Impulse Points Research Articles

Impulsive Control of Nonlinear Systems With Time-Varying Delay and Applications.

Impulsive control of nonlinear delay systems is studied in this paper, where the time delays addressed may be the constant delay, bounded time-varying delay, or unbounded time-varying delay. Based on the impulsive control theory and some analysis techniques, a new theoretical result for global exponential stability is derived from the impulsive control point of view. The significance of the presented result is that the stability can be achieved via the impulsive control at certain impulse points despite the existence of impulsive perturbations which causes negative effect to the control. That is, the impulsive control provides a super performance to allow the existence of impulsive perturbations. In addition, we apply the theoretical result to the problem of impulsive control of delayed neural networks. Some results for global exponential stability and synchronization control of neural networks with time delays are derived via impulsive control. Three illustrated examples are given to show the effectiveness and distinctiveness of the proposed impulsive control schemes.

A Statistical Salt-and-Pepper Noise Removal Algorithm

This paper proposes a novel approach to remove salt-and-pepper noise from a given noisy image. The proposed algorithm is based on statistical quantities such as mean and standard deviation. It determines the intensity to be placed on the impulse point by calculating the eligibility of the nearby points in a very simple way. This method works iteratively and removes all the impulse points restoring the edges and minute details. The proposed algorithm is very efficient and gives better results than various existing algorithms. The performance of the proposed method are compared with other existing methods with images of noise density as high as 99% and is found to perform better.

On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations

A linear two-point boundary value problem for a system of loaded differential equations with impulse effect is investigated. Values in the previous impulse points are taken into consideration in the conditions of impulse effect. The considered problem is reduced to an equivalent multi-point boundary value problem for the system of ordinary differential equations with parameters. A numerical implementation of parametrization method is offered using the Runge–Kutta method of 4th-order accuracy for solving the Cauchy problems for ordinary differential equations. The constructed numerical algorithms are illustrated by examples.

On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses

In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.

Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations

For a new linear non-instantaneous impulsive differential equations, we introduce the notation of non-instantaneous impulsive Cauchy matrix and analyze its exponential structure in terms of eigenvalues of matrix via the distance between impulsive points and junction points. Many useful criteria for asymptotic stability of linear non-instantaneous impulsive problems are derived, which allow us to establish a uniform framework to deal with asymptotic stability of linear non-instantaneous impulsive differential equations with perturbation under mild sufficient conditions. In particular, two examples are given to demonstrate the application of theoretical results for linear part. In addition, we study nonlinear non-instantaneous impulsive equations and investigate existence, uniqueness of their solutions and Ulam–Hyers–Rassias stability under the restriction of exponential growth or stable conditions for non-instantaneous impulsive Cauchy matrix, respectively, which provide an approach to find approximate solution to nonlinear non-instantaneous impulsive equations in the sense of Ulam–Hyers–Rassias stability. The obtained results cover the standard results for instantaneous impulsive differential equations, which are essentially extend the theory in the previous literatures.

Global fuel consumption optimization of an open-time terminal rendezvous and docking with large-eccentricity elliptic-orbit by the method of interval analysis

By defining two open-time impulse points, the optimization of a two-impulse, open-time terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit is proposed in this paper. The purpose of optimization is to minimize the velocity increment for a terminal elliptic-reference-orbit rendezvous and docking. Current methods for solving this type of optimization problem include for example genetic algorithms and gradient based optimization. Unlike these methods, interval methods can guarantee that the globally best solution is found for a given parameterization of the input. The non-linear Tschauner- Hempel(TH) equations of the state transitions for a terminal elliptic target orbit are transformed form time domain to target orbital true anomaly domain. Their homogenous solutions and approximate state transition matrix for the control with a short true anomaly interval can be used to avoid interval integration. The interval branch and bound optimization algorithm is introduced for solving the presented rendezvous and docking optimization problem and optimizing two open-time impulse points and thruster pulse amplitudes, which systematically eliminates parts of the control and open-time input spaces that do not satisfy the path and final time state constraints. Several numerical examples are undertaken to validate the interval optimization algorithm. The results indicate that the sufficiently narrow spaces containing the global optimization solution for the open-time two-impulse terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit can be obtained by the interval algorithm (IA). Combining the gradient-based method, the global optimization solution for the discontinuous nonconvex optimization problem in the specifically remained search space can be found. Interval analysis is shown to be a useful tool and preponderant in the discontinuous nonconvex optimization problem of the terminal rendezvous and docking than GA.

Solvability of a linear boundary value problem for a fredholm integro-differential equation with impulsive inputs

Abstract—We suggest a method for the study and solution of a linear boundary value problem for a Fredholm integro-differential equation with impulsive inputs at given times. The method is based on a partition of the interval and the introduction of auxiliary parameters as the values of the solution at the initial points of subintervals. For each partition containing impulsive points, we construct a Fredholm integral equation of the second kind. We introduce the definition of a regular partition and show that the set of regular mappings is nonempty. By using the resolvent of the constructed integral equation, the fundamental matrix of the differential part, and the original data of the problem, we form a system of linear equations for the introduced parameters. The equivalence of the solvability of that system and the considered linear boundary value problem are proved. We obtain necessary and sufficient conditions for the solvability and unique solvability.

Two-impulse transfer between coplanar elliptic orbits using along-track thrust

The two-body two-impulse orbit transfer problem between two coplanar elliptic orbits using only along-track thrust is studied in this paper. Both impulse vectors are constrained to be on the along-track directions at the impulse points. First, the relationship between transfer angle and final true anomaly is derived. For a given final true anomaly, the range of the transfer angle is obtained for different cases. Then, the transfer angle is solved by the secant method in its feasible range. With the transfer angle and the transfer semilatus rectum, the velocity vectors of the transfer orbit at impulse points are obtained. Two impulse points and the corresponding impulse magnitudes for a 180\(^{\circ }\) transfer are also analyzed. Finally, the minimum-energy and minimum-transit-time solutions are obtained by optimizing the final true anomaly. Numerical tests demonstrate the effectiveness of the proposed method for two-impulse along-track orbit transfers. The results show that the minimum-energy transfer with two along-track impulses is close to the “true” minimum-energy transfer with two free-direction impulses.

Optimal two-impulse rendezvous with terminal tangent burn considering the trajectory constraints

The two-impulse orbital rendezvous problem with a terminal tangent burn between coplanar elliptical orbits is studied by considering a lower bound on perigee radius and an upper bound on apogee radius for the transfer orbit. This problem requires that two spacecraft arrive at the rendezvous point with the same arrival flight-path angle after the same flight time. The admissible range of the final true anomaly that meets the perigee and apogee constraints is obtained in closed form. The revolution number of the transfer orbit is expressed as a function of the true anomaly and the revolution numbers of the initial and final orbits. All the feasible solutions are obtained with a bound on the revolution number of the final orbit. Then, the minimum-fuel one is determined by comparing their costs. Finally, two numerical examples are provided to obtain all the feasible solutions for given initial impulse points and the optimal solution with the initial coasting arc. Numerical results show that the minimum-fuel solution for the terminal tangent burn rendezvous is close to that for the cotangent rendezvous when the rendezvous time is long enough; however, the cotangent rendezvous may not exist when the rendezvous time is short.

Response to “Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014;19:401–3.

This paper is a response to “Comments on the concept of existence of solution for impulsive fractional differential equations” by Wang et al. (2014) [1]. Recently, Wang et al. (2014) [1] made some comments on our paper (Fečkan et al., 2012) [2] and claimed that “The objective of this note to indicate the mistake in these counterexamples and show the plausibility of the previous results”. To achieve their aim, they used classical Caputo fractional derivative and changed it in each subintervals by keeping the impulses which start the lower bounded from different impulsive points. However, we (Fečkan et al., 2012) [2] mean a different one, generalized Caputo derivative, by keeping in each impulses which start the lower bounded from zero. In support of our view-points, we present some scripts to address and discuss the comments.

Tangent-Impulse Interception for a Hyperbolic Target

The two-body interception problem with an upper-bounded tangent impulse for the interceptor on an elliptic parking orbit to collide with a nonmaneuvering target on a hyperbolic orbit is studied. Firstly, four special initial true anomalies whose velocity vectors are parallel to either of the lines of asymptotes for the target hyperbolic orbit are obtained by using Newton-Raphson method. For different impulse points, the solution-existence ranges of the target true anomaly for any conic transfer are discussed in detail. Then, the time-of-flight equation is solved by the secant method for a single-variable piecewise function about the target true anomaly. Considering the sphere of influence of the Earth and the upper bound on the fuel, all feasible solutions are obtained for different impulse points. Finally, a numerical example is provided to apply the proposed technique for all feasible solutions and the global minimum-time solution with initial coasting time.

Minimum-Time Interception with a Tangent Impulse

AbstractThe minimum-time coplanar interception problem with an upper-bounded tangent impulse for a nonmaneuvering target is studied. This problem requires the same flight time for two spacecraft and a transfer orbit tangent to the initial orbit. For a given impulse point, the transfer time for any conic orbit is a function only of the true anomaly of the target orbit; then the minimum-time orbit is determined by comparing all feasible solutions that are obtained by the secant method in the range of solution existence. Moreover, the true-anomaly range of the target orbit is derived for an upper bound on the magnitude of the tangent impulse. Finally, the global minimum-time solution with initial coasting time is obtained by numerical optimization algorithms. An example is given to apply the proposed technique for all the minimum revolution-number solutions at different impulse points and the global minimum-time solution.

Dissipative operators with impulsive conditions

In this paper a singular dissipative impulsive boundary value problem with \(n\)-impulsive points is investigated. In particular, using the Lidskiĭ’s theorem it is proved that all eigen and associated functions of this problem is complete in the Hilbert space.

Bifurcation of positive periodic solutions of first-order impulsive differential equations

We give a global description of the branches of positive solutions of first-order impulsive boundary value problem: which is not necessarily linearizable. Where is a parameter, are given impulsive points. Our approach is based on the Krein-Rutman theorem, topological degree, and global bifurcation techniques. MSC:34B10, 34B15, 34K15, 34K10, 34C25, 92D25.

Existence and solution sets of impulsive functional differential inclusions with multiple delay

In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: \[ \begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, y(t)=\phi(t), & t \in [-r,0],\end{cases} \] where \(J:=[0,b]\) and \(0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b\) (\(m \in \mathbb{N}^*\)), \(F\) is a set-valued map and \(g\) is single map. The functions \(I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k=1,...,m\)). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuousdependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.

A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form, $$\begin{array}{lll} \quad \qquad D^{\alpha}_{*}y(t) & \in & F(t, y(t)), \quad\; {\rm a.e.}\ t\, \in \, J{\backslash} \{t_{1}, \ldots, t_{m}\}, \ \alpha\, \in \, (0,1], \\ y(t^{+}_{k}) - y(t^{-}_{k}) & = & I_{k}(y(t^{-}_{k})), \quad k = 1, \ldots, m, \\ \qquad \qquad y(0) & = & a,\end{array}$$ where J = [0, b], \({D^{\alpha}_{*}}\) denotes the Caputo fractional derivative and F is a set-valued map. The functions Ikcharacterize the jump of the solutions at impulse points tk (\({k = 1, \ldots , m}\)). In addition, several existence results are established, under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on a nonlinear alternative of Leray-Schauder type and on Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.

First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions: { ( y ′ − A y ) ( t ) ∈ F ( t , y ( t ) ) , a.e. t ∈ J ∖ { t 1 , … , t m } , y ( t k + ) − y ( t k − ) = I k ( y ( t k − ) ) , k = 1 , … , m , y ( 0 ) = y ( b ) where J = [ 0 , b ] and 0 = t 0 < t 1 < ⋯ < t m < t m + 1 = b ( m ∈ N ∗ ) A is the infinitesimal generator of a C 0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions I k characterize the jump of the solutions at impulse points t k ( k = 1 , … , m ). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T ( b ) . Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.

Impulsive differential inclusions with fractional order

In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=1,…,m). Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Īk(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, and D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a, where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b), where φ:J×R→P(R) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.

Filippov's theorem for impulsive differential inclusions with fractional order

In this paper, we present an impulsive version of Filippov’s Theorem for fractional differential inclusions of the form: D ∗ y(t) ∈ F (t, y(t)), a.e. t ∈ J\{t1, . . . , tm}, α ∈ (1, 2], y(t+k )− y(t − k ) = Ik(y(t − k )), k = 1, . . . ,m, y(t+k )− y (t−k ) = Ik(y (t−k )), k = 1, . . . ,m, y(0) = a, y′(0) = c, where J = [0, b], D ∗ denotes the Caputo fractional derivative and F is a setvalued map. The functions Ik, Ik characterize the jump of the solutions at impulse points tk (k = 1, . . . ,m).

Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions y 0 (t) �y(t) 2 F(t,y(t)), a.e. t 2 J\{t1,...,tm}, y(t + ) y(t k ) = Ik(y(t k )), k = 1,2,... ,m, y(0) = y(b), where J = [0,b] and F : J × R n ! P(R n ) is a set-valued map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1,2,... ,m). The topological structure of solution sets as well as some of their geometric properties (contractibility and R�-sets) are studied. A continuous version of Filippov’s theorem is also proved.