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Response lags are necessary for most physical systems. For the sake of saving time and costs, the main aim of this paper is to design the feedback control term based on the response lags varying in a certain interval and the discrete-time observations of both the system states and the Markovian states to stabilize the controlled hybrid systems. The control principles are established, which permit the control function only depends on the partial information of the states and the modes. The upper bound on the sum of the upper bound τ of response lags, and the duration τ between two consecutive observations is obtained. Some examples and numerical experiments are given to illustrate our theory. [ABSTRACT FROM AUTHOR]

This research aims to investigate the stabilization of highly nonlinear hybrid stochastic differential delay equations (HSDDEs) with Lévy noise by delay feedback control. The coefficients of these systems satisfy a more general polynomial growth condition instead of classical linear growth condition. Precisely, an appropriate Lyapunov functional is constructed to analyze the stabilization of such systems in the sense of H∞-stability and asymptotic stability. The theoretical analysis indicates that the delay can affect the stability of highly nonlinear hybrid stochastic systems. [ABSTRACT FROM AUTHOR]

Little work seems to be known about stabilization results of highly nonlinear stochastic time-varying delay systems (STVDSs) with Markovian switching and Poisson jump. This paper is concerned with the stabilization problem for a class of STVDSs with Markovian switching and Poisson jump. The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. The aim of this paper is to design a delay feedback controller to make an unstable highly nonlinear STVDSs with Markovian switching and Poisson jump $H_\infty$-stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results.

This article aims to design a linear delay feedback control to stabilise an unstable hybrid stochastic delay differential equation in distribution. Under the global Lipschitz condition, sufficient criteria are established to guarantee the stability of the controlled system. Then LMI techniques are employed to design the control law in two structure forms: state feedback and output injection. [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with the exponential stabilisation of highly nonlinear hybrid neutral stochastic differential equation (NSDE) by variable delay feedback control. There have been so far few results on the stabilization of variable delay stochastic systems by variable delay feedback control, although the stabilisation of neutral stochastic systems on the current state has been investigated. Finally, an illustrative example is provided for verifying our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the feedback stabilization of non‐homogeneous delayed bilinear systems in Hilbert state space. More precisely, under an observability‐like assumption, we prove the exponential and strong stability of the solution by using bounded feedback control. Partial stabilization is discussed as well. The proofs of the main results are based on the decomposition method. The decay estimates of the stabilized solutions are obtained. Finally, some examples are presented. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the stabilization of stochastic regime-switching Poisson jump equations (also known as stochastic differential equations with Markovian switching and Poisson jumps, abbreviated as SDEwMJs). The aim of this paper is to design a feedback controller with delay δ ( δ > 0 $\delta >0$ ) to make an unstable SDEwMJ become stable. It is proved that the delay δ is bounded by a constant δ̄. Moreover, an implicit lower bound for δ , ¯ $\bar{\delta ,}$ which can be computed numerically, is provided. As a product, the almost sure exponential stability of the controlled SDEwMJ is obtained. Besides, an example is given to demonstrate the theoretical results.

This paper aims to design the feedback control based on past states to stabilise a class of nonlinear stochastic differential equations driven by G-Brownian motion. By building up the connection between the delay feedback control and the control function of no-delay, sufficient conditions are established in terms of size of the time delay ensuring both the pth moment exponential stability and quasi-surely exponential stability of the delay feedback controlled system. Moreover, methods for determining the upper bound of the length of the time delay and further designing the delay feedback controller are provided. A numerical example is presented to demonstrate our new theory. [ABSTRACT FROM AUTHOR]

This paper addresses the bifurcation control of a fractional-order Lokta-Volterra predator– prey model by using delay feedback control. By employing time delay as a bifurcation parameter, the conditions of bifurcation are gained for controlled systems. Then, it indications that the onset of bifurcation can be postponed as feedback gain decreases. An example numerical results are ultimately exploited to validate the correctness of the the proposed scheme.

This paper is concerned with the stabilization of stochastic regime-switching Poisson jump equations (also known as stochastic differential equations with Markovian switching and Poisson jumps, abbreviated as SDEwMJs). The aim of this paper is to design a feedback controller with delay δ (δ > 0 ) to make an unstable SDEwMJ become stable. It is proved that the delay δ is bounded by a constant δ̄. Moreover, an implicit lower bound for δ , ¯ which can be computed numerically, is provided. As a product, the almost sure exponential stability of the controlled SDEwMJ is obtained. Besides, an example is given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

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Abstract This paper is concerned with chaos control and bifurcations of the Leslie–Gower type generalist predator model in a tri-trophic food web system with the time-delayed feedback control. First, the distribution of the roots of the related characteristic equations is analyzed by the polynomial theorem, the conditions to guarantee the existence of Hopf bifurcation are given by choosing the time delay as a bifurcation parameter. Then, the explicit formula for direction of Hopf bifurcation and stability of periodic solutions bifurcating are determined by using the normal form theory and center manifold theorem. Finally, the correctness of our theoretical analysis is verified by some numerical simulation.

The stabilization problem of a class of nonlinear stochastic delay differential equations driven by G? Brownain motion(G?SDDEs) was studied. Firstly,a delay feedback control was designed in the drift term of an unstable nonlinear G?SDDE, and the controlled system was therefore obtained. Then, with the Lyapunov technique,sufficient conditions for the asymptotical stability of the controlled system were given. Finally,two examples were presented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

A novel approach for designing the feedback control based on past states is proposed for hybrid stochastic differential equations (SDEs). This new theorem builds up the connection between the delay feedback control and the control function without delay terms, which enables one to construct the delay feedback control using the existing results on stabilities of hybrid SDEs. Methods for finding the upper bound of the length of the time delay are also investigated. Numerical simulations are presented to demonstrate the new theorem. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the problem of non-fragile delay feedback control for neutral stochastic Markovian jump systems (NSMJSs) with time-varying delays. The main aim is to design non-fragile and mode-dependent delay feedback controller (DFC) both in the drift part and in the diffusion part to realize closed-loop NSMJS is stochastically stable. By constructing mode-dependent and delay-dependent comprehensive Lyapunov-Krasovskii functional, the stabilization conditions are provided in terms of linear matrix inequalities (LMIs). Simulation examples including a lossless transmission line model (LTLM) are utilized to verify the validity and usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the difference between price and quantity competition in a mixed duopoly game. We describe the behavior of a duopolistic Bertrand competition market with environmental taxes. There are two cases. In the first, the public firm is privatized and in the second, it is not privatized. In case I, private duopoly (postprivatization) where players use different production methods and choose their prices with (bounded rationality and naive). In case II, mixed duopoly (preprivatization) in this case there are two levels for the market including standard objective of the private firm is to maximize profits and including another objective function of the public firm namely "private welfare maximization". We study numerically the dynamical behaviors of the models. The Nash equilibrium loses stability through a period-doubling bifurcation and the market in the end gets to be disordered. The disordered behavior of the market has been controlled by using feedback control method. [ABSTRACT FROM AUTHOR]

This paper mainly considers the stability of highly nonlinear neutral stochastic delay differential equations driven by G-Brownian motion (G-NSDDEs) via delay feedback control (DFC). Actually, such a system is maybe unstable since its coefficients satisfy polynomial growth condition, which yields a fact that the existence and uniqueness of the solutions are not ensured. Hence, up to now, it is still an unsolved problem on the feedback control of the given unstable G-NSDDEs when its coefficients satisfy polynomial growth condition. Our aim is to find a delay feedback control that makes the controlled nonlinear G-NSDDEs stable. By using the methods of Lyapunov functionals and delay feedback control, a series of conditions for the H ∞ -stability and asymptotic stability of the control system are obtained. Finally, we provide an example to support our findings. [ABSTRACT FROM AUTHOR]

The vibro-impact capsule system is a self-propelled mechanism that has abundant coexisting attractors and moves rectilinearly under periodic excitation when overcoming environmental resistance. In this paper, we study the control of coexisting attractors in this system by using a delayed feedback controller (DFC) with a constant delay. The aim of our control is to steer this complex system toward an attractor with preferable performance characteristics among multiple coexisting attractors, e.g., a periodically fast forward progression. For this purpose, we give an example of a feedback-controlled transition from a period-3 motion with low progression speed to a period-1 motion with high progression speed at the system parameters where both responses coexist. The effectiveness of this controller is investigated numerically by considering its convergence time and the required control energy input to achieve transition. We combine pseudo-spectral approximation of the delay, event detection for the discontinuities and path-following (continuation) techniques for non-smooth delay dynamical systems to carry out bifurcation analysis. We systematically study the dynamical performance of the controlled system when varying its control gain and delay time. Our numerical simulations show the effectiveness of DFC under a wide range of system parameters. We find that the desired period-1 motion is achievable in a range of control delays between a period-doubling and a grazing bifurcation. Therefore, two-parameter continuation of these two bifurcations with respect to the control delay and control gain is conducted to identify the delay-gain parameter region where the period-1 motion is stable. The findings of this work can be used for tuning control parameters in experiments, and similar analysis can be carried out for other non-smooth dynamical systems with a constant delay term. • Control of coexisting attractors in a vibro-impact capsule system is studied. • Delayed feedback control with a constant delay is adopted for attractor switching. • Capsule velocity and energy consumption are considered as performance indices. • Path-following methods are employed to identify the effective control regions. • The method can be used for analysis of non-smooth systems with a constant delay term. [ABSTRACT FROM AUTHOR]

This paper considers a class of neutral-type stochastic functional differential equations with infinite delay (IDNSFDEs) and highly nonlinear coefficients (i.e., coefficients do not satisfy the linear growth condition). These systems are often unstable. Main aim of this paper is to design a delay feedback control to make them become stable with general decay rate. This general decay stability contains the exponential stability and the polynomial stability. Finally, to illustrate our results more clearly, as examples, this paper also introduces unstable scalar neutral-type stochastic integro-differential equations and discusses their exponential and polynomial stabilisation by delay feedback controls, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of exponential mean-square stabilization of hybrid neutral stochastic differential delay equations with Markovian switching by delay feedback control. A delay feedback controller is designed in the drift part so that the controlled system is mean-square exponentially stable. We discussed two types of structure controls; that is, state feedback and output injection. The stabilization criteria are derived in terms of linear matrix inequalities. [ABSTRACT FROM AUTHOR]

To study the signal control strategy for reducing traffic emission theoretically, we first presented a kind of discrete traffic flow model with relative speed term based on traditional coupled map car-following model. In the model, the relative speed difference between two successive running cars is incorporated into following vehicle’s acceleration running equation. Then we analyzed its stability condition with discrete control system stability theory. Third, we designed a delay-feedback controller to suppress traffic jam and decrease traffic emission based on modern controller theory. Last, numerical simulations are made to support our theoretical results, including the comparison of models’ stability analysis, the influence of model type and signal control on CO 2 emissions. The results show that the temporal behavior of our model is superior to other models, and the traffic signal controller has good effect on traffic jam suppression and traffic CO 2 emission, which fully supports the theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper established a Hotelling model with bounded rationality, and analyzed the complexity of the dynamic system. By theory solution and numerical simulation we can get the conclusion that under bounded rationality, the coefficients of the price adjustment, the place, the deviation cost of the products and other parameter values determined the stability, the output bifurcation, profit bifurcation and chaotic state of the dynamic system. When the coefficients of price adjustment are in a proper range, Hotelling equilibrium can be realized as a stable dynamic equilibrium. Otherwise, the equilibrium will become unstable, bifurcation and chaos will occur to the economic system. If the economic system enters into chaotic state, a small variation in the initial value will cause a huge fluctuation of the price, then the market become unpredictable and the players can't make a reasonable decision of the future price; Delay feedback control can make the system from a chaotic state to a stable state. Demonstrated enterprises horizontal differentiation can increase the system stability. [ABSTRACT FROM AUTHOR]

We investigate the local Hopf bifurcation in Shimizu–Morioka chaotic system with delayed feedback control. We choose the delay as the parameter, and the existence of local Hopf bifurcations are verified. By using the normal form theory and the center manifold theorem, we obtain the explicit formulae for determining the stability and direction of bifurcated periodic solutions. Numerical simulations indicate that delayed feedback control plays an effective role in control of chaos. [ABSTRACT FROM AUTHOR]

Abstract: Traffic flow system’s stability analysis can be seen as a typical control system problem, which has three kinds of damping ratio properties from the viewpoint of control theory. To discover the damping ratio characteristics of traffic flow system under sudden outer impulse, firstly, we proposed a new method to build the traffic flow model based on state space theory. Secondly, we deduced one form of state transfer matrix from system closed-loop transfer function reversely. Lastly, we fully investigated the proportional–differential effects in the traffic car-following model system analytically and numerically. For unsteady traffic flow system with the proportional–differential (PD) term, in order to suppress the traffic jam, we derived a kind of decoupled delay-feedback control algorithm, which used -norm and the numerical simulation method to find the feedback gain and delay time . The traffic flow system’s transient performance formula was presented with and without PD term, and we found out the important impact of PD term on traffic flow system’ stability. The numerical simulation results substantiate our theoretical and analytical results. [Copyright &y& Elsevier]

Symmetry breaking is a ubiquitous and important phenomenon arising in a wide range of physical systems. We propose the use of the harmonic balance in combination with homotopy continuation to investigate symmetry breaking occurrence in the periodically excited systems involving time delay. Two numerical examples are given to show the details. When the Hopfield neural network is subject to external excitation, we investigate the relation of the magnitude of excitation versus the amplitude of the bias term and analyze the effect of time delay on the steady state response. The second example concerns the delayed feedback control of a nonlinear beam subject to moving load. The relationship of the position feedback gain, external excitation frequency and time delay versus the amplitudes of steady state responses are studied analytically. The symmetry breaking points are accurately predicted. In addition, the Runge-Kutta numerical simulation results are used to cross-check the efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a method for directly analyzing the stability of complex-DDEs on the basis of stability switches. Two novel criteria are developed for the stability of a class of complex-DDEs. These results not only generalize some known results in literature but also greatly reduce the complexity of analysis and computation. To validate the effectiveness of the proposed criteria, the stabilization problem of the extended time delay auto-synchronization (ETDAS) control and n time delay auto-synchronization (NTDAS) control are then further investigated, respectively. The numerical simulations are consistent with the above theoretical analysis. [ABSTRACT FROM AUTHOR]

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state. [ABSTRACT FROM AUTHOR]

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated. [ABSTRACT FROM AUTHOR]

Given an unstable hybrid neutral stochastic differential equation (NSDE), can we design a delay feedback control to make the controlled hybrid NSDE become stable? It has been proved that this is possible under the linear growth condition. However, there is no answer to the question if the drift and diffusion coefficients of the given unstable NSDE satisfy highly nonlinear growth condition. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid NSDEs whose coefficients satisfy the polynomial growth condition. [ABSTRACT FROM AUTHOR]