Asymptotic gain results for attractors of semilinear systems

Abstract

<p style='text-indent:20px;'>We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee–Infante equations, for example.</p>

Well-posedness and stability of non-autonomous semilinear input-output systems

“…In fact, this would naturally continue and extend the results from [41] about autonomous semilinear systems of parabolic type with more general nonlinearities. And finally, it would be interesting to strengthen the presented uniform global stability result to an abstract input-tostate stability result, in order to systematically obtain non-autonomous analoga of the autonomous input-to-state stability results discussed in [22], [41], [38], [8], or [36], for instance.…”

“…Stability properties like uniform global stability are, of course, already interesting in themselves, but they also play an important role in establishing input-to-state stability. See [23], [32], [22], [41], [38], [8], [36], for instance, where the case of autonomous systems is treated.…”

Robust Stability of Global Attractors for Evolutionary Systems without Uniqueness

“…But from the application point of view it is important to prove robust stability, i.e., stability with respect to disturbances so called Input-to-State Stability (ISS) [17][18][19][20]. For single-valued evolutionary systems with non-trivial global attractors ISS theory was developed in [21][22][23]. In the present paper we generalize these results to general multi-valued case.…”

Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory

“…Stability properties of global attractors, including impulsive perturbations, can be found in [1]- [5], [9]. Recently in [6], [21] there have been obtained results about input-to-state stability and asymptotic gain properties with respect to global attractors of semi linear heat and wave equations in L 2 space. This results requires that the corresponding non autonomous problem generated semi process family with uniform attractor [3] which tends to the global attractor of undisturbed system.…”