Stability, stabilization and robustness of non-autonomous infinite-dimensional systems

• Applicant: Professor Dr. Vitalii Slynko
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 492875426

Stability and robustness are fundamental properties of dynamical systems and are very essential for the proper and effective operation of practical processes modeled by them. Analysis of these properties lead to increasing difficulties due the growing complexity of the practical systems in view of the large-scaleness, heterogeneity (discrete and continuous dynamics), non-stationary disturbances and non-linearity. Many modern applications are modeled by coupled systems including infinite dimensional ones. They are often nonlinear and possess hybrid dynamics due to switching or impulsive actions. Mathematical analysis and control of such systems require mathematical methods from functional analysis, partial differential equations, semi-group theory, Lie algebra theory and related numerical techniques. Lyapunov methods are the most powerful especially in case of nonlinear equations. We will study several classes of non-autonomous infinite dimensional systems in view of their stability. Lyapunov methods will be developed for this purpose by means of Lie algebraic tools and commutator calculus. Stability conditions will be established and explicit constructions of Lyapunov functions will be developed. Stabilizing controllers will be derived to guarantee desired stability and robustness properties for several classes of infinite dimensional systems. In particular we will consider the problem of stabilization of an abstract bilinear differential equation with impulsive action. Based on the identities of the commutator calculus we will construct a Lyapunov function. Then the problem of stabilization will be reduced to the study of an abstract linear comparison equation with Lie brackets of higher orders. The proposed approach allows to synthesize a stabilizing controller for the original system. These results will be applied to solve the stabilization problem for bilinear PDE systems of the parabolic type. We will extend the proposed approach for coupled abstract differential equations. Also we will consider a particular case of coupled parabolic PDE and ODE systems.We will propose new methods of construction of Lyapunov func for abstract differential equations in a Hilbert space. These results will be applied to switched infinite dimensional systems.The proposed Lyapunov functions will be used to investigate the robustness properties for nonlinear non-autonomous abstract differential equations with inputs and to the study the stability of non-autonomous PDE and related interconnections of ODE and PDE systems. We will formulate and prove a comparison principle for essentially nonlinear impulsive systems when the moments of impulsive action satisfy an averaged-dwell-time (ADT) condition. This comparison principle reduces the original problem to the stability investigation of a nonlinear impulsive system with constant dwell-times. These results will be then used to the study of critical cases in the theory of stability of impulsive systems.

• DFG Programme: Research Grants