Period: 04/2019 - 03/2021
Project Title: Robust stabilization of interconnected infinite-dimensional systems with boundary couplings
Funding: German Research Foundation (DFG)
Project leader: Dr. Andrii Mironchenko

Modern applications of control theory to chemical reactors, traffic networks, multi-body systems (e.g., robotic arms, flexible elements in aircraft), fluid-structure interactions, etc. require methods for robust stabilization of coupled heterogeneous systems, described by partial differential equations (PDEs). The complexity of such systems, the necessity to apply controls only to the boundary of their spatial domain, and the need to ensure efficiency and reliability of a control design in spite of actuator and observation errors and external disturbances, make the stabilization of coupled PDE systems a highly challenging problem. To approach this problem, we base ourselves on the input-to-state stability (ISS) theory, which plays a prominent role in robust nonlinear control theory. ISS unifies the notions of uniform asymptotic stability and external (input-output) stability, and provides powerful methods to study the stability of coupled nonlinear control systems by means of 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 and semigroup theories. In this proposal, we develop tools for the systematic design of robust boundary feedback controllers for linear and nonlinear PDEs and for interconnections of heterogeneous distributed parameter systems with boundary couplings. This will be achieved via the synergy of ISS theory for distributed parameter systems with the methods for controller design for partial differential equations with boundary controls. We proceed in several steps. First, we radically improve the available toolkit for ISS analysis of boundary control systems and time-delay systems. We extend the applicability of the classical Lyapunov and Lyapunov-Krasovskii methods and develop a brand-new methodology of non-coercive Lyapunov functions as well as apply methods of monotone control systems. We apply our results to several important classes of nonlinear PDEs, including parabolic equations with nonlinear boundary conditions, hyperbolic systems and nonlinear KdV equation. Next, we combine the developed ISS Lyapunov theory for boundary control systems with the methods to design boundary controllers for PDE systems, most notably with PDE backstepping technique. Finally, we combine the obtained results with the small-gain methodology to study the robust stabilization problem for the coupled heterogeneous systems, including interconnections of ordinary and partial differential equations (so-called ODE-PDE cascades) and couplings of time-delay and PDE equations (delay- PDE cascades).