Filippov Differential Inclusion Research Articles

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

Output-Feedback Adaptive Neural Network Control for Uncertain Nonsmooth Nonlinear Systems With Input Deadzone and Saturation.

Nonsmooth nonlinear systems can model many practical processes with discontinuous property and are difficult to be stabilized by classical control methods like smooth nonlinear systems. This article considers the output-feedback adaptive neural network (NN) control problem for nonsmooth nonlinear systems with input deadzone and saturation. First, the nonsmooth input deadzone and saturation is converted to a smooth function of affine form with bounded estimation error by means of the mean-value theorem. Second, with the help of approximation theorem and Filippov's differential inclusion theory, the given nonsmooth system is converted to an equivalent smooth system model. Then, by introducing a proper logarithmic barrier Lyapunov function (BLF), an output-feedback adaptive NN strategy is set up by constructing an appropriate observer and adopting the adaptive backstepping technique. A new stability criterion is established to guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, comparative simulations through Chua's oscillator are offered to verify the effectiveness of the proposed control algorithm.

Command filter‐based adaptive neural network controller design for uncertain nonsmooth nonlinear systems with output constraint

SummaryThis paper investigates the command filter‐based adaptive neural network tracking control problem for uncertain nonsmooth nonlinear systems. First, an integral barrier Lyapunov function is introduced to deal with the symmetric output constraint and make the output comply with prescribed restrictions. Second, by the Filippov's differential inclusion theory and approximation theorem, the considered nonsmooth nonlinear system is converted to an equivalent smooth nonlinear system. Third, the Levant's differentiator is used to deal with the “explosion of complexity” problem. An error compensation mechanism is established to attenuate the effect of the filtering error on control performance. Then, an adaptive neural network controller is set up by resorting to the backstepping technique. It is strictly mathematically proved that the tracking error can converge to an arbitrarily small neighborhood of the origin and all the signals in the closed‐loop system are semi‐globally uniformly ultimately bounded. Finally, a numerical example and an application example of the robotic manipulator system are provided to demonstrate the availability of the proposed control strategy.

Observer‐based bipartite Mittag‐Leffler time‐varying formation for fractional multi‐agent systems with discontinuous inherent dynamics

SummaryIn this article, the bipartite Mittag‐Leffler time‐varying formation issue of fractional multi‐agent systems is discussed, where the dynamical nodes of followers are modeled to be discontinuous and nondecreasing. A observer‐based adaptive distributed control scheme, which is dependent on output information, is designed to realized the bipartite formation control goal. Meanwhile, the efficient algorithm with respect to the control protocol also is presented. Under fractional Filippov differential inclusion framework, by constructing Lur'e Postnikov‐type Lyapunov functional and applying Clarke's non‐smooth analysis technique, the bipartite Mittag‐Leffler time‐varying formation conditions are derived in terms of linear matrix inequalities. At last, a simulation example is carried out to verify the effectiveness of the proposed scheme.

Aperiodically intermittent saturation consensus on multi-agent systems with discontinuous dynamics

This paper mainly investigates the exponential consensus problem for the multi-agent systems (MASs) with nonlinear discontinuous dynamics and time-varying delay. A novel aperiodically intermittent distributed control strategy is proposed to force the state of each agent to the common trajectory, where the configuration of control width and rest width can be aperiodic. Meanwhile, in order to limit the control effects into certain reasonable ranges, an improved saturation algorithm is proposed, which effectively reduces the non-smoothness of the control signal. Sufficient conditions for the exponential consensus on the discontinuous MASs are obtained through the Filippov differential inclusion (FDI), the Lie derivative method (LDM) and the measurement selection theorem (MST). Finally, the validity and feasibility of the main theories is demonstrated by numerical simulations.

Fixed-Time Synchronization of Delayed Memristive Neural Networks with Discontinuous Activations

In this paper, the fixed-time synchronization problem for a class of memristive neural networks with discontinuous neuron activation functions and mixed time-varying delays is investigated. With the help of the fixed-time stability theory, under the framework of Filippov solution and differential inclusion theory, several new and useful sufficient criteria for fixed-time synchronization are obtained by designing two types of energy-saving and simple controllers for the considered systems. Compared with the traditional fixed-time synchronization controller, the controllers used in this paper only have one power exponent term, which is a function of the system state error rather than a constant. Moreover, some previous relevant works are especially improved. Finally, two numerical examples are given to show the correctness and the effectiveness of the obtained theoretical results.

Global Mittag-Leffler consensus for fractional singularly perturbed multi-agent systems with discontinuous inherent dynamics via event-triggered control strategy

In this paper, the global Mittag-Leffler consensus tracking issue is considered for fractional singularly perturbed multi-agent systems (FSPMASs) based on event-triggered control strategy, where the inherent dynamic is modeled to be a discontinuous function with nondecreasing property. Firstly, a differential inequality with respect to fractional-order derivative of convex function is developed. As the special cases, the inequalities about fractional-order derivative of three known functions are also addressed. Secondly, a distributed event-triggered control scheme is designed to guarantee that the considered FSPMASs can achieve the global Mittag-Leffler consensus. Moreover, the Mittag-Leffer convergence speed of tracking the leader for followers can be adjusted to any desired values in advance. In addition, under fractional Filippov differential inclusion framework, by applying Lur’e Postnikov-type Lyapunov functional with variable upper limit integral item and Clarke’s non-smooth analysis technique, the global Mittag-Leffler consensus conditions are addressed in terms of matrix inequalities (MIs). Finally, two numerical simulations are provided to illustrate the validity of the proposed design method and theoretical results.

Adaptive control for discontinuous variable-order fractional systems with disturbances

This paper deals with the global asymptotic stabilization and finite-time stabilization issues for variable-order fractional systems with partial a priori bounded disturbances by designing an appropriate adaptive controller. Via the inductive method and Arzela-Ascoli theorem, the existence and uniqueness of the solution for the considered system is firstly verified under the proposed control strategy. By applying Lyapunov stability theory, non-smooth analysis and inequality technique, sufficient stabilization criteria are established under the framework of variable-order fractional Filippov differential inclusion. Finally, two numerical simulations are given to demonstrate the validity of the proposed method.

Group Consensus in Finite Time for Fractional Multiagent Systems With Discontinuous Inherent Dynamics Subject to Hölder Growth.

This article is concerned with the global Mittag-Leffler group consensus and group consensus in finite time for fractional multiagent systems (FMASs), where the inherent dynamics is modeled to be discontinuous, and subject to the local Hölder nonlinear growth in a neighborhood of continuous points. First, a fractional differential inequality on convex functions and a global convergence principle in finite time for absolutely continuous functions are developed, respectively. Second, two new distributed control protocols are designed to realize the consensus between the follower agents in each subgroup and respective leaders. In addition, under the fractional Filippov differential inclusion framework, by applying the Lur'e Postnikov-type convex Lyapunov functional approach and Clarke's nonsmooth analysis technique, some sufficient conditions with respect to the global Mittag-Leffler group consensus and group consensus in finite time are addressed in terms of linear matrix inequalities (LMIs), respectively. Moreover, the settling time for the group consensus in finite time is estimated accurately. Finally, two simulation examples are provided to illustrate the validity of the proposed scheme and theoretical results.

Stabilization of non-smooth variable order switched nonlinear systems

The paper discusses the issue of global asymptotic stabilization for non-smooth variable order nonlinear switched systems with partial unstable modes. The existence and uniqueness of solution for the considered system is firstly verified by utilizing Gronwall–Bellman inequality and the inductive method. Then, a slow switching strategy is performed for the stable modes and the unstable modes are handled by a fast switching mechanism. Under the framework of Filippov differential inclusion, sufficient stabilization criteria are derived by applying the mode-dependent average dwell time and dwell time schemes. Finally, some objects in real life are introduced and a numerical simulation is offered to show the validity of the obtained results.

Global dissipativity and exponential synchronization of mixed time-varying delays neural networks with discontinuous activations

Abstract In this paper, the matters of dissipativity and synchronization for non-autonomous Hopfield neural networks with discontinuous activations are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. The global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, the global exponential synchronization of the addressed network drive system and the response system is realized by utilizing inequality and some analysis techniques and designing the discontinuous state feedback controller. Finally, several numerical examples are given to verify the validity of the theoretical results.

Event-triggered synchronization in fixed time for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity

This paper is concerned with on the event-triggered synchronization in fixed time for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity. Firstly, the principle of the global convergence in fixed time with respect to nonlinear systems with semi-Markov switching is developed. Secondly, in order to realize the global stochastic synchronization goal in fixed time, a novel hybrid controller, which is composed of the event-triggered controller and the switching state-feedback controller, is designed. Under Filippov differential inclusion framework, by applying Lyapunov functional method and inequality analysis technique, the global stochastic synchronization conditions in fixed time are achieved in the form of linear matrix inequalities (LMIs). In addition, the upper bound of the stochastic settling time, which is adjusted in advance by choosing the controller parameters, is estimated accurately. Finally, the correctness of the theoretical results and the feasibility of the designed controller are verified by an example.

Event-triggered stochastic synchronization in finite time for delayed semi-Markovian jump neural networks with discontinuous activations

In this paper, the global stochastic synchronization in finite time is discussed for discontinuous semi-Markovian switching neural networks with mixed time-varying delays under stochastic disturbance based on event-triggered non-fragile control scheme. By applying the novel hybrid controller, which is composed of the event-triggered controller, the non-fragile controller and the switching state-feedback controller, the global stochastic synchronization goals in finite time are achieved. Under Filippov differential inclusion framework, based on non-smooth analysis theory, general free-weighting matrix method, Lyapunov–Krasovskii functional approach and inequality analysis technique, the global stochastic synchronization conditions in finite time are addressed in terms of linear matrix inequalities. Moreover, the expressions about the upper bound of stochastic settling time are explicitly developed. Finally, two numerical examples are provided to illustrate the feasibility of the proposed control scheme and the validity of theoretical results.

Global exponential stability of delayed complex-valued neural networks with discontinuous activation functions

This paper studies the global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. By introducing the complex-valued Filippov differential inclusion, we construct the framework of studying the dynamical behaviors of complex-valued neural networks with discontinuous bivariate activation functions. By employing the Leray–Schauder alternative theorem and choosing an appropriate Lyapunov function, we prove the global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. The numerical example provided shows the effectiveness of the obtained results.

Global dissipativity and finite-time synchronization of mixed time-varying delayed memristor-based neural networks with discontinuous activations

In this paper, the matters of dissipativity and finite time synchronization for memristor-based neural networks (MNNs) with mixed time-varying discontinuities are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. Then, the global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, some novel sufficient conditions are introduced to guarantee the finite-time synchronization of the drive-response MNNs based on a simple Lyapunov function and two different feedback controllers. Finally, several numerical examples are given to verify the validity of the theoretical results.

Non-fragile robust finite-time synchronization for fractional-order discontinuous complex networks with multi-weights and uncertain couplings under asynchronous switching

This paper focus on the global robust non-fragile finite-time synchronization issue for fractional-order complex networks with multi-weights and uncertain couplings under asynchronous switching topology, where nonlinear dynamic nodes are discontinuous and non-decreasing. Firstly, the non-fragile controller with switching gains is designed. Secondly, under the fractional Filippov differential inclusion framework and the network topologies with no-delayed and delayed couplings, by adopting multiple Lur’e Postnikov-type Lyapunov functional, nonsmooth analysis method and the average dwell time technique, the global robust non-fragile finite-time synchronization conditions are achieved in terms of linear matrix inequalities (LMIs), respectively. Finally, the validity of the theoretical results developed in this paper is verified by two numerical simulations.

Global Nonfragile Synchronization in Finite Time for Fractional-Order Discontinuous Neural Networks With Nonlinear Growth Activations.

This paper is concerned with the global nonfragile Mittag-Leffler synchronization and the global synchronization in finite time for fractional-order discontinuous neural networks, where activation functions are discontinuous at 0, or modeled as a local Hölder functions with the nonlinear growth property in a neighborhood of 0. First, two lemmas concerned with the convergence with respect to an absolutely continuous function are developed. Second, a new property, which introduces an inequality of the fractional derivative for the variable upper limit integral with respect to the nonsmooth integrable function, is presented and applied in the synchronization results' analysis. In addition, under the fractional Filippov differential inclusion framework, by utilizing the Lur'e Postnikov-type Lyapunov functional, nonsmooth analysis method, and the convergence properties developed in this paper, the synchronization conditions are derived in the form of linear matrix inequalities. Moreover, the upper bound of the setting time for the global nonfragile synchronization in finite time is calculated accurately. Finally, two illustrations are presented to verify the correctness of the theoretical results.

Non-fragile chaotic synchronization for discontinuous neural networks with time-varying delays and random feedback gain uncertainties

This paper is concerned with the non-fragile synchronization issue for neural networks with discontinuous activation functions, time-varying delays and random feedback gain uncertainties, where the randomly occurring phenomena are modeled by stochastic variables satisfying the Bernoulli distribution. The appropriate non-fragile controllers are designed to ensure that the global synchronization can be achieved easily. Under the extended Filippov differential inclusion framework, by applying non-smooth analysis theory with a generalized Lyapunov–Krasovskii functional with multiple integral terms and Wirtinger-based multiple integral inequality analysis technique, the global asymptotical stochastic stability of the synchronization error dynamical system is analytically proved, and the non-fragile synchronization conditions are addressed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the feasibility of the proposed non-fragile controller and the validity of the theoretical results.

Finite-Time Lag Synchronization for Memristive Mixed Delays Neural Networks with Parameter Mismatch

This paper is devoted to study the finite time lag synchronization problem of memristive mixed delays neural networks with switching jumps and parameter mismatch. The main objective of this paper is to design simple linear feedback control such as the drive response systems can realize synchronization in the finite time. Several theoretical aspects are addressed, mean Filippov differential inclusion and set-valued map. Based on the Gronwall’s inequality and properties of differential operation, a new finite time lag synchronization scheme is proposed to guarantee that coupled memristive mixed delays neural networks are in a state of lag synchronization in finite time. In addition, finite lag synchronization criteria of two identical memristive mixed delays neural networks are also considered. Finally, numerical examples are presented to illustrate the effectiveness of the our proposed theorems.

Finite-time stability and synchronization for memristor-based fractional-order Cohen-Grossberg neural network

In this paper, we study the finite-time stability and synchronization problem of a class of memristor-based fractional-order Cohen-Grossberg neural network (MFCGNN) with the fractional order α ∈ (0,1 ]. We utilize the set-valued map and Filippov differential inclusion to treat MFCGNN because it has discontinuous right-hand sides. By using the definition of Caputo fractional-order derivative, the definitions of finite-time stability and synchronization, Gronwall’s inequality and linear feedback controller, two new sufficient conditions are derived to ensure the finite-time stability of our proposed MFCGNN and achieve the finite-time synchronization of drive-response systems which are constituted by MFCGNNs. Finally, two numerical simulations are presented to verify the rightness of our proposed theorems.