Sufficient Conditions For Stability Research Articles

Novel results on asymptotic stability and synchronization of fractional-order memristive neural networks with time delays: The [formula omitted] case

This paper investigates the asymptotic stability and synchronization of fractional-order (FO) memristive neural networks with time delays. Based on the FO comparison principle and inverse Laplace transform method, the novel sufficient conditions for the asymptotic stability of a FO nonlinear system are given. Then, based on the above conclusions, the sufficient conditions for the asymptotic stability and synchronization of FO memristive neural networks with time delays are investigated. The results in this paper have a wider coverage of situations and are more practical than the previous related results. Finally, the validity of the results is checked by two examples.

Stability Analysis of Switched Linear Singular Systems with Unstable and Stable Modes.

In this paper, stability is studied for a class of switched singular systems containing both stable and unstable modes. By introducing a time-varying piecewise Lyapunov function (TVPLF) and a mode-dependent average dwell time (ADT) switching rule, the computable sufficient conditions for system stability are derived. The time-varying piecewise Lyapunov functions are piecewise continuously differentiable on every mode (but may not be differentiable at the interpolating points of the dwell time). This Lyapunov function method is particularly advantageous in overcoming the limitations of traditional multiple Lyapunov function (MLF) methods, which may not have a feasible solution when dealing with switched systems containing only unstable modes. As such, the TVPLF offers greater flexibility in application. Compared with the conventional ADT switching rule, the mode-dependent ADT switching rule not only enables each mode to have its own ADT but also allows for its own switching strategy. Specifically, the stable mode adopts a slow switching strategy while the unstable mode adopts a fast one, thereby reducing the conservatism of the ADT switching rule. Furthermore, based on the stability analysis, the time-varying controllers are proposed to stabilize the switched singular system, which can be expressed as the sequential linear combination of a series of linear state feedback on each mode. The proposed controllers are continuous for each mode, which are different from the controllers designed through the traditional MLF and MDLF methods, where the controllers designed by traditional MLF are the time-invariant linear state feedback in each mode while the controllers designed by the MDLF are piecewise continuous for each mode.

CONVERGENCE AND STABILITY OF GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS OF ADVANCED TYPE

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

Impact of Voltage-Loop Feedforward Terms on the Stability of Grid-Forming Inverters and Remedial Actions

This paper reveals the impact of voltage control loop feedforward terms on the stability of grid-forming inverters (GFMIs) connected to a grid. By deriving a mathematical model, this paper shows that unless the GFMI current control loop is fast enough, the feedforward terms in the voltage control loop may threaten the GFMI small-signal stability. To this end, a necessary and sufficient condition for GFMI stability is proposed, provided that the grid impedance at the point of connection is known. Furthermore, two easy-to-implement yet effective remedial actions are proposed to enhance stability. Aiming to preserve the control structure, the first remedial action re-tunes the voltage controller to satisfy the obtained stability condition, while the second remedial action incorporates simple gains to attenuate the feedforward terms. Moreover, to demonstrate the effectiveness of the second remedial action, the obtained necessary and sufficient stability condition is modified to include the proposed attenuation gain. Eventually, the accuracy of the stability analysis and the effectiveness of the proposed remedial actions are evaluated in simulation and experiment.

Novel Results on Output-Feedback LQR Design

This paper provides novel developments in output-feedback stabilization for linear time-invariant systems within the linear quadratic regulator (LQR) framework. First, we derive the necessary and sufficient conditions for output-feedback stabilizability in connection with the LQR framework. Then, we propose a novel iterative Newton's method for output-feedback LQR design and a computationally efficient modified approach that requires solving only a Lyapunov equation at each iteration step. We show that the proposed modified approach guarantees convergence from a stabilizing state-feedback to a stabilizing output-feedback solution and succeeds in solving high dimensional problems where other, state-of-the-art methods, fail. Finally, numerical examples illustrate the effectiveness of the proposed methods.

Set Stability of Probabilistic Logical Networks and Its Application to State-Based Games

Set stability of probabilistic multi-valued logical networks (PMLNs) is investigated in this paper. Different from the existing concepts of set stability, the set stability considered in this paper is starting from a given initial set to a given destination set. Firstly, an algebraic expression of PMLNs is established by using the semi-tensor product method. Then, a matrix method is proposed to obtain the largest invariant subset contained in the given destination set. Based on the invariant subset, a necessary and sufficient condition for set stability from a given initial set to a destination set is presented. Finally, application to state-based games is considered, and the criterion proposed for PMLNs is applied to verify the stability of recurrent state equilibrium.

Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

Classical sufficient conditions for ensuring the robust stability of a dynamical system in feedback with a nonlinearity include passivity, small gain, circle, and conicity theorems. We present a generalized version of these results for arbitrary semi-inner product spaces. Our result is purely algebraic, and holds even when the conventional discrete or continuous-time causal dynamical systems are replaced by general nonlinear relations, where there need not exist a notion of time. Our result clarifies when the sufficient conditions for robust stability are also necessary, and explains why stronger assumptions such as linearity and time-invariance are typically needed to prove necessity in the conventional dynamical systems setting.

Cooperative Formation Control for Multiple AUVs With Intermittent Underwater Acoustic Communication in IoUT

Autonomous Underwater Vehicle (AUV) system has played an important role in complex Internet of Underwater Things (IoUT). As we know, single AUV is inefficient in collecting data in large-scale IoUT. In order to improve the timeliness of data, this paper uses multiple AUVs with specific formation shape to collect sensory data. However, the intermittent and unreliable characteristics of underwater acoustic communication between AUVs seriously affects the performance of formation control. To solve this issue, this paper mainly investigates formation control problem of leader-follower structured AUVs with state prediction estimation under the condition of unreliable underwater acoustic channel. Firstly, a novel formation control law derived from backstepping sliding mode method is proposed, and then a suitable Lyapunov functional is constructed to prove sufficient stability conditions. Secondly, Gaussian prediction model considering time delay and packet dropout has been designed to estimate the intermittent communication channel. Moreover, a new metric named formation uniform degree (FUD) is proposed to measure the degree of queue uniformity in a quantitative manner. Finally, extensive simulation results compared with traditional methods based on ideal channel model demonstrate the effectiveness of proposed formation control method.

Stability of a stochastic ring network

In this paper we establish a necessary and sufficient stability condition for a stochastic ring network. Such networks naturally appear in a variety of applications within communication, computer, and road traffic systems. They typically involve multiple customer types and some form of priority structure to decide which customer receives service. These two system features tend to complicate the issue of identifying a stability condition, but we demonstrate how the ring topology can be leveraged to solve the problem.

Robust stability of fractional-order systems with mixed uncertainties: The [formula omitted] case

This paper addresses the problems of robust stability of fractional-order systems with mixed uncertainties, including multi-parameter uncertainties and norm-bounded uncertainties. The problems are relevant because on the one hand, uncertainties are common in real systems and the uncertainties in different components of systems may be of different types, and on the other hand, the non-convex and decoupling problems for fractional-order systems with α∈(0,1) and mixed uncertainties need to be thoroughly resolved. The core approaches utilized in this paper include the robust nonsingularity analysis of uncertain matrices, error-free determinant transformation and μ-analysis methods. Based on the above approaches, necessary and sufficient conditions for the robust stability of fractional-order systems with mixed uncertainties are proposed. Then, by estimating the spectral radius of the uncertain matrices, novel robust stability conditions for such uncertain fractional-order systems are obtained. After that, the robust stability bounds for such uncertain fractional-order systems are established, based on which an algorithm for solving the robust stability bounds is provided by utilizing the traversal method in the finite frequency interval. The derived results in this paper are more universal and less conservative than the existing works. Finally, two numerical examples including a fractional-order electrical circuit example are provided to demonstrate the applicability and effectiveness of the developed methods.

Long time decay analysis of complex-valued fractional abstract evolution equations with delay

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.

FINITE-TIME STABILITY OF WOLBACHIA-DRIVEN MOSQUITOES BASED ON STOCHASTIC DIFFERENTIAL EQUATIONS WITH TIME-VARYING DELAY

It is well known that various environmental factors, such as temperature, rainfall and humidity, strongly influence the development and reproduction of mosquito populations and thus the transmission dynamics of mosquito-borne diseases. In this paper, a stochastic noise is introduced to describe the effects of environmental changes on mosquito population dynamics. Considering the waiting period of wild mosquitoes from mating to emergence, the finite-time stability of wild mosquitoes by releasing Wolbachia-infected mosquitoes was studied using a stochastic differential equation with time-varying delay. Finite-time stability describes the phenomenon that the bound of the state does not exceed a specified threshold at a fixed time interval. Sufficient conditions for the finite-time stability are obtained by employing the Lyapunov function and stochastic comparison theorem. Numerical simulations are also provided to illustrate our theoretical results.

Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays

In this work, we establish sufficient conditions of the uniform asymptotic stability (UAS) of solutions to second-order and third-order of Volterra integro-differential equations (VIDE) with delay. Here, we prove two new theorems on the UAS of the solutions of the considered VIDEs. Our approach is based on Lyapunov’s second method. Our results improve and form a complement to some known recent results in the literature. Two illustrative examples are considered to support the results and two graphs are drawn to illustrate the asymptotic stability of the zero solution for the considered numerical equations. The obtained results are new and original.

Positivity and decentralized [formula omitted] control of nonlinear interconnected switched positive systems under MDDT constraint

The article is focused on the problems of positivity and decentralized L1 control for nonlinear interconnected switched positive systems(NISPSs) under mode-dependent dwell time(MDDT) constraint, which includes mode-dependent constant dwell time(MDCDT), mode-dependent minimum dwell time(MDMDT) and mode-dependent ranged dwell time(MDRDT) constraints. Both nonlinear subsystems and nonlinear interconnections are included in the considered NISPSs. First, a sufficient and necessary condition of positivity is presented for NISPSs. Next, by applying a novel discretized linear copositive Lyapunov function method, a sufficient condition of exponential stability with L1-gain performance is provided for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. Further, by applying the matrix decomposition technology to the decentralized controller gain matrix, an effective mode-dependent piece-wise design scheme of decentralized L1 controller in the solvable linear programming(LP) form is proposed for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. The proposed decentralized L1 controller design scheme for NISPSs under MDDT constraint can be degenerated into the one in mode-independent constraint case accordingly. At last, four examples with comparisons are given to illustrate the importance and effectiveness of the obtained results.

Bacterial Competition in the Presence of a Virus in a Chemostat

We derive a mathematical model that describes the competition of two populations in a chemostat in the presence of a virus. We suppose that only one population is affected by the virus. We also suppose that the substrate is continuously added to the bioreactor. We obtain a model taking the form of an “SI” epidemic model using general increasing growth rates of bacteria on the substrate and a general increasing incidence rate for the viral infection. The stability of the steady states was carried out. The system can have multiple steady states with which we can determine the necessary and sufficient conditions for both existence and local stability. We exclude the possibility of periodic orbits and we prove the uniform persistence of both species. Finally, we give some numerical simulations that validate the obtained results.

Output containment control in heterogeneous multi-agent systems without exchange of controller states

In this paper, we propose an observer-based distributed protocol for solving the output containment problem in linear heterogeneous multi-agent systems under the external disturbances. The distinct feature of the proposed protocol is that it does not require exchange of the internal controller states, which significantly reduces or completely eliminates communication burden. A sufficient local stability condition is derived and a novel controller design procedure is proposed based on tools from H∞ theory and algebraic Riccati equations. An extensive solvability analysis of the output containment control problem under the proposed protocol is carried out and theoretical guarantees for the existence of a solution for a wide class of agents’ dynamics and graph topologies are established. Numerical simulations that demonstrate the effectiveness of the proposed protocol are performed.

Stability and Hopf Bifurcation Analysis for a Phage Therapy Model with and without Time Delay

This study proposes a mathematical model that accounts for the interaction of bacteria, phages, and the innate immune response with a discrete time delay. First, for the non-delayed model we determine the local and global stability of various equilibria and the existence of Hopf bifurcation at the positive equilibrium. Second, for the delayed model we provide sufficient conditions for the local stability of the positive equilibrium by selecting the discrete time delay as a bifurcation parameter; Hopf bifurcation happens when the time delay crosses a critical threshold. Third, based on the normal form method and center manifold theory, we derive precise expressions for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to verify our theoretical analysis.

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered Ψ-Caputo fractional derivative of order 0<α<1, where the right side may depend on time, the solution, or its tempered Ψ-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the Ψ-exponential stability of trivial solutions are proven for these kinds of differential equations.

Exponential stability of time‐delay systems with flexible delayed impulse

AbstractThe problem of exponential stability for time‐delay systems with flexible delayed impulse control is investigated. Unlike the conventional Razumikhin‐type inequality, variable parameter based on the flexible impulsive gain is introduced. Sufficient conditions for exponential stability are developed for a class of time‐delay systems, for which the size of delay is not limited by impulsive intervals. A flexible impulse control scheme is developed by utilizing the flexible impulsive gain and time‐varying delays. When the system is disturbed by the impulse interferences, both the impulse gain and impulse delay can be adjusted accordingly to make the system exponential stable. The effectiveness of the present results is illustrated by two numerical examples and an application.