Stabilität und Robustheit von Attraktoren nichtlinearer unendlich-dimensionaler Systeme bei Störungen
Zusammenfassung der Projektergebnisse
A combination of discrete and continuous behavior of systems appears in many modern applications due to a sampling of signals, event triggered controllers or just mechanical collisions. Another relevant example is pandemic evolution subjected to vaccination. The corresponding processes can be modeled by impulsive dynamical systems and are often of a large-scale or infinite dimensional. External disturbances are inevitable in many cases and can affect the behavior of the model drastically decreasing the performance of the corresponding practical system. Both stability and robustness are important for a reasonable operation of such systems. Mathematical tools to verify and quantify these properties are instrumental and can be used for the design or stabilization purposes. In this project we have developed such tools, which can be used in practical applications and for further theoretical investigations, for example for the design of stabilizing controllers. Moreover, the question of a stable and robust interaction of interconnected systems is relevant in modeling and design of modern networks and multi-agent systems. This question was also studied in this project. Stability is usually understood with respect to an equilibrium point. A more general notion is a global attractor to which all solutions converge. In this project we have investigated the influence of external disturbances on the globally attracting sets. Our main achievements include: • Conditions for the existence and stability of globally attracting sets for impulsive systems without external perturbations. • Conditions under which solutions are globally attracted to a vicinity of attractors in case of external perturbations. • Derivations of ISS-type estimates for solutions to different classes of systems. • Conditions assuring input-to-state stability for impulsive systems that can cover the case of simultaneous instability of the flow and jumps constituting the system. Recall, that such conditions do not exist in the ISS framework even in the finite dimensional case. • An approach to construct a Lyapunov function assuring asymptotic stability for interconnected not-strictly hyperbolic system, which allows to establish stability in cases, where any of known approaches cannot be applied. • Stabilizing impulsive actions were designed to stabilize a perturbed ensemble on the base of a scalar output. Potential applications of these results include stabilization of multicomponent systems, stability analysis and regulation of boundary control systems, analysis and design of large-scale multi-agent systems. These problems naturally appear in biology, chemical engineering, robotics, aerospace engineering, additive manufacturing as well as climate modeling.
Projektbezogene Publikationen (Auswahl)
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A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations. Mathematics of Control, Signals, and Systems, 32(3), 309-326.
Dashkovskiy, Sergey; Kapustyan, Oleksiy & Schmid, Jochen
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Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input. International Journal of Robust and Nonlinear Control, 31(1), 225-249.
Almeida, R.; Hristova, S. & Dashkovskiy, S.
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Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System. Mathematical Problems in Engineering, 2021, 1-7.
Dashkovskiy, S.; Kapustian, O. A.; Kapustyan, O. V. & Gorban, N. V.
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Stability conditions for impulsive dynamical systems. Mathematics of Control, Signals, and Systems, 34(1), 95-128.
Dashkovskiy, Sergey & Slynko, Vitalii
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Stability of uniform attractors of impulsive multi-valued semiflows. Nonlinear Analysis: Hybrid Systems, 40, 101025.
Dashkovskiy, Sergey; Kapustyan, Oleksiy & Perestyuk, Yuriy
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Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces. European Journal of Control, 67, 100703.
Dashkovskiy, Sergey & Zaitsev, Vasilii
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Asymptotic gain results for attractors of semilinear systems. Mathematical Control and Related Fields, 12(3), 763.
Schmid, Jochen; Kapustyan, Oleksiy & Dashkovskiy, Sergey
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Robustness of global attractors: Abstract framework and application to dissipative wave equations. Evolution Equations and Control Theory, 11(5), 1565.
Dashkovskiy, Sergey & Kapustyan, Oleksiy
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Dwell-time stability conditions for infinite dimensional impulsive systems. Automatica, 147, 110695.
Dashkovskiy, Sergey & Slynko, Vitalii
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Lyapunov Functions for Linear Hyperbolic Systems. IEEE Transactions on Automatic Control, 68(11), 6496-6508.
Atamas, Ivan; Dashkovskiy, Sergey & Slynko, Vitalii
