arXiv subjects
Publications and source records attributed to S. Dashkovskiy.
The paper is devoted to obtaining conditions for the roughness of dichotomy in the Banach spaces. Deep analysis of the well known papers was considered. The main results also works for the case of unbounded operators.
This work studies stability and robustness of a nonlinear system given as an interconnection of an ODE and a parabolic PDE subjected to external disturbances entering through the boundary conditions of the parabolic equation. To this end we develop an approach for a construction of a suitable coercive Lyapunov function as one of the main results. Based on this Lyapunov function we establish the well-posedness of the considered system and establish conditions that guarantee the ISS property. ISS estimates are derived explicitly for the particular case of globally Lipschitz nonlinearities.
The small gain condition is sufficient for input-to-state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of a large size of the system. To facilitate this procedure we aggregate the subsystems and the gains between the subsystems that belong to certain interconnection patterns (motifs) using three heuristic rules. These rules are based on three motifs: sequentially connected nodes, nodes connected in parallel and almost disconnected subgraphs. Aggregation of these motifs keeps the main structure of the mutual influences between the subsystems in the network. Furthermore, fulfillment of the reduced small gain condition implies ISS of the large network. Thus such reduction allows to decrease the number of computations needed to verify the small gain condition. Finally, an ISS-Lyapunov function for the large network can be constructed using the reduced small gain condition. Applications of these rules is illustrated on an example.