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Mironchenko, Andrii ; Prieur, Christophe:
Input-to-state stability of infinite-dimensional systems : Recent results and open questions.
In: SIAM Review.
Bd. 62
(2020)
Heft 3
.
- S. 529-614.
ISSN 1095-7200
DOI: https://doi.org/10.1137/19M1291248
| Rez.: |
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Abstract
In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows for the estimation of the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this paper, starting from classic results for nonlinear ordinary differential equations, we motivate the study of the ISS property for distributed parameter systems. Then fundamental properties are given, such an ISS superposition theorem and characterizations of (global and local) ISS in terms of Lyapunov functions. We explain in detail the functional-analytic approach to ISS theory of linear systems with unbounded input operators, with special attention devoted to ISS theory of boundary control systems. The Lyapunov method is shown to be very useful for both linear and nonlinear models, including parabolic and hyperbolic partial differential equations. Next, we show the efficiency of the ISS framework in studying the stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows for the reduction of the stability analysis of complex networks, by considering the stability properties of its components and the interconnection structure between the subsystems. An extra section is devoted to ISS theory of time-delay systems with the emphasis on techniques that are particularly suited for this class of systems. Finally, numerous applications are considered for which ISS properties play a crucial role in their study. The survey contains recent as well as classical results on systems theory and suggests many open problems.
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| Publikationsform: | Artikel in einer Zeitschrift |
|---|---|
| Begutachteter Beitrag: | Ja |
| Zusätzliche Informationen: | Contents:
1 Introduction 1.1 Input-to-State Stability 1.2 Outline of the Paper 1.3 ISS of Finite-Dimensional Systems 1.4 Main Definitions 2 Fundamental Properties of ISS Systems 2.1 ISS Superposition Theorem 2.2 ISS Lyapunov Functions 2.3 Characterization of Local ISS 2.4 Converse Lyapunov Theorems 2.5 Integral Input-to-State Stability 3 ISS of Linear Systems 3.1 ISS of Linear Systems with Bounded Input Operators 3.2 ISS of Linear Systems with Unbounded Input Operators 3.3 Integral ISS of Linear Systems with Unbounded Input Operators 3.4 Lyapunov Functions for Linear Systems 3.5 Diagonal Systems 3.6 Bilinear Systems 4 Boundary Control Systems 4.1 Boundary Control Systems as Systems with Admissible Operators 4.2 Spectral-Based Methods and Related Techniques 4.3 Applications to Riesz-Spectral Systems 4.4 Remark on Nonlinear Boundary Control Systems 5 Lyapunov Methods for ISS Analysis of PDE Systems 5.1 ISS Lyapunov Methods for Parabolic Systems 5.2 Lyapunov Methods for Semilinear Parabolic Systems with Boundary Inputs 5.3 ISS Lyapunov Methods for Stabilization of Stationary Hyperbolic Systems 5.4 ISS Lyapunov Methods for Time-Varying Hyperbolic Systems 6 Interconnected Systems 6.1 Interconnections of Control Systems 6.2 Small-Gain Theorems in a Trajectory Formulation 6.3 Small-Gain Theorems in a Lyapunov Formulation 6.4 Interconnections of Integral ISS Systems 6.5 Cascade Interconnections 6.6 Example 6.7 Interconnections of an Infinite Number of Systems 7 Input-to-State Stability of Time-Delay Systems 7.1 Retarded Differential Equations 7.2 ISS Lyapunov Theory for Time-Delay Systems 7.3 Small-Gain Theorems: Trajectory Formulation 7.4 Small-Gain Theorems: Lyapunov Formulation 8 Applications 9 Further Topics 9.1 Strong and Weak Input-to-State Stability 9.2 Input-to-StatePracticalStability 9.3 Lur’e Systems and the Circle Criterion 9.4 Numerical Computation of ISS Lyapunov Functions 9.5 ISS for Monotone Parabolic Systems 9.6 ISS of Infinite-Dimensional Impulsive Systems 10 Open Problems 10.1 Infinite-Dimensional Integral-ISS Theory 10.2 Input-to-Output Stability Theory 10.3 ISS of Fully Nonlinear PDEs 10.4 Robust Control Design 11 Conclusion and Discussion 12 Appendix 12.1 Inequalities 12.2 Glossary References |
| Fachklassifikationen: | Mathematics Subject Classification Code: 93D25 93-02 (34H05 35Q93 37B25 37L15 93A15 93B35 93B52 93C10 93C15 93C20 93C25 93C43 93D05 93D09 93D30)
arXiv Subjects: math.OC (math.AP, math.DS) |
| Institutionen der Universität: | Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik V (Angewandte Mathematik) Profilfelder > Advanced Fields > Nichtlineare Dynamik |
| Titel an der UBT entstanden: | Nein |
| Themengebiete aus DDC: | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Eingestellt am: | 17 Mär 2025 09:26 |
| Letzte Änderung: | 17 Mär 2025 09:26 |
| URI: | https://eref.uni-bayreuth.de/id/eprint/92855 |