## UNI-PORTAL FÜR

# Input-to-State Stability and Stabilization of Distributed Parameter Systems

**Period: **10/2015-10/2018**Project Title:** Input-to-State Stability and Stabilization of Distributed Parameter Systems**Funding:** German Research Foundation (DFG)

## General Introduction

When looking closely enough, almost every real-world system turns out to be infinite-dimensional. The flow of the water in the oceans and of blood in our cells is described by the equations of hydrodynamics. Deformation of bodies is described by equations of elasticity theory. Chemical reactions and models of population dynamics can be successfully modeled by reaction-diffusion equations. All these and many other systems are infinite-dimensional and governed by partial differential equations. A central position in analysis of such systems is occupied by the question of stability and robustness: engineers want to build reliable nuclear and chemical reactors; in the aerospace industry aircraft are desired, which can sustain a pressure arising during the motion; and the biologists eager to know whether a distribution of the species in a given area is stable.

In this project we develop efficient tools for study of stability of infinite-dimensional systems, with a bias on systems of partial differential equations. In order to describe stability properties of dynamical systems we exploit the celebrated input-to-state stability (ISS) theory, which unifies internal and input-output stability paradigms. The methods which we develop, as Lyapunov theory, small-gain theorems and various characterizations of ISS property help the researchers to prove robust stability of infinite-dimensional systems, to design robust controllers, which stabilize unstable systems and to study stability of large-scale systems, consisting of many subsystems with their specific dynamics.

## Introduction for control theorists

The notion of input-to-state stability (ISS) has been introduced by E. Sontag in 1989. It combines two different types of stability behavior: stability in the sense of Lyapunov and input-output stability. The unified treatment of external and internal stability has made ISS a central tool in robust stability analysis of nonlinear control systems. ISS plays an important role in constructive nonlinear control; in particular, in robust stabilization of nonlinear systems, stabilization via controllers with saturation, design of robust nonlinear observers, nonlinear detectability, ISS feedback redesign, stability of nonlinear networked control systems, supervisory adaptive control and others.

In recent years the interest in this theory is rapidly growing because of modern attempts to control processes described by PDEs. On the other hand, during the last decade several effective stabilization methods for infinite-dimensional systems have been proposed, as continuum backstepping, stabilizer design for port-Hamiltonian systems etc. This paves the way for the development of robust stabilization methods for infinite-dimensional systems, provided that another prerequisite is available: the theory of ISS for distributed parameter systems. Thus a strong theoretical background for ISS theory of distributed parameter systems is needed from the viewpoint of ISS theory itself, as well as in view of applications. It is the aim of our project to develop such a comprehensive theory beginning with its foundations and to apply it to the robust stabilization of PDEs.

## Project description

It is the aim of this project to develop a firm basis for input-to-state stability (ISS) theory of distributed parameter systems and to introduce systematic methods for ISS stabilization of important classes of PDEs. More specifically, we focus on three groups of questions:

- Development of an ISS theory for linear and bilinear distributed parameter systems with unbounded input operators:
- Relations between ISS and integral ISS for linear systems with admissible input operators
- Integral ISS of bilinear systems
- Characterizations of ISS for linear systems generated by sectorial operators
- Generalization of fundamental nonlinear results from ISS theory of ordinary differential equations to general infinite-dimensional systems:
- Lyapunov characterizations of the ISS property
- Characterizations of ISS in terms of other stability properties
- Criteria for local and practical ISS
- Small-gain theorems for interconnections of an infinite number of ISS systems.
- Development of methods for analysis of ISS and for ISS stabilization of important classes of partial differential equations. In particular, we are going:
- to develop methods to check ISS of nonlinear parabolic systems with boundary inputs
- to prove that PDE backstepping method is robust with respect to actuator errors
- to develop methods of finite-time stabilization of PDEs
- to design ISS stabilizers for port-Hamiltonian systems.

## Publications

### Articles

- Nabiullin, R. and Schwenninger, F..
**Strong input-to-state stability for infinite dimensional linear systems.***Submitted, 2017.*pdf - Jacob, B.; Schwenninger, F. and Zwart, H.
**L_infty-admissibility for analytic semigroups and applications to input-to-state stability.***Submitted, 2017.*pdf - Mironchenko, A. and Wirth, F.
**Lyapunov characterization of input-to-state stability fo semilinear control systems over Banach spaces.***Submitted to Systems & Control Letters, 2017.*pdf - Mironchenko, A.; Karafyllis, I. and Krstic, M.
**Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances.***Submitted to SIAM Journal on Control and Optimization, 2017***.**pdf - Mironchenko, A. and Wirth, F.
**Characterizations of input-to-state stability for infinite-dimensional systems.***Accepted to IEEE Transactions on Automatic Control, 2017.* - Mironchenko, A.
**Uniform weak attractivity and criteria for practical uniform asymptotic stability.***Systems & Control Letters, 105: 92-99, 2017.*pdf - Mironchenko, A.
**Criteria for input-to-state practical stability.***Submitted to IEEE Transactions on Automatic Control, 2017.*pdf - Jacob, B.; Nabiullin, R.; Partington, J.R. and Schwenninger, Felix L.
**Infinite-dimensional input-to-state stability and Orlicz spaces.***Submitted, 2016.*arXiv. - Mironchenko, A. and Ito, H.
**Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions.***Mathematical Control and Related Fields, 6 (3): 447-466, 2016.*pdf - Mironchenko, A. and Wirth, F.
**Non-coercive Lyapunov functions for infinite-dimensional systems.***Submitted, 2016.*pdf - Mironchenko, A.
**Local input-to-state stability: Characterizations and counterexamples.***Systems & Control Letters, 87: 23-28, 2016.*pdf

### In Proceedings

- Dashkovskiy, S. and Pavlichkov, S.
**Decentralized Stabilization of Inﬁnite Networks of Systems with Nonlinear Dynamics and Uncontrollable Linearization.***Proc. of the IFAC 20th World Congress, pages 1728--1734, 2017.* - Dashkovskiy, S.; Hristova, S.; Kichmarenko, O. and Sapozhnikova, K.
**Behavior of solutions to systems with maximum.***Proc. of the IFAC 20th World Congress, pages 13467--13472, 2017* - Mironchenko, A. and Wirth, F.
**A non-coercive Lyapunov framework for stability of distributed parameter systems.***Accepted to the 56th IEEE Conference on Decision and Control, 2017.*pdf - Mironchenko, A. and Wirth, F.
**Input-to-state stability of time-delay systems: criteria and open problems.***Accepted to the 56th IEEE Conference on Decision and Control, 2017.*pdf - Jacob, B.; Nabiullin, R.; Partington, J.R. and Schwenninger, Felix L.
**On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems.***Proc. of the 55th IEEE Conference on Decision and Control, pages 2265-2269, 2016.* - Mironchenko, A. and Wirth, F.
**Restatements of input-to-state stability in infinite dimensions: what goes wrong?***Proc. of the 22th International Symposium on Mathematical Theory of Systems and Networks, pages 667-674, 2016.*pdf - Mironchenko, A. and Wirth, F.
**Global converse Lyapunov theorems for infinite-dimensional systems.***Proc. of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), pages 909-914, 2016.*pdf