Input-to-state stability and stabilization of distributed parameter systems

• Applicants: Professor Dr. Sergey Dashkovskiy; Professorin Dr. Birgit Jacob; Professor Dr. Fabian Wirth
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 281417092

Robust stability and stabilization of control systems are fundamental and challenging problems in control theory and its applications. One of the milestones of stability theory is the theory of input-to-state stability (ISS), developed over the last two decades for ordinary differential equations and more general finite dimensional systems. This concept is especially useful for the analysis of robust stability of nonlinear systems, control design for nonlinear systems and the dynamics of interconnected systems.However, in the area of infinite-dimensional systems the theory is far from being complete. In modern applications, a crucial role is played by distributed parameter systems (DPS), both linear and nonlinear. ISS theory for such systems is becoming increasingly popular in recent years, but it is still fragmentary and considerably less developed than the finite dimensional case. Furthermore, over the last decade novel methods for stabilization of infinite-dimensional systems have been proposed; most notably, a continuum backstepping method. These approaches yield ISS-based methods for the design of robust and adaptive controllers for linear and nonlinear distributed parameter systems. To obtain powerful methods for control, however, major steps in theunderstanding and development of these methods are still required.In this project we are going to build a firm basis for the investigation of input-to-state stability and stabilization of distributed parameter systems. More specifically, our aims are:1. To develop an ISS theory for linear and bilinear distributed parameter systems, including criteria for input-to-state stability and stabilizability of linear and bilinear DPS and sufficient conditions for robustness of ISS.2. To obtain the infinite-dimensional counterparts of fundamental nonlinear results from ISS theory of finite-dimensional systems. In particular, Lyapunov characterizations of the ISS property, small-gain theorems for DPS and characterizations of ISS in terms of other stability properties.3. To develop methods for robust stabilization of infinite-dimensional systems, namely, a robust version of continuum backstepping, finite-time robust stabilization of partial differential equations and design of ISS stabilizers for port-Hamiltonian systems.In order to obtain these aims expertise is required in ISS theory, functional analysis, semigroup theory, infinite-dimensional systemstheory, partial differential equations, backstepping design and Lyapunov theory. Therefore we will work on this project as a team,consisting of three groups with complementary knowledge covering all of the above topics. It is the aim of the project to establish a solid basis for the long term development of ISS theory as a fundamental tool for a wide range of nonlinear infinite-dimensional systems.

• DFG Programme: Research Grants