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Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

Generalized solutions to hybrid dynamical systems

Ricardo G. Sanfelice, Rafal Goebel, Andrew R. Teel (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise in the system is taken into account.

(h0, h)-boundedness of the solutions of differential systems with impulses.

G. K. Kulev, D. D. Bainov (1990)

Publicacions Matemàtiques

In the present paper the question of boundedness of the solutions of systems of differential equations with impulses in terms of two measures is considered. In the investigations piecewise continuous auxiliary functions are used which are an analogue of the classical Lyapunov's functions. The ideas of Lyapunov's second method are combined with the newest ideas of the theory of stability and boundedness of the solutions of systems of differential equations.

Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative

Ait Dads, E., Benchohra, M., Hamani, S. (2009)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 34A37.In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential inclusions involving the Caputo fractional derivative. Both cases of convex and nonconvex valued right-hand side are considered. The topological structure of the set of solutions is also considered.

Impulsive semilinear neutral functional differential inclusions with multivalued jumps

Nadjet Abada, Ravi P. Agarwal, Mouffak Benchohra, Hadda Hammouche (2011)

Applications of Mathematics

In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Impulsive stabilization of high-order nonlinear retarded differential equations

Juan Liu, Xiaodi Li (2013)

Applications of Mathematics

In this paper, impulsive stabilization of high-order nonlinear retarded differential equations is investigated by using Lyapunov functions and some analysis methods. Our results show that several non-impulsive unstable systems can be stabilized by imposition of impulsive controls. Some recent results are extended and improved. An example is given to demonstrate the effectiveness of the proposed control and stabilization methods.

Integral Manifolds and Perturbations of the Nonlinear Part of Systems of Autonomous Differential Equations with Impulses at Fixed Moments

Stamov, G. (1995)

Serdica Mathematical Journal

* This investigation was supported by the Bulgarian Ministry of Science and Education under Grant MM-7.Sufficient conditions are obtained for the existence of local integral manifolds of autonomous systems of differential equations with impulses at fixed moments. In case of perturbations of the nonlinear part an estimate of the difference between the manifolds is obtained.

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