Type Lyapunov Function Research Articles

Stabilization of nonlinear safety-critical systems by relaxed converse Lyapunov-barrier approach and its applications in robotic systems

Combining safety objectives with stability objectives is crucial for safety-critical systems. Existing studies generally unified these two objectives by constructing Lyapunov-type barrier functions. However, insufficient analysis of key set relationships within the system may render the proposed safety and stability conditions conservative, and these studies also did not provide how to use such conditions to design safety-stability control strategies. This paper proposed a feasible and constructive design to achieve stabilization of safety-critical systems by a relaxed converse Lyapunov-barrier approach. By analyzing the relationships between a series of sets associated with the safety-critical system, the stability and safety conditions can be appropriately relaxed. Then, with the help of relaxed converse control Lyapunov-barrier functions (RCCLBFs), a theoretical result was obtained for the stability of affine nonlinear systems with safety constraints. Subsequently, a constructive method was developed for a second-order strict-feedback system to transform the process of solving RCCLBFs into a Lyapunov-like stabilization problem. Finally, the proposed safety-stability control method is exerted on a robotic system and demonstrated by simulations.

Dynamics of the epidemiological Predator-Prey system in advective environments.

This paper aims to establish the existence of traveling wave solutions connecting different equilibria for a spatial eco-epidemiological predator-prey system in advective environments. After applying the traveling wave coordinates, these solutions correspond to heteroclinic orbits in phase space. We investigate the existence of the traveling wave solution connecting from a boundary equilibrium to a co-existence equilibrium by using a shooting method. Different from the techniques introduced by Huang, we directly prove the convergence of the solution to a co-existence equilibrium by constructing a special bounded set. Furthermore, the Lyapunov-type function we constructed does not need the condition of bounded below. Our approach provides a different way to study the existence of traveling wave solutions about the co-existence equilibrium. The existence of traveling wave solutions between co-existence equilibria are proved by utilizing the qualitative theory and the geometric singular perturbation theory. Some other open questions of interest are also discussed in the paper.

Stability and synchronization of octonion-valued neural networks with leakage and mixed delays on time scales

There has been a great deal of interest in the last few years for neural networks (NNs) with values in multidimensional domains. The most popular models are complex-valued neural networks (CVNNs), followed by quaternion-valued neural networks (QVNNs), and, more recently, by Clifford-valued neural networks (ClVNNs). However, also very recently, a different type of NNs were put forward, namely octonion-valued neural networks (OVNNs). OVNNs are defined on the 8D octonion algebra, and they are not a special type of ClVNNs, because Clifford numbers are associative, whereas octonions are not. Moreover, beside the complex and quaternion algebras, the only other normed division algebra over the reals is the algebra of octonions, which makes OVNNs a direct generalization of CVNNs and QVNNs from this point of view, raising interest for applications handling high-dimensional data. On the other hand, systems defined on time scales were proposed as a generalization of both discrete time and continuous time systems, or any type of hybrid combination between the two. Finally, time delays appear as a consequence of implementing NNs in real life circuits. Taking all these into consideration, this paper studies the fundamental properties of exponential stability and exponential synchronization for OVNNs with leakage and mixed delays defined on time scales. In order to avoid the problems raised by the non-associativity of the octonion algebra, the OVNN model is decomposed into a real-valued one. Then, two different Lyapunov-type functionals are defined and the particularities of time scale calculus are used in order to deduce sufficient conditions expressed as scalar and linear matrix inequalities (LMIs) for the exponential stability of the proposed models, based on Halanay-type inequalities suitable for time scale systems. Afterwards, a state feedback controller is used to deduce sufficient criteria given as scalar inequalities and LMIs for the exponential synchronization of the same type of models. The generality of the model, given by the definition on the algebra of octonions, the use of different types of delays, and the definition on time scales represent an important advantage. Also, for less general models, it is possible to particularize the obtained results. With the aim to demonstrate each of the paper’s four theorems, four numerical examples are provided.

Fixed-time synchronization of discontinuous fuzzy competitive neural networks via quantized control

This article concentrates on fixed-time synchronization for discontinuous fuzzy competitive neural networks including time-varying delays. Firstly, to cut down the signal transmission cost, two kinds of nontrivial control strategies, namely quantized control protocols without the linear part, are developed to investigate fixed-time synchronization of discontinuous fuzzy competitive neural networks based on two different types of Lyapunov functions. Then, with the aid of inequality skills and differential inclusion theory, several uncomplicated synchronization criteria are attained and the synchronized time possessing less conservativeness is effectively evaluated. A numerical example is presented at last to testify the validity of theoretical analysis.

Quantized Communications in Containment Maneuvering for Output Constrained Marine Surface Vehicles: Theory and Experiment

The aim of this paper is to derive a containment maneuvering coordination strategy for a team of marine surface vehicles (MSVs) to track one parameterized path guided by two virtual leaders, where the tracking errors are constrained and the communications among MSVs are quantized. A two-level coordinated framework for containment maneuvering is proposed. For the kinematic level, to obtain the performance of tracking error constraints, guidance laws are derived by developing parameterized line-of-sight (LOS)-based symmetrical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ln$</tex-math></inline-formula> -type Lyapunov functions, where the control functions are injected into the guidance laws to prevent constraint violations. For the coordination level, a communication protocol is proposed to reduce the amount of communication data by developing a quantized control strategy. Compared with the current containment maneuvering strategy, the whole design can bind the along-track and cross-track errors into the user-defined regions, increase the convergence speed of tracking errors, save bandwidth of the whole network and reduce the sensitivity of the initial positions of MSVs. Theoretical analysis shows that all signals are bounded. Experimental examples are also provided to verify the proposed containment maneuvering strategy.

General stability for a Cohen–Grossberg neural network system

Of concern is a Cohen–Grossberg neural network (CGNNs) system taking into account distributed and discrete delays. The class of delay kernels ensuring exponential stability existing in the previous papers is enlarged to an extended class of functions guaranteeing more general types of stability. The exponential and polynomial (or power type) type stabilities becomes particular cases of our result. This is achieved using appropriate Lyapunov-type functionals and the characteristics of the considered class.

Observer-Based Fixed-Time Adaptive Fuzzy Consensus DSC for Nonlinear Multiagent Systems.

This article studies the output-feedback fixed-time fuzzy consensus control problem for nonlinear multiagent systems (MASs) under the directed communication topologies. Since the controlled systems contain the unmeasurable states and unknown dynamics, the unmeasurable states are reconstructed via linear state observers, and fuzzy logic systems are utilized to identify the unknown internal dynamics. By constructing the integral type Lyapunov function, a fixed-time adaptive fuzzy consensus control scheme is developed by introducing the nonlinear filter technique into the backstepping recursive technique adaptive control algorithm. The presented consensus control method can not only guarantee the controlled system is semi-global practical fixed-time stable (SGPFTS), but also avoid the singular problem in existing backstepping recursive control design methods. Finally, an application of unmanned surface vehicles is provided to verify the effectiveness of the presented fixed-time fuzzy consensus control method.

Dynamical Analysis of an Age-Structured SVEIR Model with Imperfect Vaccine

Based on the spread of COVID-19, in the present paper, an imperfectly vaccinated SVEIR model for latent age is proposed. At first, the equilibrium points and the basic reproduction number of the model are calculated. Then, we discuss the asymptotic smoothness and uniform persistence of the semiflow generated by the solutions of the system and the existence of an attractor. Moreover, LaSalle’s invariance principle and Volterra type Lyapunov functions are used to prove the global asymptotic stability of both the disease-free equilibrium and the endemic equilibrium of the model. The conclusion is that if the basic reproduction number Rρ is less than one, the disease will gradually disappear. However, if the number is greater than one, the disease will become endemic and persist. In addition, numerical simulations are also carried out to verify the result. Finally, suggestions are made on the measures to control the ongoing transmission of COVID-19.

Fixed-time formation fault tolerant control for unmanned surface vehicle systems with intermittent actuator faults

This article studies the fixed-time fuzzy formation tracking control problem for multiple unmanned surface vehicle (USV) systems with intermittent actuator faults. Since each USV is subject to unknown motion dynamics and the external disturbances. The fuzzy state observers are developed to estimate the positions and velocity in multiple USV systems, and disturbance observers are developed to estimate the external disturbances, and fuzzy logic systems are utilized to identify the unknown motion dynamics. Then, to compensate for the influence of intermittent actuator faults, the novel fixed-time adaptive output-feedback formation fault-tolerant controllers are constructed by constructing the integral type Lyapunov function and co-designing the last virtual controller. The presented formation control method can guarantee all the signals in multiple USV systems are bounded in fixed-time. Finally, simulation results and comprehensive comparisons are provided to verify the effectiveness of the presented formation control method.

Stabilization of a Non-linear Fractional Problem by a Non-Integer Frictional Damping and a Viscoelastic Term

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the possibilities of stabilizing the system by a lower-order fractional term and/or a memory term involving the Laplacian. We prove a global Mittag–Leffler stability result in case a fractional frictional damping is active and a local Mittag–Leffler stability result when the material is viscoelastic in case of small relaxation functions. Unlike the integer-order problems, additional serious difficulties arise in the present case. These difficulties are highlighted clearly in the introduction. They are mainly due to the memory dependence of the fractional derivatives which is the cause of the invalidity of the product rule in particular. We utilize several properties in fractional calculus. Moreover, we introduce new Lyapunov-type functionals in the context of the multiplier technique.

Event-Triggered Adaptive Fuzzy Asymptotic Tracking Control of Nonlinear Pure-Feedback Systems With Prescribed Performance.

This article considers the problem of fixed-time prescribed event-triggered adaptive asymptotic tracking control for nonlinear pure-feedback systems with uncertain disturbances. The fuzzy-logic system (FLS) is introduced to deal with the unknown nonlinear functions in the system. By constructing a new type of Lyapunov function, the restrictive requirement that the upper bounds of the partial derivative of the unknown system functions need to be known is relaxed during the controller design process. At the same time, by developing a novel fixed-time performance function (FPF), the fixed-time prescribed performance (FPP) can be achieved, that is, the tracking error can converge to the neighborhood of the origin in a fixed time and finally converges to zero asymptotically. In addition, the event-triggered strategy is developed to reduce the waste of communication resources. The proposed control law can ensure that all the signals of the system are bounded. Meanwhile, the Zeno behavior can be effectively avoided. Finally, an example is provided to prove the effectiveness of the proposed scheme.

Global attractive set of neural networks with neutral item

This paper investigates the global attractive set of neural networks with neutral item. To better deal with the neutral terms, different types of activation functions are considered. Based on matrix measures, inequality techniques, and Lyapunov theory, three new types of Lyapunov functions are designed to find the global attractive set of the system. We give out a simulation example to verify the validity of theory results. The result is very inclusive, whether the system has equilibrium or not. As long as the system is stable, we can find its global attractive set.

Stability analysis of sampled-data systems with sector-bounded input nonlinearity

This work proposes a method for the stability analysis of aperiodic sampled-data control systems with sector and slope bounded input nonlinearities. The stability conditions are derived by using a hybrid system representation and a timer-dependent Lur'e type Lyapunov function. Considering a polynomial timer-dependence, the stability conditions are cast in sum-of-squares optimization problems aiming at computing the largest range of sampling intervals or the largest sector bounds on the nonlinearity for which the origin of the closed-loop system is globally asymptotically stable.

Methods of Nonhyperbolic Shadowing

This paper is a survey of some recent results giving sufficient conditions of shadowing for dynamical systems in the absence of hyperbolicity. The main topics of the survey are as follows: method of pairs of Lyapunov type functions, shadowing in a neighborhood of a nonisolated fixed point, conditional multiscale shadowing for sequences of mappings of a Banach space, conditional shadowing for dynamical systems on so-called simple time scales. The paper contains a new result on conditional multiscale shadowing in the case of an infinite family of projections of the phase space.

On the regularization and matrix Lyapunov functions for fuzzy differential systems with uncertain parameters

&lt;abstract&gt;&lt;p&gt;In this paper, for a regularized fuzzy system, a generalization of the direct Lyapunov method is adapted on the base of matrix-valued Lyapunov-like functions. First, the new concept of a regularization scheme for fuzzy systems is discussed and the matrix-valued Lyapunov function technique is introduced. Then, sufficient conditions are established for the boundedness and stability of the equilibrium set of solutions of the regularized fuzzy system of differential equations. Scalar and vector Lyapunov-type functions are used based on an auxiliary matrix-valued function. Finally, a discussion is offered for the future directions of the proposed approach. Since the strategies for the analysis of the stability of fuzzy models are very important in numerous aspects, we expect that our results will inspire researchers to develop the introduced concept.&lt;/p&gt;&lt;/abstract&gt;

Neutral-Type and Mixed Delays in Fractional-Order Neural Networks: Asymptotic Stability Analysis

The lack of a conventional Lyapunov theory for fractional-order (FO) systems makes it difficult to study the dynamics of fractional-order neural networks (FONNs). Instead, the existing literature derives necessary conditions for various dynamic properties of FONNs using Halanay-type lemmas. However, when these lemmas are used, the results are frequently more conservative than those produced for integer-order neural networks (NNs). In order to provide sufficient criteria that are less conservative than those found in other research, a novel application of the Halanay-type lemma is made within this study. Thus, for extremely general FONNs containing neutral-type, time-varying, and distributed delays, sufficient conditions presented by way of linear matrix inequalities (LMIs) and algebraic inequalities are achieved. For the FO scenario, a model this broad and including so many different kinds of delays is developed for the first time. Additionally, a novel form of Lyapunov-like function is built, which results in less stringent algebraic inequalities. One of the first times in the setting of FONNs, the free-weighting matrix method is also used to further lower the conservativeness of the obtained conditions. Based on different Lyapunov-type functions, three theorems are developed regarding the asymptotic stability of the proposed networks. Three numerical simulations are used to demonstrate the theoretical developments.

Synchronization of Fractional-Order Neural Networks with Time Delays and Reaction-Diffusion Terms via Pinning Control

This paper introduces a novel synchronization scheme for fractional-order neural networks with time delays and reaction-diffusion terms via pinning control. We consider Caputo fractional derivatives, constant delays and distributed delays in our model. Based on the stability behavior, fractional inequalities and Lyapunov-type functions, several criteria are derived, which ensure the achievement of a synchronization for the drive-response systems. The obtained criteria are easy to test and are in the format of inequalities between the system parameters. Finally, numerical examples are presented to illustrate the results. The obtained criteria in this paper consider the effect of time delays as well as the reaction-diffusion terms, which generalize and improve some existing results.

Khasminskii-type theorem for a class of stochastic functional differential equations

Abstract This paper is concerned with the existence and uniqueness theorems for stochastic functional differential equations with Markovian switching and jump, where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Lyapunov-type functions.

Inverse Optimal Control of a Class of Nonlinear Multi-Agent Systems With Applications to Ship Slew Rate Control

Abstract This paper deals with the inverse optimal leader-following consensus problem of a class of high-order nonlinear multi-agent systems in strict-feedback form under directed communication topology. By employing the backstepping methodology and introducing an integral type Lyapunov function, a nonlinear distributed control law is constructed using the relative information between neighboring agents. The proposed control law solves the inverse optimal leader-following consensus problem under directed communication graph that contains a spanning tree with the root node being the leader agent. A practical example of ship slew rate control system is given to verify the effectiveness of the theoretical results.

Generalized dissipative state estimation for discrete-time nonhomogeneous semi-Markov jump nonlinear systems

The problem of generalized dissipative state estimation for discrete-time nonhomogeneous semi-Markov jump nonlinear systems is concerned in this paper. In this paper, we consider the semi-Markov renewal chain is nonhomogeneous and the states of the system are inaccessible. The aim of this paper is to propose the estimator-designed method to ensure that the system is σ-mean-square stable and satisfy extended dissipative performance. By using the semi-Markov kernel method and polytopic approach, and constructing a new type of Lyapunov function, which not only depends on the sojourn-time but also on the stochastic switching rules, the state estimator gains can be obtained. At last, a numerical example is adopted to verify the superiority of the presented control strategy.