Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances

• Applicant: Professor Dr. Sergey Dashkovskiy
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 405685496

The aim of this project is to study stability and robustness of global attractors of infinite-dimensional nonlinear systems subject to impulsive actions and external perturbations that can be distributed in the domain on which the system is considered or at the boundary of this domain. Due to impulsive actions the continuous dependence of solutions with respect to the initial conditions fails to hold. This makes the application of classical methods of global attractors theory impossible and requires for new tools and methods to study properties of global attractors. For rather wide classes of systems in infinite-dimensional state spaces we will propose sufficient conditions for the existence, stability and robustness of global attractors. These conditions will be used to study asymptotic behavior of impulsive systems generated by evolution inclusions of parabolic type with multivalued right-hand side, parabolic systems of reaction-diffusion type with non-smooth interaction functions and weakly nonlinear hyperbolic equations. Two types of impulsive actions will be considered. One of them leads to an instantaneous state transition from one given subset of the states space to another one, which is also fixed and given. This happens, for example, when the energy of a system jumps from one fixed level to another one. The other one is such that the instantaneous change of the state depends on the state value before the jump only. Such behavior appears, for example, when elastic shock happens due to a collision of a solid with a rigid body. A suitable stability concept for global attractors of such systems will be elaborated for this purpose.After this we will investigate the influence of external perturbations on global attractors. In particular the deviation of the perturbed systems flow from the unperturbed one will be studied in dependence on the size of the disturbance. We are going to establish input-to-state stability properties of global attractors for nonlinear parabolic equations and inclusions.Moreover we will consider interconnected infinite-dimensional systems having stable global attractors and ask whether the interconnection possess a global attractor as well and whether it is still input-to-state stable or not. Small-gain type conditions will be developed to answer these questions. The interactions between small-gain and dwell-time conditions that are typically needed in case of impulsive systems will be also investigated for interconnections subject to impulsive effects.

• DFG Programme: Research Grants
• International Connection: Ukraine
• Partner Organisation: The State Fund for Fundamental Researches
• Cooperation Partner: Professor Dr. Oleksiy Kapustyan