Abstract
In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka–Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.
Highlights
Fractional differential systems have attracted the attention of many researchers, due to their generalizations and wide range of applications in science and technologies
We will state our main practical stability and boundedness criteria for impulsive systems with FLDs. These results extend and generalize the results in [29,31,32,33,34,36,40,41] for different classes of differential, functional differential and fractional differential equations, and are first contributions to the stability theory of impulsive equations with FLDs
Note that boundedness results for fractional differential equations are very rare [49] in the existing literature
Summary
Fractional differential systems have attracted the attention of many researchers, due to their generalizations and wide range of applications in science and technologies. To the best of our knowledge, practical stability results with respect to manifolds have not been established for fractional-like differential systems under impulsive perturbations. We will apply the Lyapunov technique and extend the practical stability results for differential equations with FLDs to the impulsive case To this end, we elaborate the definition of FLDs of piecewise continuous Lyapunov-type functions. 2. Practical stability and boundedness results with respect to a manifold defined by a specific function for a fractional-like impulsive system are proposed. We will need the following lemma whose proof is similar to the proofs of corollaries 5.3 and 5.4 in [25] using the generalized definition for FLDs. Similar results for equations with fractional Caputo-type derivatives are given in [36].
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