Stochastic averaging on infinite time interval for a class of nonlinear systems with stochastic perturbation

2009 ◽  
Author(s):  
Shu-Jun Liu ◽  
Miroslav Krstic
Keyword(s):  
Nonlinear Systems ◽  
Stochastic Perturbation ◽  
Stochastic Averaging ◽  
Time Interval ◽  
Infinite Time ◽  
Infinite Time Interval

DECAY/GROWTH RATE ESTIMATION USING INSTANTANEOUS LYAPUNOV EXPONENT

2012 ◽  
Vol 22 (03) ◽  
pp. 1250047
Author(s):  
YUSUKE TOTOKI ◽  
TAKAMI MATSUO
Keyword(s):  
Growth Rate ◽  
Lyapunov Exponent ◽  
Nonlinear Systems ◽  
Growth Rates ◽  
Jacobian Matrix ◽  
Time Interval ◽  
Comparison Function ◽  
Infinite Time ◽  
Infinite Time Interval ◽  
The Mean

The Lyapunov exponent gives a measure of the mean decay/growth rates of the flows of nonlinear systems. However, the Lyapunov exponent needs an infinite time interval of flows and the Jacobian matrix of system dynamics. In this paper, we propose an instantaneous decay/growth rate that is a kind of generalized Lyapunov exponent and call the instantaneous Lyapunov exponent (ILE) with respect to a decay function. The instantaneous Lyapunov exponent is one of the measures that estimate the decay and growth rates of flows of nonlinear systems by assigning a comparison function and can apply a stable system whose decay rate is slower than an exponential function. Moreover, we propose a synchronization measure of two signals using the ILE.

scholarly journals Problems of Optimal Control for a Class of Linear and Nonlinear Systems of the Economic Model of a Cluster

2020 ◽  
Vol 07 (02) ◽  
pp. 109-127
Author(s):  
Zainelkhriet Murzabekov ◽  
Marek Milosz ◽  
Kamshat Tussupova ◽  
Gulbanu Mirzakhmedova
Keyword(s):  
Optimal Control ◽  
Nonlinear Systems ◽  
Control Object ◽  
Quadratic Functional ◽  
Time Interval ◽  
Infinite Time ◽  
Optimal Stabilization ◽  
Infinite Time Interval ◽  
The Mathematical Model ◽  
Linear And Nonlinear Systems

For the mathematical model of a three-sector economic cluster, the problem of optimal control with fixed ends of trajectories is considered. An algorithm for solving the optimal control problem for a system with a quadratic functional is proposed. Control is defined on the basis of the principle of feedback. The problem is solved using the Lagrange multipliers of a special form, which makes it possible to find a synthesizing control. The problem of optimal stabilization for a class of nonlinear systems with coefficients that depend on the state of the control object is considered. The results obtained for nonlinear systems are used in the construction of control parameters for a three-sector economic cluster on an infinite time interval.

scholarly journals Stability of Nonlinear Systems of Fractional Order Differential Equations

2010 ◽  
Vol 7 (4) ◽  
pp. 1458-1461
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Order ◽  
Finite Time ◽  
Time Interval ◽  
Sufficient Condition ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Fractional Order Differential Equations ◽  
Infinite Time Interval

In this paper, a sufficient condition for stability of a system of nonlinear multi-fractional order differential equations on a finite time interval with an illustrative example, has been presented to demonstrate our result. Also, an idea to extend our result on such system on an infinite time interval is suggested.

On the Interrelation of Two Linear NonStationary Problems with Multiple Evaders

2015 ◽  
Vol 17 (04) ◽  
pp. 1550013 ◽  
Author(s):  
N. N. Petrov ◽  
K. A. Shchelchkov
Keyword(s):  
Finite Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Pursuit Problem ◽  
Infinite Time Interval ◽  
Admissible Controls

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

On the Placement of Active Bars and Optimal Feedback Gain in Random Adaptive Structures for Vibration Suppression

2005 ◽  
Author(s):  
F. Yang ◽  
R. Sedaghati ◽  
D. Younesian ◽  
E. Esmailzadeh
Keyword(s):  
Optimization Technique ◽  
Optimal Placement ◽  
Feedback Gain ◽  
Time Interval ◽  
Infinite Time ◽  
Random Load ◽  
Adaptive Structures ◽  
Optimal Velocity ◽  
Optimal Feedback ◽  
Infinite Time Interval

This paper addresses active vibration control of adaptive structures using piezoelectric active structural elements with built-in sensing and actuation function. Optimal placement and feedback gain of these active members are important issues, which are discussed in this study. An efficient optimal placement strategy has been developed using minimum control energy dissipating over infinite time interval. Optimal location of active members has been found through minimization of total control energy dissipating over infinite time interval using Simulated Annealing (SA) algorithm. Moreover, the effect of structural randomness and random load in optimal feedback gain has been investigated. To accomplish this, a mathematical model with reliability constraints on the stress and displacement is developed and the optimal velocity feedback gain of active members, using probabilistic optimization technique, is obtained. Illustrative examples are presented to demonstrate the effectiveness of this methodology. It is shown that deterministic approach can not provide reliable optimum design values.

scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Power Law ◽  
Memory Function ◽  
Time Interval ◽  
Infinite Time ◽  
Suggested Approach ◽  
Infinite Time Interval ◽  
History Of ◽  
With Memory

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

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