Abstract
The state-dependent impulsive dynamical system with boundary constraints is a kind of special but common system in nature. But because of the complexity of the geometry or topological structures of the impulsive surface, it is hard to determine when an event or an impulsive surface is reached. Therefore, a general state-dependent impulsive nonlinear dynamical system is rarely studied. This paper presents a class of state-dependent impulsive dynamical systems with boundary constraints. We obtain the existence and continuation of their viable solutions and provide sufficient conditions for the existence and uniqueness of the viable solutions to the system. Finally, two examples are given to illustrate the effectiveness of the results.
Highlights
1 Introduction The impulsive conditions are involved in ordinary differential equations, these conditions may be involved in fractional differential equations as well as in partial differential equations [1, 2]
Impulsive differential equations (IDEs) are basic dynamical models to describe the dynamics of kinds of evolution processes which experience a change of state suddenly, such as harvesting, vibro-impact, natural disasters
Impulsive differential equations play a very important role in the model construction and analysis of impulsive problems in electrical, mechanical, population dynamics, industrial robotics, biotechnology, optimal control, pharmacokinetics, economic and social sciences, and so on [6, 7], and they have been extensively studied in the past several years [8,9,10,11,12]
Summary
The impulsive conditions are involved in ordinary differential equations, these conditions may be involved in fractional differential equations as well as in partial differential equations [1, 2]. Motivated by the above discussions, this paper further studies the viability problem of solutions for general state-dependent impulsive autonomous differential systems with state constraints by combining the relevant research methods in the book Discontinuous Dynamical Systems. We take a reasonable control strategy on the state of systems when the evolution of state x(t) reaches the boundary of K, that is, when x(t) reaches M ⊂ ∂K at time tk(M), x(t) is reset to x(t+) = J(x(t–)) Under this strategy, we consider state-dependent impulsive autonomous differential systems with state constraints that are governed by the following:. 3, sufficient conditions for the existence and continuation of viable solution of state-dependent impulsive autonomous differential system (1.1) with state constraints are presented and proved.
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