Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results

Integral Input-to-State Stability of Nonlinear Time-Delay Systems with Delay-Dependent Impulse Effects

This paper studies integral input-to-state stability (iISS) of nonlinear impulsive systems with time-delay in both the continuous dynamics and the impulses. Several iISS results are established by using the method of Lyapunov-Krasovskii functionals. For impulsive systems with iISS continuous dynamics and destabilizing impulses, we derive two iISS criteria that guarantee the uniform iISS of the whole system provided that the time period between two successive impulse moments is appropriately bounded from below. Then we provide an iISS result for systems with unstable continuous dynamics and stabilizing impulses. For this scenario, it is shown that the iISS properties are guaranteed if the impulses occur frequently enough. For impulsive systems with stabilizing impulses and stable continuous dynamics for zero input, we obtain an iISS result which shows that the entire system is uniformly iISS over arbitrary impulse time sequences. As applications, iISS properties of a class of bilinear systems are studied in details with simulations to demonstrate the presented results