Lyapunov Inequality Research Articles

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

<abstract><p>In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.</p></abstract>

K-contraction analysis for discrete-time systems

The definition of k-contraction promises a useful generalization of the classical notion of contraction for dynamical systems. However, most of the k-contraction literature focuses on continuous-time systems. In this work, we derive conditions for k-contractivity of discrete-time dynamics. Our first result follows traditional lines for k-contraction analysis, and provides Lyapunov-like sufficient conditions based on matrix compounds and state-dependent metrics. However, our subsequent results avoid the complexities related to matrix compounds. Inspired by recent findings in the context of k-contraction for continuous-time systems, we provide conditions on the system's dynamics that rely on generalized Lyapunov inequalities and quadratic cone fields. The proposed conditions are also shown to be necessary for linear time-invariant systems.

A Matrix Pencil Formulation for Nonconservative Realization of Scaling-Based Controllers for Feedforward-Like Systems

We develop a general matrix pencil based approach for efficient non-conservative realization of dual dynamic high-gain scaling based control designs for a class of uncertain feedforward-like nonlinear systems. It is shown that the control design can be cast into a set of matrix pencil based sub-problems capturing the detailed system structure, state dependence structure of uncertain terms, and precise roles of design freedoms in the context of detailed structures of Lyapunov inequalities. The design freedoms in the dynamic high-gain scaling based design are extracted in terms of eigenvalues of the formulated matrix pencil structures. It is seen that the proposed matrix pencil based approach greatly reduces design conservatism and algebraic complexity compared to prior results on dynamic high-gain scaling based control designs.

Stability of conformable fractional time-varying linear systems

In this paper, we investigate the exponential and asymptotic stability analysis of conformable fractional linear time-varying systems. We propose sufficient conditions for testing the asymptotic and exponential stability of conformable differential systems by using the notion of stable functions and differential Lyapunov inequalities. Furthermore, we apply the developed approach to the problem of conformable stabilisation and observer design for conformable fractional linear time-varying systems.

Event-triggered pinning passivity and synchronization of multiple spatial diffusion coupled reaction–diffusion neural networks

In this paper, the problem of the event-triggered pinning passivity and synchronization for multiple spatial diffusion coupled reaction–diffusion neural networks is investigated. Two types of multiple spatial diffusion coupled reaction–diffusion neural networks, with undirected and directed topologies, are considered. Combining the advantages of event-triggered and pinning control methods, event-triggered pinning controllers for these two networks are constructed. By imposing appropriate Lyapunov functionals and inequality techniques, several passivity and synchronization conditions are established for these two networks under the designed event-triggered pinning controllers. Moreover, the non-occurrence of the Zeno behavior in the proposed control is proved. Two numerical examples are illustrated to verify the effectiveness of the proposed criteria.

Graph-based conditions for feedback stabilization of switched and LPV systems

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: (i) the robust case, in which the active mode is completely unknown and unobservable, and (ii) the mode-dependent case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs). Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

Event‐triggered pinning control for passivity and synchronization of directly coupled delayed reaction‐diffusion neural networks

SummaryThis article solves event‐triggered pinning control for passivity and synchronization problems of directly coupled delayed reaction‐diffusion neural networks (CDRDNNs). Outer coupling matrix of the presented model can be direct, while previous literature referred to reaction‐diffusion neural networks usually assumed this matrix undirected simply. First, through exploiting appropriate event‐triggered pinning protocol, passivity and synchronization criteria are acquired for directly CDRDNNs on account of Lyapunov functional and inequality techniques. After that, we also deal with event‐triggered pinning passivity and synchronization for undirected CDRDNNs. In the end, the effectiveness of these obtained results is verified by given examples.

Extended dissipative performance for a complex dynamical networks based on adaptive event‐triggered filter method subject to hybrid attacks

SummaryThe challenge of adaptive event‐triggered based filter design for complex dynamical networks with time‐varying coupling delays under hybrid attack is studied in this research. A new model of complex dynamical networks with input constraints, involving denial‐of‐service (DoS) attack, replay attack, and deception attack is designed. An adaptive event‐triggered scheme is applied to filter design, which adaptively modifies the communication threshold level in order to reduce the effect of the transmission load. By using Lyapunov theory and inequality techniques, a novel necessary condition is proposed that guarantees mean‐square asymptotically stability with extended dissipative performance. The proposed filter parameters are then given an explicit expression, which is solved using a set of linear matrix inequalities. Finally, two numerical results are presented to demonstrate the validity of the theoretical result suggested.

Global Asymptotic Stability and Synchronization of Fractional-Order Reaction–Diffusion Fuzzy BAM Neural Networks with Distributed Delays via Hybrid Feedback Controllers

In this paper, the global asymptotic stability and global Mittag–Leffler stability of a class of fractional-order fuzzy bidirectional associative memory (BAM) neural networks with distributed delays is investigated. Necessary conditions are obtained by means of the Lyapunov functional method and inequality techniques. The hybrid feedback controllers are then developed to ensure the global asymptotic synchronization of these neural networks, resulting in two additional synchronization criteria. The derived conditions are applied to check the fractional-order fuzzy BAM neural network’s Mittag–Leffler stability and synchronization. Three examples are given to demonstrate the effectiveness of the achieved results.

Integral input-to-state stability for delayed networks control systems and its applications

Integral input-to-state stability (iISS) for delayed networks control systems (DNCS) is studied. Based on Kirchhoff’s matrix tree theorem in graph theory, a novel Lyapunov-Krasovaskii functional with integral terms is constructed via the Lyapunov function of each node control system in DNCS and its corresponding topological structure. According to Lyapunov method, some analysis and inequality skills, we raise two sufficient criterions assuring iISS for DNCS, which are the Lyapunov-type criterion and the coefficient-type criterion, respectively. Then, our approach is used in some classical DNCS. Finally, some numerical examples and simulation process are given to verify the effectiveness and applicability of the theoretical results.

New results for dynamical analysis of fractional-order gene regulatory networks with time delay and uncertain parameters

In this paper, a novel class of networks named fractional-order gene regulatory networks with time delay and uncertain parameters (FGRNTDUP) are formulated, and some new results are obtained for the dynamics of FGRNTDUP. Firstly, a general fractional-order differential equality is built to supply a new viewpoint about the study of finite-time stability, stabilization, and synchronization for many fractional systems. Secondly, based on contraction mapping principle, the existence and uniqueness of the equilibrium point are strictly proved for the FGRNTDUP. Meanwhile, by virtue of fractional Lyapunov method and inequality techniques, some sufficient conditions are derived to guarantee the global Mittag–Leffler stability of FGRNTDUP. Moreover, based on a newly developed inequality, some novel criteria are yielded to realize the finite-time synchronization of FGRNTDUP by designing a suitable adaptive controller. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

Receding horizon control for continuous‐time mean‐field systems

AbstractIn this paper, we study the receding horizon control (RHC) stabilization of continuous‐time mean‐field system. An explicit RHC controller is obtained by introducing a cost function with two terminal matrices and the conditions that make the system stable are also obtained; that is, when two coupled Lyapunov inequalities are satisfied, the system controlled by RHC is stabilized. Finally, a simulation example is given to verify the effectiveness of the proposed strategy.

Periodic Intermittent Adaptive Control with Saturation for Pinning Quasi-Consensus of Heterogeneous Multi-Agent Systems with External Disturbances.

A periodic intermittent adaptive control method with saturation is proposed to pin the quasi-consensus of nonlinear heterogeneous multi-agent systems with external disturbances in this paper. A new periodic intermittent adaptive control protocol with saturation is designed to control the internal coupling between the follower agents and the feedback gain between the leader and the follower. In particular, we use the saturation adaptive law: when the quasi-consensus error converges to a certain range, the adaptive coupling edge weight and the adaptive feedback gain will not be updated. Furthermore, we propose three saturated adaptive pinning control protocols. The quasi-consensus is achieved through its own pinning as long as the agents remain connected to each other. Using the Lyapunov function method and inequality technique, the convergence range of the quasi-consensus error of a heterogeneous multi-agent system is obtained. Finally, the rationality of the proposed control protocol is verified through numerical simulation. Theoretical derivation and simulation results show that the novel proposed periodic intermittent adaptive control method with saturation can successfully be used to achieve the pinning of quasi-consensus of nonlinear heterogeneous multi-agent systems.

Linear formation of Cucker–Smale model with distributed time delays

In this paper, linear formation problem of Cucker–Smale model with distributed time delays is investigated. By using Lyapunov functional method and inequality techniques, combining with the connection between primitive system and projection system, the consistency of individual velocity is verified. Then, sufficient conditions for forming flocking and linear formations are obtained. In addition, the theoretical results are illustrated by some examples. The results obtained extend the application range of the classical Cucker–Smale model.

Quantized pinning bipartite synchronization of fractional-order coupled reaction–diffusion neural networks with time-varying delays

Neural synchronization not only has a significant theoretical role for understanding brain function, but also is important for artificial neural network development. In this paper, a novel and more general directed signed network model, consisting of a set of fractional reaction–diffusion delay neural networks, is articulated. Moreover, we also consider the coexistence of cooperation and competition as a coupling scheme among neurons, which is a mechanism found in biological neural interactions. By designing a new quantized pinning controller based on depth-first algorithm and logarithmic quantization, the sufficient conditions for the bipartite synchronization of the addressed network are given by using Lyapunov method, inequality technique and Green’s formula. In addition, using M-matrix theory, the more applicable bipartite synchronization criteria in the form of low-dimensional linear matrix inequality and the form of network coupling strength threshold are given respectively. This work enriches and improves the previous works. At last, simulation experiments are offered to verify the correctness of our theoretical results.

Self-stabilizing economic model predictive control without pre-calculated steady-state optima: Stability and robustness

We propose a new economic nonlinear model predictive control (eNMPC) formulation that tracks the optimality conditions of the real-time optimization problem rather than any specific steady states. The proposed formulation maintains its nature of optimizing economic performance and assured stability properties with the Lyapunov inequality constraint for the closed-loop control. Under general assumptions, we prove that the proposed controller is asymptotically stable without process disturbances and is input-to-state stable when there is a process disturbance. The proposed eNMPC is demonstrated on two case studies and compared against setpoint-tracking NMPC with setpoints determined by the steady-state real-time optimizer to show improved dynamic performance. We also highlight the capability of self-stabilization of the new eNMPC with parameter updates in the process model.

A Generalized Lyapunov Inequality for a Pantograph Boundary Value Problem Involving a Variable Order Hadamard Fractional Derivative

The authors obtain existence and uniqueness results for a nonlinear fractional pantograph boundary value problem containing a variable order Hadamard fractional derivative. This type of model is appropriate for applications involving processes that occur in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem considered. Their results are obtained by the fractional calculus and Krasnosel’skii’s fixed point theorem. An example is given to illustrate their approach.

Fixed-time stability of Cohen-Grossberg BAM neural networks with impulsive perturbations

This article concerns the problem of fixed-time stability (FXTS) of a Cohen-Grossberg bidirectional associative memory neural network (CGBAMNN) with destabilizing impulsive effects. A novel sufficient condition for the impulsive dynamical systems (IDSs) to be FXTS for destabilizing impulses is obtained. Different from the usual Lyapunov inequality for FXTS of IDSs, we have applied a new Lyapunov inequality to obtain the results under impulsive perturbations. Based on the average impulsive interval (AII) and the comparison principle, we have derived the results of this paper. Two types of continuous controllers: one with signum terms and another without signum terms, based on a new Lyapunov inequality, FXTS of CGBAMNN have been studied. The settling-time functions obtained in this article depend on the parameters of the impulsive sequences. Finally, two numerical examples, one is a cyber-physical system with deception attacks and another is a neural network, are given to validate the efficiency of our obtained theoretical results.

Finite/nearly fixed-time stability of nonlinear impulsive systems with destabilizing impulses and its application to neural networks

This paper investigated the finite-time stability (FTS) and nearly fixed-time stability (nFxTS) of nonlinear impulsive systems with destabilizing impulses. The concept of nFxTS is different from FxTS, as the former may be just asymptotically stable but the latter must be FTS. A Lyapunov inequality with linear terms has been used to derive sufficient conditions based on the dwell time (DT) and average dwell time (ADT) properties of impulsive sequences to ensure FTS/FxTS of the system. Theorems are constructed based on the initial state of the system, which belongs to either δ-neighborhood(nbh) or outside δ-nbh of the origin. When the system starts within δ−nbh of the origin then FTS can be achieved by the system following conditions on DT and ADT of impulsive sequences. The same is true for nFxTS of the system when it starts outside δ-nbh of the origin. We investigated the condition on DT for which nFxTS is not possible, but FTS is possible only if δ>0 is bounded. The main results of this paper are applied to the synchronization problem of impulsive neural networks with destabilizing impulses. Finally, a numerical example of an impulsive neural network is given to illustrate the validity of the proposed theoretical results.

Lyapunov Inequality for Second-Order Equation with Operator of Distributed Differentiation

Lyapunov Inequality for Second-Order Equation with Operator of Distributed Differentiation