Discontinuous Differential Equations Research Articles

Time Reparametrization and Event Location for Discontinuous Differential Algebraic Equations

In this paper, we consider numerical methods for the event location of differential algebraic equations. The event corresponds to cross a discontinuity surface, beyond which another differential algebraic equation holds. The methods are based on a particular change of the independent variable time, called time reparametrization or time transformation, reducing the equation to another equation where the event time is known in advance. From a numerical point of view, these methods never cross the discontinuity surface and reach it in a fixed number of steps. The methods works also for differential algebraic equations of index higher than one.

Heteroclinic solutions in singularly perturbed discontinuous differential equations

We derive Melnikov type conditions for the persistence of heteroclinic solutions in perturbed slowly varying discontinuous differential equations extending to these equations similar results for continuous differential equations.

Second–order discontinuous ODEs and billiard problems

We present an existence principle for boundary value problems involving discontinuous ordinary differential equations of the second order using the Krasovskii regularization technique. Especially we obtain sufficient conditions of transversality type for Krasovskii solutions to be also Carathéodory solutions of the original problem. This result is applied on a certain billiard problem, which can be thought as an ordinary differential equation with state-dependent impulses that is equivalent to certain discontinuous differential equation. In particular, we obtain new existence and multiplicity results for Dirichlet problems in billiard spaces with time-varying boundaries.

Correction to: Periodic Solutions in Slowly Varying Discontinuous Differential Equations: A Non-Generic Case

Correction to: Periodic Solutions in Slowly Varying Discontinuous Differential Equations: A Non-Generic Case

Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses

<abstract><p>In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} _{t}^{C} \mathcal {D}^{\mathfrak{R}}_{0^{+}}\Lambda(t) = \mathcal {E}(t)\digamma(t, \Lambda(t)), \ a.e.\ t\in Q, \\ \triangle \Lambda|_{t = t_{{\kappa}}} = \Phi_{{\kappa}}(\Lambda(t_{{\kappa}})), \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ \triangle \Lambda'|_{t = t_{{\kappa}}} = 0, \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ {\vartheta} \Lambda(0)-{\chi} \Lambda(1) = \int_{0}^{1}\varrho_{1}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \\ {\zeta} \Lambda'(0)-\delta \Lambda'(1) = \int_{0}^{1}\varrho_{2}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ {\vartheta} > {\chi} > 0, \ {\zeta} > \delta > 0 $, $ \Phi_{{\kappa}}\in C(\mbox{ $\mathbb{R}$ }^{+}, \mbox{ $\mathbb{R}$ }^{+}) $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \geq 0 $ a.e. on $ Q = [0, 1] $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \in L^{1}(0, 1) $ and $ \digamma:[0, 1]\times \mbox{ $\mathbb{R}$ }^{+}\rightarrow \mbox{ $\mathbb{R}$ }^{+} $, $ \mbox{ $\mathbb{R}$ }^{+} = [0, +\infty) $. By using Krasnosel skii's fixed point theorem for discontinuous operators on cones, some sufficient conditions for the existence of single or multiple positive solutions for the above discontinuous differential system are established. An example is given to confirm the main results in the end.</p></abstract>

Fixed and preassigned-time synchronization of drive-response networks with discontinuous activation functions

Aiming at the problem of fixed and preassigned-time synchronization of drive-response networks with discontinuous activation function, we derive a fixed-time stability theorem with more accurate stable time estimation for discontinuous systems by comparison principle and variable substitution. We design a fixed-time synchronization controller without linear term and a preassigned-time synchronization controller with limited control gain. Based on the proposed theorem and controller, together with the theory of discontinuous differential equations, we obtain the sufficient conditions for realizing fixed-time synchronization of drive-response networks with discontinuous activation function, and the sufficient condition for realizing the preassigned-time synchronization of drive-response networks with discontinuous activation function. Wherein, the synchronization time can be prespecified according to the actual situation, and it is independent of the initial value of the system and the controller parameters. The numerical simulation results verify the correctness of the stability theorem and the accuracy of the synchronization time estimation.

Multiple Solutions for a Class of BVPs of Second-Order Discontinuous Differential Equations with Impulse Effects

This paper deals with a class of boundary value problems of second-order differential equations with impulses and discontinuity. The existence of single or multiple positive solutions to discontinuous differential equations with impulse effects is established by using the nonlinear alternative of Krasnoselskii’s fixed point theorem for discontinuous operators on cones. Finally, an example is given to illustrate the main results.

On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations

We investigated the finite/fixed-time stability (FNTS/FXTS) of discontinuous differential equations (DDEs) in this paper. To cope with differential equations that were discontinuous on the right-hand side, we utilized the Filippov solution, which is widely used in engineering. Under the framework of the Filippov solution, we transformed this issue into an FNTS/FXTS problem in the corresponding functional differential inclusion. We proposed some new FNTS/FXTS criteria, which will have important applications in the field of control engineering. It is worth mentioning that the coefficient function in the inequality satisfied by the Lyapunov function (LF) could be indefinite. Moreover, our paper gave a new estimation for the settling time (ST). Finally, two illustrative examples were given to demonstrate the validity and feasibility of the proposed criteria.

Periodic Solutions in Slowly Varying Discontinuous Differential Equations: A Non-Generic Case

Periodic Solutions in Slowly Varying Discontinuous Differential Equations: A Non-Generic Case

Numerical event location techniques in discontinuous differential algebraic equations

In this paper, we consider numerical methods for the location of events of differential algebraic equations of index one. These events correspond to cross a discontinuity surface, beyond which another differential algebraic equation holds. Convergence theorems of the numerical event time and event point to the true event time and event point are given. It is proved that, for integrations by semi-implicit methods or Rosenbrock methods, the order of convergence of the numerical event location is the order of convergence of the continuous extension and, for integration by implicit Runge-Kutta methods, the order of convergence is the order of convergence of the implicit RK method.

Lyapunov Instability for Discontinuous Differential Equations

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

Fixed-time synchronization of delayed impulsive inertial neural networks with discontinuous activation functions via indefinite LKF method

This article investigates the fixed time synchronization (FXTSY) problem of time-varying delayed impulsive inertial neural networks (INNs) with discontinuous activation functions. First, the addressed delayed discontinuous INNs are converted into a first-order differential equation using a generalized variable transformation with suitable tunable variables. Due to the existence of the discontinuities, the delayed discontinuous differential equations are transformed into the differential inclusions by using the differential inclusion theory and set-valued map concepts. Furthermore, by designing the suitable centralized impulsive control and discontinuous control, constructing the novel indefinite type Lyapunov functionals, new algebraic conditions are derived to realize the FXTSY for the leader-following impulsive INNs. Moreover, the settling time is explicitly calculated. Finally, the developed theoretical results are verified by two numerical simulation results.

Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

Periodic solutions of superlinear and sublinear state-dependent discontinuous differential equations

A classical, second-order differential equation is considered with state-dependent impulses at both the position and its derivative. This means that the instants of impulsive effects depend on the solutions and they are not fixed beforehand, making the study of this problem more difficult and interesting from the real applications point of view. The existence of periodic solutions follows from a transformation of the problem into a planar system followed by a study of the Poincaré map and the use of some fixed point theorems in the plane. Some examples are presented to illustrate the main results.

On the Boundedness of Solutions of Discontinuous Differential Equations

By using the notion of Carathéodory solution to differential equations, the present work studies the boundedness of solutions of discontinuous differential equations. For these discontinuous systems determined by discontinuous differential equations, results are obtained that guarantee sufficient conditions to boundedness of solutions in terms of nonsmooth Lyapunov functions.

Generalized solutions to bounded-confidence models

Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations (ODEs) with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at the boundary. Various works in the literature analyzed properties of solutions, such as barycenter invariance and clustering. On the other side, the problem of giving a precise definition of solution, from an analytical point of view, was often overlooked. However, a rich literature proposing different concepts of solution to discontinuous differential equations is available. Using several concepts of solution, we show how existence is granted under general assumptions, while uniqueness may fail even in dimension one, but holds for almost every initial conditions. Consequently, various properties of solutions depend on the useddefinition and initial conditions.

Global stability of a SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy

We consider a SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis, the basic reproductive number [Formula: see text] is proved to be a sharp threshold value which completely determines the dynamics of the model. If [Formula: see text], then there exists a disease-free equilibrium which is globally stable. If [Formula: see text], the disease-free equilibrium becomes unstable and there exists an endemic equilibrium which is globally stable. We discuss that the disease will die out in a finite time which is impossible for the corresponding SEIR model with continuous treatment. Furthermore, the numerical simulations indicate that strengthening treatment measure after infective individuals reach some level is beneficial to disease control.

A Unified Variational Approach to Discontinuous Differential Equations

A direct variational technique involving Clarke generalized gradient is used to treat general boundary value problems with discontinuous nonlinearities. Based on the theory of positive definite symmetric operators it is established the nonsmooth variational form of the regularized inclusions which give the Filippov solutions of the discontinuous problems. These solutions reduce to classical solutions in case that a transversality condition on the set of discontinuities is satisfied. The results apply to a wide class of concrete boundary value problems of different orders. Two illustrative examples are given.

Model Reference Adaptive Control of Switched Dynamical Systems with Applications to Aerial Robotics

This paper presents an adaptive control law for unknown nonlinear switched plants that must follow the trajectory of user-defined linear switched reference models. The effectiveness of the proposed control architecture is proven in two alternative frameworks, that is, analyzing Caratheodory and Filippov solutions of discontinuous differential equations. Numerical and experimental data verify the applicability of the theoretical results to problems of practical interest. The proposed numerical simulation involves the design of a model reference adaptive control law to regulate the roll dynamics of a reconfigurable delta-wing aircraft. The proposed flight tests involve an aerial robot tasked with autonomously mounting a camera of unknown inertial properties to a vertical surface.

Existence and uniqueness of solutions for systems of discontinuous differential equations under localized Bressan–Shen transversality conditions

We present new results on existence and uniqueness of absolutely continuous solutions for systems of discontinuous ordinary differential equations. Our existence result complements an earlier theorem by Bressan and Shen. Basically, we show that a global transversality condition assumed by Bressan and Shen need only be imposed on the sets where the nonlinear part is discontinuous. Our proof, completely different to the one given by Bressan and Shen, uses Krasovskij solutions as a first step. We illustrate the applicability of our result with several examples not covered by the previous literature. The second part of this paper concerns uniqueness. Specifically, we prove uniqueness of solutions for discontinuous systems of differential equations with piecewise Lipschitz continuous nonlinearities and assuming localized Bressan–Shen transversality conditions on the boundaries between different Lipschitz continuity domains. Our uniqueness result appears to be new even in the classical case of continuous nonlinearities.