Discontinuous Right-hand Side Research Articles

Guaranteed strategies with memory for alternative pursuit

A pursuit game is called alternative if, at any state of the game, a pursuer can terminate it on any of two given target sets and the Boltz payoff function differs on these sets only in the terminal part. By assumption, the optimal universal feedback strategies of the pursuer are known for each fixed terminal alternative. Endeavoring to decrease its payoff via the choice of an alternative, the pursuer may compare guaranteed results (e.g., at the initial and current states) and apply one of two fixed-alternative strategies. If both players optimize the choice of their alternatives depending on the position, the state of the pursuit game can enter some neighborhood of a hypersurface with equivalent alternatives (in the sense of payoffs) and stay there for a finite period of time. In different formalizations, the solutions generated by an appropriate differential equation with discontinuous right-hand side (ideal solutions, limits of Euler polygonal lines, generalized solutions) lead to different outcomes. For certain states, they even increase the payoff in comparison with the pursuit strategy based on the preferable alternative chosen at the initial state (due to sliding mode occurrence). The author introduces a method of constructing pursuit strategies with memory and a finite number of admissible switchings for the target alternative. The proposed method allows estimating the guaranteed result of the pursuer by confining to the solution concept of the limits of Euler polygonal lines.

Global exponential stability in a Lagrange sense for memristive recurrent neural networks with time-varying delays

In this paper, we consider the global exponential stability in a Lagrange sense for memristive recurrent neural networks with time-varying delays. Here, we adopt nonsmooth analysis and control theory to handle memristive neural networks with discontinuous right-hand side, and by constructing proper Lyapunov functionals and using inequality technique, several new sufficient conditions in linear matrix inequality form are given to ensure the ultimate boundedness and global exponential attractivity of the memristor-based neural networks in the sense of Filippov solutions. In addition, these conditions do not require the connection weight matrices to be symmetric and the delay functions to be differentiable. Finally, numerical simulations illustrate the effectiveness of our results.

Dissipativity analysis of memristor-based complex-valued neural networks with time-varying delays

In this paper, the problem of dissipativity analysis for memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is investigated extensively. Dissipativity analysis is an important concept in control system theory and its applications. The analysis in the paper employ results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. By using the framework of Filippov solution, differential inclusion theory, an appropriate Lyapunov–Krasovskii functional and linear matrix inequality (LMI) technique, several new sufficient conditions for global dissipativity, global exponential dissipativity and strictly (Q,S,R)-dissipativity are derived in the form of complex-valued as well as real-valued LMIs. Both real and complex-valued LMIs guarantee feasibility results for addressed MCVNNs. These LMIs can be solved by using standard available numerical packages. Moreover, the global attractive sets which are positive invariant are obtained. Finally, three numerical examples are established to illustrate the effectiveness of the proposed theoretical results.

Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays

In this paper, the problem of finite-time stability of fractional-order complex-valued memristor-based neural networks (NNs) with time delays is extensively investigated. We first initiate the fractional-order complex-valued memristor-based NNs with the Caputo fractional derivatives. Using the theory of fractional-order differential equations with discontinuous right-hand sides, Laplace transforms, Mittag-Leffler functions and generalized Gronwall inequality, some new sufficient conditions are derived to guarantee the finite-time stability of the considered fractional-order complex-valued memristor-based NNs. In addition, some sufficient conditions are also obtained for the asymptotical stability of fractional-order complex-valued memristor-based NNs. Finally, a numerical example is presented to demonstrate the effectiveness of our theoretical results.

Exponential Stabilization of Memristor-based Chaotic Neural Networks with Time-Varying Delays via Intermittent Control.

This paper is concerned with the global exponential stabilization of memristor-based chaotic neural networks with both time-varying delays and general activation functions. Here, we adopt nonsmooth analysis and control theory to handle memristor-based chaotic neural networks with discontinuous right-hand side. In particular, several new sufficient conditions ensuring exponential stabilization of memristor-based chaotic neural networks are obtained via periodically intermittent control. In addition, the proposed results here are easy to verify and they also extend the earlier publications. Finally, numerical simulations illustrate the effectiveness of the obtained results.

Passivity Analysis of Memristor-Based Complex-Valued Neural Networks with Time-Varying Delays

In this paper, the model of memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is established and the problem of passivity analysis for MCVNNs is considered and extensively investigated. The analysis in this paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. By employing the appropriate Lyapunov---Krasovskii functional, differential inclusion theory and linear matrix inequality (LMI) approach, some new sufficient conditions for the passivity of the given MCVNNs are obtained in terms of both complex-valued and real-value LMIs, which can be easily solved by using standard numerical algorithms. Numerical examples are provided to illustrate the effectiveness of our theoretical results.

Adaptive Finite-Time Complete Periodic Synchronization of Memristive Neural Networks with Time Delays

This paper is concerned with the adaptive finite-time complete periodic synchronization issue for memristive based neural networks with time delays. Under the framework of Filippov solutions of the differential equations with discontinuous right-hand side, based on Mawhin-like coincidence theorem in set-valued analysis theory, the existence of periodic solution is proved. By applying Lyapunov---Krasovskii functional approach, adaptive controller is designed and unknown control parameters are determined by adaptive update law. A novel and useful finite-time complete synchronization condition is obtained in terms of linear matrix inequalities to ensure the synchronization goal. An illustrative example is given to demonstrate the effectiveness of the theoretical results.

Impact of discontinuous harvesting on fishery dynamics in a stock-effort fishing model

We consider a stock-effort fishing model with discontinuous harvesting strategies. Under some reasonable assumptions on the discontinuous harvesting function, we are able to confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that the solutions of the fishing model can arrive at a sliding mode in finite time. A qualitative analysis shows that the goal of maintaining the system at a sustainable equilibrium and optimizing the harvesting can be achieved by introducing the discontinuous harvesting strategies. From the viewpoint of optimal harvesting, we can obtain that discontinuous harvesting strategies are superior to continuous harvesting strategies, which are usually adopted in previous literature. The main difficulty resides in the discontinuity of the model, and is conquered by exploiting the theory of differential equations with discontinuous righthand sides and variable structure system theory.

On the periodic dynamics of memristor-based neural networks with time-varying delays

In this paper, we study the existence, uniqueness and stability of periodic solution for a wide class of memristor-based neural networks with time-varying delays. By employing the topological degree theory in set-valued analysis, differential inclusions theory and a new Lyapunov function method, we prove that the neural network has a unique periodic solution, which is globally exponentially stable. Moreover, we prove the existence, uniqueness and global exponential stability of equilibrium point for time-varying delayed memristor-based neural networks with constant coefficients. The obtained results improve and extend previous works on memristor-based or usual neural network dynamical systems with continuous or discontinuous right-hand side. Finally, two numerical examples are provided to show the applicability and effectiveness of our main results.

Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control

This paper investigates the exponential synchronization of coupled memristor-based chaotic neural networks with both time-varying delays and general activation functions. And here, we adopt nonsmooth analysis and control theory to handle memristor-based chaotic neural networks with discontinuous right-hand side. In particular, several new criteria ensuring exponential synchronization of two memristor-based chaotic neural networks are obtained via periodically intermittent control. In addition, the new proposed results here are very easy to verify and also complement, extend the earlier publications. Numerical simulations on the chaotic systems are presented to illustrate the effectiveness of the theoretical results.

Optimal control processes associated with a class of discontinuous control systems: applications to sliding mode dynamics

This paper presents a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right-hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of constrained OCPs is used as an analytic background for designing numerically tractable schemes and computational methods for their solutions. The proposed analytic method guarantees consistency of the resulting approximations related to the original infinite-dimensional optimization problem and leads to specific implementable algorithms.

Almost periodic dynamical behaviors for generalized Cohen–Grossberg neural networks with discontinuous activations via differential inclusions

In this paper, we investigate the almost periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides, time-varying and distributed delays. By means of retarded differential inclusions theory and nonsmooth analysis theory with generalized Lyapunov approach, we obtain the existence, uniqueness and global stability of almost periodic solution to the neural networks system. It is worthy to pointed out that, without assuming the boundedness or monotonicity of the discontinuous neuron activation functions, our results will also be valid. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.

The Hamiltonian property of the flow of singular trajectories

Pontryagin's maximum principle reduces optimal control problems to the investigation of Hamiltonian systems of ordinary differential equations with discontinuous right-hand side. An optimal synthesis is the totality of solutions to this system with a fixed terminal (or initial) condition, which fill a region in the phase space one-to-one. In the construction of optimal synthesis, singular trajectories that go along the discontinuity surface of the right-hand side of the Hamiltonian system of ordinary differential equations, are crucial. The aim of the paper is to prove that the system of singular trajectories makes up a Hamiltonian flow on a submanifold of . In particular, it is proved that the flow of singular trajectories in the problem of control of the magnetized Lagrange top in a variable magnetic field is completely Liouville integrable and can be embedded in the flow of a smooth superintegrable Hamiltonian system in the ambient space.Bibliography: 17 titles.

Stability notions and Lyapunov functions for sliding mode control systems

The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory.

On optimal control of a sweeping process coupled with an ordinary differential equation

We study a special case of an optimal control problem governed by a differential equation and a differential rate--independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process.  &nbspWe perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.

Stability Analysis of Retarded Differential Inclusions

Retarded differential inclusions have drawn more and more attention, due to the development of feedback control systems with delays and dynamical systems determined by retarded differential equations with a discontinuous right-hand side. The purpose of this paper is to establish a result on the stability and asymptotical stability for retarded differential inclusions. Comparing with the previous results, the main result obtained in this paper allows Lyapunov functions to be nonsmooth. Moreover, to deal with the asymptotical stability, it is not required that Lyapunov functions should have an infinitesimal upper limit, but this condition is needed in most of the previous results. To demonstrate applicability, we use the main result in the analysis of asymptotical stability of a class of neural networks with discontinuous activations and delays.

Discontinuous Lyapunov Functions for Nonasymptotic Stability Analysis

The paper presents mathematical tools required for finite-time stability analysis of discontinuous control systems using discontinuous Lyapunov functions. Elements of Filippov theory of differential equations with discontinuous right-hand sides and stability notions are briefly observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples.

Periodicity and global exponential stability of generalized Cohen–Grossberg neural networks with discontinuous activations and mixed delays

In this paper, we investigate the periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides, time-varying and distributed delays. By means of retarded differential inclusions theory and the fixed point theorem of multi-valued maps, the existence of periodic solutions for the neural networks is obtained. After that, we derive some sufficient conditions for the global exponential stability and convergence of the neural networks, in terms of nonsmooth analysis theory with generalized Lyapunov approach. Without assuming the boundedness (or the growth condition) and monotonicity of the discontinuous neuron activation functions, our results will also be valid. Moreover, our results extend previous works not only on discrete time-varying and distributed delayed neural networks with continuous or even Lipschitz continuous activations, but also on discrete time-varying and distributed delayed neural networks with discontinuous activations. We give some numerical examples to show the applicability and effectiveness of our main results.

Opinion consensus of modified Hegselmann–Krause models

We consider the opinion consensus problem using a multi-agent setting based on the Hegselmann–Krause (H–K) Model. Firstly, we give a sufficient condition on the initial opinion distribution so that the system will converge to only one cluster. Then, modified models are proposed to guarantee convergence for more general initial conditions. The overall connectivity is maintained with these models, while the loss of certain edges can occur. Furthermore, a smooth control protocol is provided to avoid the difficulties that may arise due to the discontinuous right-hand side in the H–K model.

Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks

The present paper introduces memristor-based fractional-order neural networks. The conditions on the global Mittag-Leffler stability and synchronization are established by using Lyapunov method for these networks. The analysis in the paper employs results from the theory of fractional-order differential equations with discontinuous right-hand sides. The obtained results extend and improve some previous works on conventional memristor-based recurrent neural networks.