Piecewise Continuous Solutions Research Articles

Positive piecewise continuous quasi-periodic solutions to logistic impulsive differential equations

ABSTRACT To prove the existence of piecewise continuous solutions to a logistic quasi-periodic differential system with impulses (whose coefficients have rationally independent periods), this system is divided into a differential equation and a difference equation. The quasi-periodicity of a function is proved by showing that this function is the uniform limit of a series of trigonometric polynomials with a finite total number of frequencies. The asymptotically stable quasi-periodic positive and piecewise continuous solution is proved to exist and to be unique. Quasi-periodic variation of the environment leads to a quasi-periodic growth of the population size in the sense that the rationally independent frequencies of the system are also frequencies of the quasi-periodic solution. The positive solutions have a repeated behavior similar to that of the quasi-periodic solution for a sufficiently long time due to asymptotical stability. The separation of the continuous-discrete system into a differential equation and a difference equation is a method of proving the existence of a quasi-periodic solution with perturbed coefficients of the impulsive system.

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

AbstractThe objective of this article is to consider a new class of fractional stochastic differential systems driven by the Rosenblatt process with impulses. We used fractional calculus, stochastic analysis, and Krasnoselskii's fixed point theorem to study the existence of piecewise continuous mild solutions for the proposed system. Further, we discussed the existence of optimal controls for the considered system. Our main results are well supported by an illustrative example.

Non-autonomous Evolution Equations of Parabolic Type with Non-instantaneous Impulses

In this paper, we study the Cauchy problem to a class of non-autonomous evolution equations of parabolic type with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. New existence result of piecewise continuous mild solutions is established under more weaker conditions. At last, as a sample of application, the abstract result is applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families

In this paper, we study the initial value problem to a class of non-autonomous parabolic evolution equations with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an noncompact evolution family. Some new existence results of piecewise continuous mild solutions are established under more weaker conditions. At last, as a sample of application, these results are applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The results obtained in this paper are a supplement to the existing literature and essentially extend some existing results in this area.

Spacecraft formation flying reconfiguration with extended and impulsive maneuvers

This paper presents the control solutions to the spacecraft formation reconfiguration problem when impulsive or extended maneuvers are considered, and the reference orbit is circular. The proposed approach for the derivation of the control solutions is based on the inversion of the linearized equations of relative motion parameterized using the mean relative orbit elements. The use of mean relative orbit elements eases the inclusion of perturbing accelerations, such as the Earth’s oblateness effects, and offers an immediate insight into the relative motion geometry. Several maneuvering schemes of practical operational relevance are considered and the performance of the derived impulsive and piecewise continuous control solutions are investigated through the numerical propagation of the nonlinear relative dynamics. Finally, the benefits of the new extended maneuvers strategies are assessed through a comparison with the corresponding impulsive one.

Two dimensional relativistic Euler equations in a convex duct

In this paper, we investigate the motion of supersonic flow governed by 2-D steady isentropic irrotational relativistic Euler system entered into a convex duct. We will give the propagation and interactions of rarefaction waves between the upward and downward wall and obtain the global existence of piecewise continuous solutions. Furthermore, in some cases the vacuum state may appear and we also give a sufficient condition to the appearance of the vacuum in finite domain.

Invariance principle for nonautonomous functional-differential equations with discontinuous right-hand sides

For nonautonomous functional-differential equations with piecewise continuous right-hand sides and solutions understood in the sense of A.F. Filippov, the method of limit differential inclusions is proposed to study the asymptotic properties of solutions. Properties of the type of the invariance of -limit sets and analogues of LaSalle’s invariance principle are proved by applying invariantly differentiable Lyapunov functionals with derivatives of constant sign.

A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications

Abstract In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order q ∈ ( 0 , 1 ) {q\in(0,1)} , see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1–4.4. We allow the nonlinearity p ⁢ ( t ) ⁢ f ⁢ ( t , x ) {p(t)f(t,x)} in fractional differential equations to be singular at t = 0 , 1 {t=0,1} and be involved a super-linear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.

Boundary value problems for impulsive Bagley–Torvik models involving the Riemann–Liouville fractional derivatives

By using the Picard iterative technique, we get the exact expression of solutions of linear fractional Bagley–Torvik model. Then the piecewise continuous solutions of this kind of models are obtained. By using these results, we convert the boundary value problems for impulsive fractional differential equations to integral equations technically. By using the weighted function Banach spaces, the completely continuous operators and the Schauder’s fixed point theorem, some existence results for solutions of the boundary value problems for impulsive Bagley–Torvik fractional differential equations are established. Examples are given to illustrate the main results.

Piecewise continuous solutions of initial value problems of singular fractional differential equations with impulse effects

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.

On piecewise continuous solutions of higher order impulsive fractional differential equations and applications

For impulsive differential equations with fractional order, we show that the formula of solutions in cited papers are incorrect. We then give exact piecewise continuous solutions (the explicit solutions) of two classes of fractional differential equations of order α∈(n−1,n) involving Caputo derivatives and Riemann–Liouville derivatives. Apply our results to obtain existence of solutions of two classes of initial value problems of singular impulsive fractional differential equations. Examples are presented to illustrate the main theorems.

Periodic boundary value problems for singular fractional differential equations with impulse effects

Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann-Liouville and Hadamard fractional derivatives with order $q \in(0,1)$. Then we obtain exact expression of solutions of impulsive fractional differential equations, i.e., exact expression of piecewise continuous solutions. Finally, four classes of integral type periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity $p(t) f(t, x)$ in fractional differential equations to be singular at $t=0,1$ and be involved a superlinear and sub-linear term. The analysis relies on Schaefer's fixed point theorem.

Existence and β-Ulam-Hyers stability for a class of fractional differential equations with non-instantaneous impulses

In this paper, we investigate a new class of fractional differential equations with non-instantaneous impulses. We give a suitable formula of piecewise continuous solutions and present the concept of β-Ulam-Hyers stability. We present existence and β-Ulam-Hyers stability results on a compact interval.

Solvability for an impulsive fractional multi-point boundary value problem at resonance

In this paper, we consider the fractional multi-point boundary value problem with impulse effects. As indicated by M Feckan, Y Zhou and JR Wang in 2012, the concepts of piecewise continuous solutions used in most of the current literature about the impulsive differential equations of fractional order are not appropriate. Based on a new concept of a piecewise continuous solution, we establish sufficient conditions for the existence of the solutions for the boundary value problem at resonance by the continuation theorem of coincidence degree theory.

Periodic BVPs for fractional order impulsive evolution equations

In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results.

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.

Nonlinear impulsive problems for fractional differential equations and Ulam stability

In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided.

On a class of iterative-difference equations

Solving the class of second order iterative-difference equations is a difficult problem, proposed as an open problem in Brillouët-Belluot [Aequationes Math. 61 (3) (2001), p. 304]. In this paper, we use three methods to approach such a class of equations, finding their affine solutions, proving the existence of their bounded continuous solutions and constructing their continuous and piecewise continuous solutions.

An analytical solution for a nonlinear time-delay model in biology

In this paper, the homotopy analysis method is applied to develop a analytic approach for nonlinear differential equations with time-delay. A nonlinear model in biology is used as an example to show the basic ideas of this analytic approach. Different from other analytic techniques, the homotopy analysis method provides a simple way to ensure the convergence of the solution series, so that one can always get accurate approximations. A new discontinuous function is defined so as to express the piecewise continuous solutions of time-delay differential equations in a way convenient for symbolic computations. It is found that the time-delay has a great influence on the solution of the time-delay nonlinear differential equation. This approach has general meanings and can be applied to solve other nonlinear problems with time-delay.

Repair-Control of Enterprise Systems Using RFID Sensory Data

Abstract This paper describes a framework for implementing real-time enterprise planning, scheduling and control processes based on information provided by RFID sensing systems. The proposed framework is based on optimal control algorithms, and interfaces with existing ERP infrastructure. The objective is to respond autonomously to changes in the enterprise using a feedback configuration that minimizes disruptions. RFID sensing systems have the potential to provide the real time data needed to implement enterprise feedback functionality. The central concept presented in the paper is a real time repair schema implemented in a distributed architecture, that utilizes dynamical models described by differential equations with piece-wise continuous solutions.