Sufficient Conditions For Stability Research Articles

Dual CAR-T cell therapy for glioblastoma: strategies to cure tumour diseases based on a mathematical model

AbstractThe CAR-T cell immunotherapy entails the genetic reprogramming of T-lymphocytes, which then engage with cancer cells, triggering an anti-tumour immune response. While this treatment has gained approval for hematological cancers, addressing solid tumours presents new obstacles. Challenges include the heterogeneity of antigen expression within solid tumours, encompassing antigen-positive non-tumoural cells, the presence of immune inhibitory molecules, and the difficulty of CAR-T cell trafficking within the tumour microenvironment. In this article, we analytically study a generalisation of a mathematical model proposed by León-Triana et al. (Cancers 13(4):703, 2021a. https://doi.org/10.3390/cancers13040703, Commun Nonlinear Sci Numer Simul 94:105570). This model focuses on the dynamics of glioblastoma, the most aggressive brain tumour, and its response to CAR-T cell treatment. We study the basic properties of the model, the dynamics of the solutions of the model when the treatment is not sustained during the time, and finally we study analytically the model when the therapy is constant, periodic and/or impulsive. We derive sufficient conditions for global stability of tumour-free equilibrium, as well as necessary and sufficient conditions for local stability of the equilibrium obtaining conditions for an effective treatment. Finally, we perform different numerical simulations to find the strategies to keep the tumour under control. The obtained results are based on a combination of different analytical techniques in differential equations, dynamical systems and numerical simulations.

Distributed control and passivity‐based stability analysis for time‐delayed DC microgrids

AbstractFor cooperation among distributed generations in a DC microgrid (MG), distributed control is widely applied. However, the delay in distributed communication will result in steady‐state bias and the risk of instability. This paper proposes a novel distributed control for time‐delayed DC MGs to achieve accurate current proportional sharing and weighted average voltage regulation. Firstly, by utilizing an advanced observer based on the PI consensus algorithm, the steady‐state bias problem is addressed. Then, using the passivity theory, stability analysis is conducted to reveal the principle of system instability caused by communication delay. On this basis, to offset the adverse effects of communication delay on the system stability, scattering transformation is introduced in the observer‐based distributed control. Moreover, considering the potential delay from the measurement stage in real‐life scenarios, the sufficient condition of the system stability is concluded by constructing the Lyapunov–Krasovskii functional. Finally, the performance of the proposed control and conclusions of stability analysis are verified by hardware‐in‐loop tests.

Sampled-data control for Markovian switching neural networks with output quantization and packet dropouts

This paper explores sampled-data control for Markovian switching neural networks (MSNNs) with dynamic output quantization and packet dropouts. The primary goal is to construct a multi-mode, quantized sampled-data controller that ensures stochastic stability and H∞ disturbance-reduction performance of the closed-loop MSNN. A Bernoulli-distributed random variable with uncertain probability is introduced to characterize the incidence of packet dropouts. To describe potential mode inconsistencies that may occur between the MSNN and controller, an exponential hidden Markov model is employed. Furthermore, the quantizer’s dynamic scaling factor is intentionally built as a piecewise function to avoid the potential division-by-zero problem. A sufficient condition for stochastic stability and H∞ disturbance-reduction performance is proposed, utilizing a mode- and time-dependent Lyapunov-type functional and several stochastic analysis tools. Then, through decoupling nonlinearities, a numerically efficient approach for determining the desired controller gains and parameter range associated with the dynamic scaling factor is developed. In order to facilitate comparisons, the situation with no uncertainty in the probability of packet dropouts is studied, and both analysis and design approaches are offered. Finally, two simulation examples are provided to validate the effectiveness and applicability of the developed approaches.

Stabilization for a class of fractional-order nonlinear reaction–diffusion systems with time-varying delay: Event-triggered boundary control approach

Based on the hybrid event-triggered mechanism (HETM), the boundary stabilization issue for fractional-order nonlinear reaction–diffusion systems (FNRDSs) with time-varying delay is studied by using two kinds of measurements. First, when the system state is measurable, a event-triggered feedback controller (ETFC) is designed directly based on the average measured output. Secondly, for the case that the state is unmeasurable, an event-triggered feedback controller based on observer framework is constructed through the boundary point measurement information. Utilizing the Lyapunov method and Wirtinger’s inequality, sufficient conditions for the asymptotic stability of the system are given in the form of linear matrix inequalities (LMIs), respectively, in which the Razumikhin theorem is used to deal with time-varying delay. Meanwhile, it is proved that Zeno behavior can be excluded by the designed HETM. Finally, numerical simulations demonstrate the validity and feasibility of the proposed control scheme.

On the positivity and asymptotic stability of the fractional 2D hybrid linear Roesser model

The aim of this paper is to introduce a new 2D fractional hybrid linear system described by the Roesser model with the conformable fractional derivative in the continuous time variables and the fractional difference in the discrete part. The solution of the considered model is derived. The positivity conditions are then extracted and the Cayley–Hamilton theorem is extended. A new necessary and sufficient conditions for the asymptotic stability of the proposed positive fractional 2D hybrid linear model are then established. Finally, some numerical examples are presented to show the applicability and the effectiveness of the presented methods.

Analysis of an intra- and interspecific interference model with allelopathic competition

In this paper, we propose an original model of two-microbial species competing for a single nutrient in the chemostat including general intra- and interspecific density-dependent growth rates with allelopathic interactions. Each species produces a toxin that affects the growth of other species as well as its own growth. The removal rates are distinct and include the specific death rate and the autotoxicity of each species. We establish an in-depth mathematical analysis by determining the multiplicity of all steady states of the three-dimensional system and their necessary and sufficient conditions of existence and local stability according to the operating parameters, which are the dilution rate and the inflowing concentration of the substrate. To describe the asymptotic behavior of the process according to these control parameters, we first determine theoretically the operating diagram. Using MATCONT software, these theoretical results are validated numerically but it reveals the cusp bifurcation that occurs by varying two parameters. The one-parameter bifurcation diagram in the dilution rate shows that there can be either transcritical or saddle-node bifurcations. Finally, we apply our results to a particular model in the literature without intra- and interspecific interference but with only allelopathic effects of the second species on the first species. We show that one of the coexistence steady states is locally exponentially stable when it exists, whereas in the literature they have not been able to demonstrate that the stability condition of this steady state is always fulfilled.

Stability and convergence rate analysis of continuous-time delay-difference systems with both point and distributed delays

This paper studies the exponential stability and convergence rate analysis of continuous-time delay-difference systems. Firstly, stability and convergence rate analysis of delay-difference systems with both point delays and distributed delays having exponential integral kernels are studied by using the weighted Lyapunov–Krasovskii functionals (LKFs) approach and the state transformation approach, respectively. Different sufficient stability conditions expressed by linear matrix inequalities (LMIs) are presented. Secondly, as a particular case, LMIs based conditions and spectral radius based conditions are given to ensure the exponential stability with a guaranteed convergence rate for delay-difference systems with both point delays and distributed delays having constant integral kernels, respectively. Finally, numerical examples illustrate the effectiveness of the obtained results.

On the local quadratic stability of T–S fuzzy systems in the vicinity of the origin

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. Discussions on the inherent limitations associated with fuzzy Lyapunov approaches are also given. To illustrate the theoretical results, we provide comprehensive examples that demonstrate the core concepts and validate the efficacy of the proposed conditions.

Almost sure exponential stability of stochastic nonlinear semi-Markov jump T-S fuzzy systems under intermittent EDF scheduling controller

This paper studies the almost sure exponential stability (ASES) with a random switching process of stochastic nonlinear semi-Markov jump T-S Fuzzy systems (SMJT-SFSs) based on an intermittent scheduling controller. The intermittent scheduling controller is established by the earliest deadline first (EDF) algorithm, where the controller is enabled to take a rest interval. By building a mode-dependent Lyapunov function and employing the Itô formula, the sufficient stability conditions of the SMJT-SFSs with intermittent scheduling controller are obtained about solvable forms of linear matrix inequalities (LMIs). Furthermore, we explore two special cases involving linear systems and deterministic systems. In addition, an application example of the nonlinear robot arm model of single-link is provided to illustrate the results.

Stability and stabilization of discrete-time linear compartmental switched systems via Markov chains

The stabilizing switching signal design of discrete-time linear compartmental switched systems (DT-LCSSs) has been heretofore unsolved. It has been proven that a DT-LCSS is stabilizable if and only if it is stabilizable by a periodic switching signal. However, it still needs to be determined whether the period of a stabilizing switching signal can be confined within a bound. Moreover, the existing design method for stabilizing periodic switching signals requires the diagonal entries of system matrices of all subsystems to be strictly positive. In this study, we propose a novel approach to solve this problem completely. We construct a discrete-time Markov chain for a given DT-LCSS, termed the associated Markov chain, and prove the equivalence of stability and stabilizability between the DT-LCSS and the associated Markov chain. Based on this, verifiable necessary and sufficient conditions for stability and stabilizability are derived. Especially, the period of a stabilizing switching signal for an n-dimensional DT-LCSS can always be chosen within the bound n2−n+1. We propose a state-independent stabilizing switching signal design method for general stabilizable DT-LCSSs. We also prove the equivalence between stabilizability by state-independent switching laws and stabilizability by state-dependent switching laws. A state-dependent global stabilizing switching signal design method is also proposed. Additionally, the proposed results are applied to the consensus analysis of discrete-time leader–follower multi-agent systems with switching communication digraphs. The effectiveness of the theoretical results is demonstrated by examples.

Tensor product model transformation-based reinforcement learning neural network controller with guaranteed stability

Despite the success of neural network (NN) controllers, the stability of closed-loop NN control systems is still a major challenge. This paper proposes a constrained reinforcement learning (RL) algorithm to design optimal NN controllers for a class of nonlinear dynamical systems while guaranteeing the stability of the control systems. By representing a controlled system as a tensor product (TP) model, a sufficient stability condition for a closed-loop NN control system is derived, which is then employed as a training constraint for the NN controller throughout the training process to ensure the stability in each training step. By showing the existence of a Lyapunov function, which is also directly obtained during the training process, the stability can be theoretically confirmed. Simulation studies on various nonlinear dynamical systems illustrate that the NN controllers trained with the proposed constrained RL algorithm outperform linear–quadratic–regulator controllers and yield roughly the same performance index values as those trained with an unconstrained RL algorithm, indicating that the imposed constraint does not significantly affect the training performance. Most importantly, while simulation results show that all controllers provide closed-loop stability, only the NN controllers trained with the proposed constrained RL algorithm theoretically guarantee stability.

COMPETITION BETWEEN PLASMID-BEARING AND PLASMID-FREE ORGANISMS IN A CHEMOSTAT WITH DISTINCT REMOVAL RATES AND AN EXTERNAL INHIBITOR

A mathematical model of competition between plasmid-bearing and plasmid-free organisms for a single limiting resource in a chemostat with distinct removal rates and in the presence of an external inhibitor is analyzed. This model was previously introduced in the special case where the growth rate functions and the absorption rate of the inhibitor follow the Monod kinetics and the removal rates are the same as the dilution rate. Here, we consider the general case of monotonic growth and absorption functions, and distinct removal rates. Through the three operating parameters of the model, represented by the dilution rate, the input concentrations of the substrate and the inhibitor, we give necessary and sufficient conditions for existence and stability of all equilibria. To better understand the richness of the model’s behavior with respect to those operating parameters, we determine the operating diagram theoretically and numerically. This diagram is very useful to understand the model from both the mathematical and biological points of view.

A new criterion on stability in distribution for a hybrid stochastic delay differential equation

A new sufficient condition for stability in distribution of a hybrid stochastic delay differential equation (SDDE) has been proposed. In the new criterion leading to stability for an SDDE, its main component only depends on the coefficients of a corresponding SDE without delay. The Lyapunov method is applied to find an upper bound, so that the SDDE is stable in distribution if the delay is less than the upper bound. Also, the criterion shows that delay terms can be impetuses toward the stability in distribution.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ > 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

Switching LPV approach for analysis and control of TCP-based cyber-physical systems under DoS attack

Cyber-physical systems (CPSs) integrate controllers, sensors, actuators, and communication networks. Tight integration with communication networks makes CPSs vulnerable to cyberattacks. In this paper, the impact of denial of service (DoS) attacks on the stability of cyber physical systems is investigated, with a focus on the transmission control protocol (TCP). A sufficient stability condition is extracted in linear matrix inequality (LMI) form. The TCP-CPS under DoS attack is modeled as a switching linear parameter-varying (LPV) time-delay system using the Markov jump model. Subsequently, a parameter-dependent stabilizing controller is designed for CPSs under DoS attacks, taking into account network parameters. Finally, the efficiency and feasibility of the findings are demonstrated through a well-known case study in the networked control systems literature.

Graph-Based Restricted and Arbitrary Switching for Switched Positive Systems via a Weak CLCLF.

This article studies the stability problem of discrete-time switched positive linear systems (SPLSs) with marginally stable subsystems. Based on the weak common linear copositive Lyapunov function (weak CLCLF) approach, the switching property and the state component property are combined to ensure the asymptotic stability of SPLSs under three types of switching signals. First, considering the transfer-restricted switching signal described by the switching digraph, novel cycle-dependent joint path conditions are proposed in combination with state component digraphs. Second, under the time interval sequence, two types of path conditions are constructed for designing switching schemes. Third, necessary and sufficient conditions for the asymptotic stability of SPLSs under arbitrary switching are established. Finally, three examples are provided to illustrate the effectiveness of the proposed method.

Computation of the exact time delay margin for vehicle platoon under generic communication topologies

Due to the limited bandwidth and transmission congestion of the vehicle platoon's communication, it is inevitable to induce time delay, which significantly degrades the control performance of the vehicle platoon, even resulting in instability. This paper focuses on analyzing the internal stability under generic communication topologies and presents a method of computing the exact time delay margin (ETDM). The proposed method can offer a necessary and sufficient internal stability condition with no conservatism. Firstly, to reduce the analytical complexity and computational burden elegantly, we decompose the closed-loop platoon dynamics into a set of individual subsystems via similarity transformation and matrix factorization. This decomposition approach is applicable for any general communication topology. Secondly, an explicit formula is deduced to compute the ETDM by surveying the characteristic roots' distribution of all these individual subsystems. It is further demonstrated that only the positive purely imaginary roots need to be considered to compute the ETDM. Finally, simulations are conducted to demonstrate the effectiveness of the theoretical claims.

Dynamics of a Predator–Prey System with Impulsive Stocking Prey and Nonlinear Harvesting Predator at Different Moments

In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management.

H∞ stabilization of singular networked cascade control systems subject to periodic DoS attack

In this paper, a [Formula: see text] controller with an event-triggered mechanism is designed for a singular networked cascade control system (SNCCS) under periodic denial-of-service (DoS) attacks. The SNCCS is treated as a system of differential-algebraic equations (DAEs) with an index of two and is modeled as a switched system. The paper first establishes sufficient stability conditions, which are formulated as linear matrix inequalities based on multiple Lyapunov functions. These conditions ensure that the system remains regular, free from impulse modes, and stable. Next, state feedback controllers are designed for both the inner loop and the outer loop. However, the outer loop is subjected to periodic DoS cyber-attacks. Finally, the proposed method’s effectiveness is demonstrated through simulation results on a practical thermal cascade control system.

Memory‐based event‐triggered control for networked control system under cyber‐attacks

AbstractThis article focuses on the problem of stability for a class of linear networked control systems (NCSs) subjected to network communication delays and random deception attacks. A new memory event‐triggered mechanism (METM) is proposed to reduce the unnecessary transmitted data through the communication channel and then enhance the network resources. In this context, a new memory stochastic state feedback controller is proposed to stabilize the closed‐loop networked control system. A new randomly occurring deception attacks model is employed to deal with the security problem of NCSs. Sufficient stability conditions are derived based on a suitable Lyapunov‐Krasovskii functional (LKF). The designed methodology is proposed in terms of linear matrix inequality to synthesize both event‐triggered parameters and controller gains, and to reduce the conservatism of the system some integral lemma are exploited to bind the time derivative of the LKF. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed method which provides a maximal upper bound value of the network‐induced delay and less transmitted packet regarding the maximal value delay obtained in other works, so less conservatism results are obtained, compared to previous ones in the literature.