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This book lays the foundation for the study of input-to-state stability (ISS) of partial differential equations (PDEs) predominantly of two classes—parabolic and hyperbolic. This foundation consists of new PDE-specific tools.

In addition to developing ISS theorems, equipped with gain estimates with respect to external disturbances, the authors develop small-gain stability theorems for systems involving PDEs. A variety of system combinations are considered:

PDEs (of either class) with static maps;

PDEs (again, of either class) with ODEs;

PDEs of the same class (parabolic with parabolic and hyperbolic with hyperbolic); and

feedback loops of PDEs of different classes (parabolic with hyperbolic).

In addition to stability results (including ISS), the text develops existence and uniqueness theory for all systems that are considered. Many of these results answer for the first time the existence and uniqueness problems for many problems that have dominated the PDE control literature of the last two decades, including—for PDEs that include non-local terms—backstepping control designs which result in non-local boundary conditions.

Input-to-State Stability for PDEs will interest applied mathematicians and control specialists researching PDEs either as graduate students or full-time academics. It also contains a large number of applications that are at the core of many scientific disciplines and so will be of importance for researchers in physics, engineering, biology, social systems and others.

### Chapter 1. Preview

A preview of the material contained in the book is given in this chapter. The technical difficulties for the extension of Input-to-State Stability (ISS) to systems containing at least one PDE are illustrated by means of various examples. The chapter also offers an overview of the topics covered in the book as well as a brief presentation of the contents of all subsequent chapters. A list of all applications contained in the book is provided. The applications include mathematical models arising in various scientific disciplines. Finally, the required background for a reader of the book is detailed.
Iasson Karafyllis, Miroslav Krstic

### Chapter 2. Existence/Uniqueness Results for Hyperbolic PDEs

The chapter provides a variety of existence and uniqueness results for a single 1 − D, first-order, hyperbolic PDEs, which is in feedback interconnection with a system of ODEs. The results are developed for various cases, some of which are not frequently encountered in the literature: Nonlinear and non-local terms as well as distributed and boundary inputs are allowed. The existence/uniqueness results are employed in two applications with non-local terms: A chemical reactor in which an exothermic chemical reaction is taking place and the study of the age distribution of the population of a microorganism.
Iasson Karafyllis, Miroslav Krstic

### Chapter 3. ISS in Spatial L p Norms for Hyperbolic PDEs

The chapter deals with the derivation of ISS estimates expressed in spatial $$L^{p}$$ norms for 1-D, first-order, hyperbolic PDEs with a constant transport velocity. Two different methodologies for deriving ISS estimates are provided. The first methodology is the use of ISS-Lyapunov Functionals (ISS-LFs). The second methodology utilizes the transformation to a system of Integral Delay Equations (IDEs). The latter methodology provides ISS estimates expressed only in the sup-norm of the state and the derivation of the ISS estimate is performed by using a Lyapunov-like function (not a functional). Finally, the differences between the two methodologies are explained in detail.
Iasson Karafyllis, Miroslav Krstic

### Chapter 4. Existence/Uniqueness Results for Parabolic PDEs

Existence and uniqueness results for 1-D, parabolic PDEs which are in feedback interconnection with a system of ODEs are provided in the chapter. The obtained results allow the presence of nonlinear and non-local terms and guarantee the existence of classical solutions. The existence/uniqueness results are utilized in two applications with non-local terms: a chemical reactor in which an exothermic chemical reaction is taking place and a water tank. Finally, the case where boundary inputs are present is also studied.
Iasson Karafyllis, Miroslav Krstic

### Chapter 5. ISS in Spatial L2 and H1 Norms

The chapter provides ISS estimates of the solutions of 1-D, parabolic PDEs with respect to boundary and distributed disturbances, which are expressed in the spatial $$L^{2}$$ and $$H^{1}$$ norms of the state. Two different methodologies for the derivation of ISS estimates are developed. The first methodology involves an eigenfunction expansion of the solution. The second methodology involves the use of an ISS Lyapunov functional and deals with special cases for the boundary conditions where boundary disturbances appear only in boundary conditions of Robin or Neumann type. The derivation of ISS estimates is illustrated in two different applications: the study of the sensitivity of the temperature distribution in a solid bar subject to variations of the air temperature and the study of the sensitivity of R–A–D PDEs subject to inlet disturbances.
Iasson Karafyllis, Miroslav Krstic

### Chapter 6. ISS in Spatial L p Norms for Parabolic PDEs

The chapter provides ISS estimates of the solutions of 1-D, parabolic PDEs with respect to boundary and distributed disturbances, which are expressed in weighted spatial $$L^{p}$$ norms of the state with $$1 \le p \le + \infty$$. A novel methodology for the derivation of the ISS estimates is presented: the use of an ISS-Lyapunov Functional Under Discretization (ISS-LFUD). The notion of the ISS-LFUD and its relation to discretized, numerical schemes for the given parabolic PDE is explained in detail. The obtained ISS estimates are employed to show ISS for Taylor–Couette flow and to show that recently proposed backstepping boundary feedback controllers for parabolic PDEs guarantee robustness with respect to control actuator errors.
Iasson Karafyllis, Miroslav Krstic

### Chapter 7. Fading Memory Input-to-State Stability

The chapter is devoted to the presentation of two basic lemmas that allow the conversion of an ISS-like inequality to an inequality with fading memory. The first lemma deals with an ISS-like inequality in the standard form (involving summation), while the second lemma deals with an ISS-like inequality in the max-formulation. Both lemmas play crucial roles in small-gain analysis.
Iasson Karafyllis, Miroslav Krstic

### Chapter 8. PDE-ODE Loops

In this chapter, we present the small-gain methodology for interconnections of PDEs with ODEs. In general, the methodology consists of the following steps:
Iasson Karafyllis, Miroslav Krstic

### Chapter 9. Hyperbolic PDE-PDE Loops

The chapter is devoted to the development of the small-gain methodology for coupled 1-D, hyperbolic, first-order PDEs under the presence of external inputs. Our aim is the derivation of sufficient conditions that guarantee ISS for a given system of coupled hyperbolic PDEs. Globally, Lipschitz nonlinear, non-local terms are allowed to be present both in the PDEs and the boundary conditions. The results are expressed in the spatial sup-norm. The chapter also includes the development of existence/uniqueness results for hyperbolic PDE-PDE loops as well as a detailed comparison with existing results in the literature.
Iasson Karafyllis, Miroslav Krstic

### Chapter 10. Parabolic PDE-PDE Loops

This chapter of the book is devoted to the study of parabolic–parabolic PDE loops by means of the small-gain methodology. The results contained in the present chapter allow the existence of non-local reaction terms (both distributed terms and boundary terms) as well as distributed and boundary inputs. The main focus is on two particular cases, which are analyzed in detail: the case of in-domain interconnections and the case of boundary interconnections. Sufficient small-gain conditions for ISS in the spatial $$L^{2}$$ norm and the spatial $$L^{\infty }$$ norm are provided. The chapter also includes the development of existence/uniqueness for initial-boundary value problems of systems of parabolic PDEs with non-local terms.
Iasson Karafyllis, Miroslav Krstic

### Chapter 11. Parabolic–Hyperbolic PDE Loops

The last chapter of the book is devoted to the study of parabolic–hyperbolic PDE loops by means of the small-gain methodology. Since there are many possible interconnections that can be considered, we focus on two particular cases, which are analyzed in detail. The first case considered in the chapter is the feedback interconnection of a parabolic PDE with a special first-order hyperbolic PDE: a zero-speed hyperbolic PDE. The study of this particular loop is of special interest because it arises in an important application: the movement of chemicals underground. Moreover, the study of this loop can be used for the analysis of wave equations with Kelvin–Voigt damping. The second case considered in the chapter is the feedback interconnection of a parabolic PDE with a first-order hyperbolic PDE by means of a combination of boundary and in-domain terms. The interconnection is effected by linear, non-local terms. For both cases, results for existence/uniqueness of solutions as well as sufficient conditions for ISS or exponential stability in the spatial sup-norm are provided.
Iasson Karafyllis, Miroslav Krstic