Lyapunov theory meets boundary control systems

• Applicant: Dr. Andrii Mironchenko
• Subject Area: Mathematics
• Term: since 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415101813

Nonlinear distributed parameter systems with both distributed and boundary inputs are used to model a broad range of phenomena, including chemical reactors, fluid and gas dynamics, traffic networks, multi-body systems, adaptive optics, fluid-structure interactions, etc. For many classes of such systems, it is known that small disturbances coming from actuator and observation errors, hidden dynamics, and external disturbances, can dramatically reduce the performance, alter the stability, or even destabilize the control system. Counteracting to these challenges requires the development of methods for the design of robust controllers and observers for nonlinear boundary control systems (BCS), which ensure the reliability and efficiency of closed-loop systems. This objective has to be achieved despite the fact, that usually infinite-dimensional systems have to be controlled using finitely many (and usually very few) actuators and sensors, which can be accessed possibly only at some discrete moments of time, and which can frequently be placed merely at the boundary of the spatial domain.To approach these problems, we base ourselves on the input-to-state stability (ISS) theory, which plays a prominent role in robust nonlinear control. ISS unifies the notions of uniform asymptotic stability and external (input-output) stability, and provides powerful tools to study the stability of coupled nonlinear control systems via Lyapunov and small-gain methods. The ISS theory of infinite-dimensional systems was intensively developing during the last five years and has evolved into a powerful interdisciplinary theory exploiting methods from nonlinear systems theory, partial differential equations, and operator theory. In this project, we develop a systematic ISS theory for nonlinear BCS using a synergy of Lyapunov analysis and operator-theoretic methods (semigroup, and admissibility theory). We analyze the applicability of classical (coercive) Lyapunov methods for ISS analysis of linear BCS and develop wide-reaching constructive converse non-coercive ISS Lyapunov theorems for linear systems. Next, we characterize well-posedness of nonlinear boundary control systems and extend our design of Lyapunov functions for linear boundary control systems to the nonlinear case. As an important application of this machinery, we propose constructions of ISS Lyapunov functions for linear and nonlinear reaction-diffusion equations, as well as for Burger's equations, which are important open problems right now. In the second part of this project, we apply our techniques to the development of robust boundary controllers and observers for PDE systems, to robust event-based control, and to robust stabilization of PDE-ODE cascades, for which Lyapunov-based methods are an indispensable tool.

• DFG Programme: Research Grants