Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems

Input-to-state stability of non-autonomous infinite-dimensional control systems

“…As a byproduct of this insight, the uniform asymptotic stability criteria of non-autonomous infinite-dimensional systems is also established. These results generalize those ones in [3,23,26]. Furthermore, we present a converse Lyapunov for a class of linear non-autonomous systems in a Hilbert space that generates an evolution operator uniformly exponentially stable.…”

“…The ISS means that the state of the system will eventually become small if the external inputs are uniformly small for each initial state. ISS were subsequently extended and studied for different types of dynamical systems, such as discrete-time, time-varying delay and infinite dimensional systems, see [2,3,5,7,14,16]. The ISS theory of ordinary differential equations (ODEs) and the robust stability analysis of partial differential equations (PDEs) motivated the development of ISS theory in the infinite-dimensional, see [11,13,14,16,17,18,15]).…”

“…Sufficient conditions are given that verifying the ISS, iISS and uniform asymptotic stability of nonlinear systems with time-delay by employing the indefinite derivative Lyapunov-Krasovskii functional, see [21]. In [3], iISS of non-autonomous evolutions equations has been investigated. Indeed, it was shown the equivalence between uniform global asymptotic stability and iISS for non-autonomous bilinear infinite-dimensional control systems.…”

Comparison principles and input-to-state stability for stochastic impulsive systems with delays

“…In recent years, the input-to-state stability (ISS) and extensions of the ISS on different systems have attracted widespread attentions in the literature (Damak, 2021;Khalil, 2002;Sontag, 2004;Sontag & Wang, 1996) due to their extensive usage in characterizing the effects of external inputs (such as sensor noise (Liu et al, 2012), actuator disturbances (Liao et al, 2014), parameter perturbations, or measurement errors) on the considered systems. The notion of ISS, firstly proposed in Sontag (1989) for continuous-time nonlinear systems, is formed to investigate how the external disturbance affects the system stability.…”

The Practical Feedback Stabilization for Evolution Equations in Banach Spaces

“…Moreover, they gave the exponential stabilizability of a class of nonlinear control systems. In recent years, nonautonomous differential equations on infinite-dimensional spaces have been studied by many researchers, see the references Damak and Hammami (2020), Damak (2021), Chen et al (2020aChen et al ( , 2020bChen et al ( , 2020cChen et al ( , 2021 and Chen (2021) for more details. In the study by Chen et al (2020b), sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are given by introducing a new Green's function and constructing a control function involving Gramian controllability operator.…”