Lyapunov theory meets boundary control systems

• Applicant: Privatdozent Dr. Andrii Mironchenko
• Subject Area: Mathematics
• Term: from 2018 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415101813

• Final Report Year: 2025

Final Report Abstract

Nonlinear distributed parameter systems with both distributed and boundary inputs model various phenomena, including chemical reactors, fluid and gas dynamics, traffic networks, multi-body systems, adaptive optics, and fluid-structure interactions. For many such systems, small disturbances—arising from actuator and observation errors, hidden dynamics, or external factors—can significantly degrade performance, compromise stability, or even destabilize control systems. Addressing these challenges requires robust controller and observer design for nonlinear boundary control systems (BCS) to ensure reliability and efficiency of closed-loop systems. This task is particularly complex since infinite-dimensional systems can often be controlled with a small number of actuators and sensors, typically placed at the system’s boundary and accessed only at discrete time instants. This project tackles these issues using input-to-state stability (ISS) theory, a key framework in robust nonlinear control. ISS integrates uniform asymptotic stability with external (input-output) stability, offering powerful tools—such as Lyapunov and small-gain methods—to analyze coupled nonlinear control systems. Over the past five years, ISS theory for infinite-dimensional systems has advanced significantly, drawing on techniques from nonlinear systems theory, partial differential equations, and operator theory. As a starting point, we have studied well-posedness and regularity properties of nonlinear boundary control systems, which is needed for the application of the infinite-dimensional ISS theory to this class of systems. Next, we have developed several results in the Lyapunov theory for infinite-dimensional systems, in particular, we have clarified the conditions under which the coercive quadratic ISS Lyapunov function for linear boundary control systems exist. Significant progress has been established in the derivation of converse Lyapunov theorems for several properties related to ISS such as outputto-state stability property, and for robust forward completeness. These results are important in their own right, but also they give an idea how a general converse ISS Lyapunov result can be achieved. Besides boundary control systems, ISS of several other classes of infinite-dimensional systems has been considered. In particular, breakthrough results have been achieved in the small-gain analysis of nonlinear infinite networks, - including first small-gain theorems for infinite networks consisting of nonlinear subsystems and with nonlinear type of interactions between subsystems. Significant progress has been achieved in the ISS theory of time-delay systems, including characterizations of the input-to-state stability and the properties of reachability sets of delay systems. Finally, a new paradigm of live systems has been proposed which gives a possibility to model systems whose state space depends on time, which opens new vistas in control theory.

Publications

• A Case Study of Port-Hamiltonian Systems With a Moving Interface. IEEE Control Systems Letters, 7, 1572-1577.
  Kilian, Alexander; Maschke, Bernhard; Mironchenko, Andrii & Wirth, Fabian
  
• A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems. IEEE Transactions on Automatic Control, 68(3), 1447-1462.
  Kawan, Christoph; Mironchenko, Andrii & Zamani, Majid
  
• Input-to-State Stability. Communications and Control Engineering, 41-115. Springer International Publishing.
  Mironchenko, Andrii
  
• Lyapunov criteria for robust forward completeness of distributed parameter systems. Systems & Control Letters, 180, 105618.
  Mironchenko, Andrii
  
• Well-posedness and properties of the flow for semilinear evolution equations. Mathematics of Control, Signals, and Systems, 36(3), 483-523.
  Mironchenko, Andrii
  
• For Time-Invariant Delay Systems, Global Asymptotic Stability Does Not Imply Uniform Global Attractivity. IEEE Control Systems Letters, 8, 484-489.
  Chaillet, Antoine; Wirth, Fabian; Mironchenko, Andrii & Brivadis, Lucas
  
• Forward Completeness Implies Bounded Reachable Sets for Time-Delay Systems on the State Space of Essentially Bounded Measurable Functions. IEEE Control Systems Letters, 8, 1667-1672.
  Brivadis, Lucas; Chaillet, Antoine; Mironchenko, Andrii & Wirth, Fabian
  
• Stability criteria for positive semigroups on ordered Banach spaces. Journal of Evolution Equations, 25(1).
  Glück, Jochen & Mironchenko, Andrii
  
• Infinite-dimensional port-Hamiltonian systems with a stationary interface. European Journal of Control, 82, 101190.
  Kilian, Alexander; Maschke, Bernhard; Mironchenko, Andrii & Wirth, Fabian