planck Equation ◽  
Reaction Diffusion ◽  
Small Scale ◽  
Model Errors ◽  
Nonlinear Stochastic Systems ◽  
Hybrid Strategy ◽  
Highly Nonlinear ◽  
Non Gaussian

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

scholarly journals Forecasting COVID-19 with Importance-Sampling and Path-Integrals

2020 ◽  
Author(s):  
Lester Ingber
Keyword(s):  
Simulated Annealing ◽  
Nonlinear Systems ◽  
Financial Markets ◽  
Path Integral ◽  
Stochastic Systems ◽  
Path Integrals ◽  
Strong Support ◽  
Stochastic Nonlinear Systems ◽  
Nonlinear Stochastic Systems ◽  
Adaptive Simulated Annealing

Background: Forecasting nonlinear stochastic systems most often is quite difficult, without giving in to temptations to simply simplify models for the sake of permitting simple computations. Objective: Here, two basic algorithms, Adaptive Simulated Annealing (ASA) and path-integral codes PATHINT/PATHTREE (and their quantum generalizations qPATHINT/qPATHTREE) are offered to detail such systems. Method: ASA and PATHINT/PATHTREE have been effective to forecast properties in three disparate disciplines in neuroscience, financial markets, and combat analysis. Applications are described for COVID-19. Results: Results of detailed calculations have led to new results and insights not previously obtained. Conclusion: These 3 applications give strong support to a quite generic application of these tools to stochastic nonlinear systems.

Sign in / Sign up

Export Citation Format

Share Document