A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

In this paper, we consider the stabilization of the nonstationary incompressible Navier–Stokes equations around a stationary solution by a boundary linear feedback control. The feedback operator is obtained from the solution of the algebraic Bernoulli equation associated with the penalized linearized Navier–Stokes equations around an unstable stationary solution and is used to locally stabilize the original nonlinear equations. We give the explicit factorized form of the stabilizing solution of the algebraic Bernoulli equation. The numerical effectiveness of this approach is demonstrated by stabilizing the vortex shedding behind a circular obstacle. Copyright © 2011 John Wiley & Sons, Ltd.     

Delayed feedback stabilization and the Huijberts–Michiels–Nijmeijer problem

A short survey on delayed feedback stabilization is given. The Huijberts–Michiels–Nijmeijer problem on the delayed feedback stabilization of unstable equilibria of two- and three-dimensional dynamical systems is considered. It is shown that the methods of delayed feedback stabilization of unstable periodic orbits can be used with advantage for the stabilization of unstable equilibria. An analytical study based on the D-decomposition method is given. Efficient necessary and/or sufficient conditions for the stabilizability of the systems in question are obtained in the form of explicit analytic expressions. These conditions define the boundaries of stabilizability domains in terms of system parameters. It follows from these conditions that the introduction of a delayed feedback control generally extends the possibilities of stationary stabilization of linear systems with delay-free feedback.     

非线性系统周期解不动点迭代法

本文提出一种求解非线性系统周期解的数值方法。首先对非线性自治系统和非自治系统给出不同的点映射定义。其次指出用线性映射逼近原非线性映射,而线性映射是由非线性映射插值获得的。继而求取线性映射的不动点,作为原系统不动点的近似解。如不满足精度则作为下次映射的初始点。本文还提出了研究周期解稳定性的相应方法。     

Control of chaos in a piecewise smooth nonlinear system

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincaré mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems.     

Finite-dimensional stabilization of stationary Navier-Stokes systems

We consider the stabilization problem for an unstable solution of an operator equation of Navier-Stokes type. We show that one can exponentially stabilize this solution by treating it as the unique solution of a stationary variational inequality; the stabilizing operator has finite-dimensional range.     

Stability Analysis of Stationary Light Transmission in Nonlinear Photonic Structures

We study optical bistability of stationary light transmission in nonlinear periodic structures of finite and semi-infinite length. For finite-length structures, the system exhibits instability mechanisms typical for dissipative dynamical systems. We construct a Leray-Schauder stability index and show that it equals the sign of the Evans function in = 0. As a consequence, stationary solutions with negative-slope transmission function are always unstable. In semi-infinite structures, the system may have stationary localized solutions with nonmonotonically decreasing amplitudes. We show that the localized solution with a positive-slope amplitude at the input is always unstable. We also derive expansions for finite size effects and show that the bifurcation diagram stabilizes in the limit of the infinite domain size.     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.     

Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.     

Periodic solutions of competition systems with delays by an average approach

A delayed competition system of Lotka-Volterra equations, with periodic coefficients, is considered. Such a differential system admits at least a periodic positive solution if and only if the corresponding autonomous, averaged system has a positive stationary solution.     

Periodic solutions of competition systems with delays by an average approach

A delayed competition system of Lotka–Volterra equations, with periodic coefficients, is considered. Such a differential system admits at least a periodic positive solution if and only if the corresponding autonomous, averaged system has a positive stationary solution.     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

Controlling chaos for the dynamical system of coupled dynamos

This work is a tutorial on the different methods to control chaotic behaviour of the coupled dynamos system. Feedback and nonfeedback control techniques are proposed to suppress chaos to unstable equilibrium or unstable periodic solution. The stabilization of unstable fixed point of the chaotic behaviours is achieved also by bounded feedback method. Stability of the controlled systems are studied by Routh–Hurwitz criterion. Nonfeedback method and a derived method based on the delay feedback control are used to control chaos to periodic orbits. Numerical simulation results are included to show the control process of the different methods.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers

This paper deals with a multi-class priority queueing system with customer transfers that occur only from lower priority queues to higher priority queues. Conditions for the queueing system to be stable/unstable are obtained. An auxiliary queueing system is introduced, for which an explicit product-form solution is found for the stationary distribution of queue lengths. Sample path relationships between the queue lengths in the original queueing system and the auxiliary queueing system are obtained, which lead to bounds on the stationary distribution of the queue lengths in the original queueing system. Using matrix-analytic methods, it is shown that the tail asymptotics of the stationary distribution is exact geometric, if the queue with the highest priority is overloaded.      

Stabilization of unstable stationary points in equations with delayed argument

We solve the problem of localization and stabilization of an unstable stationary point of a nonlinear system of ordinary differential equations (ODE) with a delayed argument for parameter values when the ODE system has chaotic dynamics. Translated from Nelineinaya Dinamika i Upravlenie, pp. 133–141, 1999.     

A numerical verification method for a periodic solution of a delay differential equation

We describe a numerical method with guaranteed accuracy to enclose a periodic solution for a system of delay differential equations. Using a certain system of equations corresponding to the original system, we derive sufficient conditions for the existence of the solution, the satisfaction of which can be verified computationally. We describe the verification procedure in detail and give a numerical example.     

The influence of parametric excitation on floating bodies

In the present study, we investigate the possibility of stabilizing a floating body that is unstable without parametric excitation. We prescribe a non–stationary geometric constraint, that is, the centre of gravity is moved periodically in vertical direction. This periodic movement causes the parametric excitation described by a non–autonomous differential equation. By using the stability chart of the Mathieu equation, we can find sets of parameters where the stabilization of the floating body is realizable. This can help in the stabilization of a canoeist, but it can also cause stable ships to capsize. Numerical simulations and an experiment were also accomplished to confirm that the stability chart derived for the case of the floating body is realistic. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.     

Control of space-time chaos in a system of equations of the FitzHugh–Nagumo type

We perform an analytic and numerical study of a system of partial differential equations that describes the propagation of nerve impulses in the heart muscle. We show that, for fixed parameter values, the system has infinitely many distinct stable wave solutions running along the spatial axis at arbitrary velocities and infinitely many distinct modes of space-time chaos, where the bifurcation parameter is the velocity of running wave propagation along the spatial axis, which does not explicitly occur in the original system of equations. We suggest an algorithm for controlling the space-time chaos in the system, which permits one to stabilize any of its unstable periodic running waves.