# 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

- 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.

- 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**

- Jacob, B., Nabiulling, R., Partington, J.R., Schwenninger, F.
**Infinite-dimensional input-to-state stability and Orlicz spaces.***SIAM Journal on Control and Optimization, 56(2):868-889, 2018,***pdf** - Mironchenko, A.
**Criteria for input-to-state practical stability.***IEEE Transactions on Automatic Control, accepted, DOI:10.1109/TAC.2018.2824983, 2018,***pdf** - Mironchenko, A., Karafyllis, I. and Krstic, M.
**Monotonicity Methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances.***SIAM Journal on Control and Optimization, submitted, 2018,***pdf** - Mironchenko, A. and Wirth, F.
**Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability.***Mathematics of Control, Signals and Systems, submitted, 2018,***pdf** - Mironchenko, A. and Wirth, F.
**Non-coercive Lyapunov functions for infinite-dimensional systems.***Journal of Differential Equations, submitted, 2018,***pdf** - Mironchenko, A. and Wirth, F.
**Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces.***Systems & Control Letters, 119:64-70, 2018*,**pdf** - Mironchenko, A. and Wirth, F.
**Characterizations of input-to-state stability for infinite-dimensional systems.***IEEE Transactions on Automatic Control, 63(6):1602-1617, 2018,*pdf - Nabiullin, R., Schwenninger, F.
**Strong input-to-state stability for inifinite dimensional linear systems.**

Mathematics of Control, Signals and Systems, 30(4), 2018,**pdf** - Schmid, J.
**Stabilization of port-Hamiltonian systems with discontinuous energy densities.***Submitted, 2018,***pdf** - Schmid, J.
**Weak input-to-state stability: characterizations and counterexamples.***Submitted, 2018,***pdf** - Schmid, J. and Zwart, H.
**Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances.***Submitted, 2018,***pdf** - Jacob, B., Schwenninger, F., Zwart, H.
**On continuity of solutions for parabolic control systems and input-to-state stability.***submitted 2017, preprint arXiv:1709.04261,***pdf** - Mironchenko, A.
**Uniform weak attractivity and criteria for practical uniform asymptotic stability.***Systems & Control Letters, 105: 92-99, 2017.*pdf - Mironchenko, A. and Ito, H.
**Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach.***SIAM Journal on Control and Optimization, 53(6):3364-3382, 2015,***pdf**

**In Proceedings**

- B. Jacob, A. Mironchenko, J. R. Partington and F. Wirth.
**Remarks on input-to-state stability and noncoercive Lyapunov functions.***Accepted to the 57th IEEE Conference on Decision and Control*(CDC 2018), Miami Beach, USA, 2018. pdf - A. Mironchenko, I. Karafyllis, M. Krstic.
**Input-to-State Stability of Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances.***Proc. of the 23rd International Symposium on Mathematical Theory of Systems and Networks*(MTNS 2018), Hong Kong, pp. 38–44, 2018. pdf - A. Mironchenko, F. Wirth.
**Integral uniform global asymptotic stability and non-coercive Lyapunov functions.***Proc. of the 23rd International Symposium on Mathematical Theory of Systems and Networks*(MTNS 2018), Hong Kong, pp. 734–741, 2018. pdf - 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. - 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. - Mironchenko, A. and Wirth, F.
**Input-to-state stability of time-delay systems: criteria and open problems.***Proc. of the 56th IEEE Conference on Decision and Control (CDC 2017), Melbourne, Australia, pp. 3719-3724, 2017.*pdf - Mironchenko, A. and Wirth, F.
**A non-coercive Lyapunov framework for stability of distributed parameter systems.***Proc. of the 56th IEEE Conference on Decision and Control (CDC 2017), Melbourne, Australia, pp. 1900-1905, 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, Las Vegas, USA, pp. 2265-2269, 2016.***pdf** - 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,*(MTNS 2016), Minneapolis, Minnesota, USA, pp. 667–674, 2016*.*pdf - Mironchenko, A. and Wirth, F.
**Global converse Lyapunov theorems for ininite-dimensional systems.***Proc. of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), Monterey, California, USA, 909-914, 2016,***pdf** - A. Mironchenko, F. Wirth.
**A note on input-to-state stability of linear and bilinear infinite-dimensional systems.***Proc. of the 54th IEEE Conference on Decision and Control*(CDC 2015), Osaka, Japan, pp. 495–500, 2015