Reply by: Irena Lasiecka.
IEEE Control Systems Magazine 39 (6) (2019), 25-27.

My first monograph lecture notes (jointly with R Triggiani) published in 1991 by Springer, provides the first overview of the literature on optimal control theory as applied to classical and physically relevant single partial differential equation models with boundary/point control and/or boundary observation. It axiomatizes the properties and criteria that need to be satisfied for a successful application of optimal-control theory. The emphasis is on feedback control and the associated Riccati equations. Since the emphasis is on boundary point control, the study leads to Riccati equations with unbounded coefficients, a notable property leading to a plethora of new phenomena. A substantial expansion and continuation of the book came with the two volumes (also jointly with R Triggiani and published in 2000 by Cambridge University Press) Encyclopedia of Mathematics and Its Applications, for a total of 1100 pages. Particular emphasis is placed on concrete examples illustrating the need for the infinite-dimensional/partial differential equation analysis, where basic mathematical questions in partial differential equations had to be first resolved.

Another book is my solo National Science Foundation/Society for Industrial and Applied Mathematics (SIAM)/Conference Board of Mathematical Sciences lecture notes, published by SIAM 2000, a work that, for the first time, is devoted to the mathematical control theory of systems of coupled partial differential equations evolving in contiguous domains and coupled at a common interface. The propagation of properties, for example, controllability and stability through the interface, is the main theme, particularly when the two partial differential equations are of different types, one parabolic and one hyperbolic, say. My 700-page Springer-Verlag monograph, jointly with Igor Chueshov, was published in 2010 and provides a dynamical system approach to nonlinear control theory. The asymptotic reduction of the physically relevant dynamics to well-structured finite-dimensional states is the goal. Particular applications include controlling flutter and vibrations in elastic solids.

1.1. From the Publisher.

This book provides, in a unified framework, an updated and rather comprehensive treatment centred on the theory of optimal control with quadratic cost functional for abstract linear systems with application to boundary/point control problems for partial differential equations (distributed parameter systems). The book culminates with the analysis of differential and algebraic Riccati equations which arise in the pointwise feedback synthesis of the optimal pair. It incorporates the critical topics of optimal irregularity of solutions to mixed problems for partial differential equations, exact controllability, and uniform feedback stabilisation. It covers the main results of the theory - which has reached a considerable degree of maturity over the last few years - as well as the authors' basic philosophy behind it. Moreover, it provides numerous illustrative examples of boundary/point control problems for partial differential equations, where the abstract theory applies. However, in line with the purpose of the manuscript, many technical proofs are referred to in the literature. Thus, the manuscript should prove useful not only to mathematicians and theoretical scientists with expertise in partial differential equations, operator theory, numerical analysis, control theory, etc., but also to those who simple wish to orient themselves with the scope and status of the theory presently available. Both continuous theory and numerical approximation theory thereof are included.

1.2. Review by: Nasir Uddi Ahmed.
Mathematical Reviews MR1132440 (92k:93009).

The authors develop linear-quadratic regular regulator theory for infinite-dimensional systems with unbounded control and observation operators. The results can be applied to a large class of systems governed by linear partial differential equations with boundary and point controls.

The authors also discuss some of their recent results on the approximation theory of the corresponding differential and algebraic Riccati equations. The results are illustrated with application to linear wave equations, beam equations and plate equations. Open problems are also mentioned in the process.

The book should be of great value to graduate students working on control theory and looking for research problems. It may also be useful to working scientists.

2.1. From the Publisher of the 2006 reprint.

Originally published in 2000, this is the first volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which is unifying across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 1 includes the abstract parabolic theory for the finite and infinite cases and corresponding PDE illustrations as well as various abstract hyperbolic settings in the finite case. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

2.2. From the Preface.

This is the first volume of a comprehensive and up-to-date three-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon and related differential (integral) anti algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. An abstract space, operator theoretic treatment is provided, which is based on semigroup methods, and which is unifying across a few basic classes of evolution. A key feature of this treatise is the wealth of concrete multidimensional PDE illustrations, which fit naturally into the abstract theory, with no artificial assumptions imposed, at both the continuous and numerical level.

Throughout these volumes. emphasis is placed on unbounded control operators or on unbounded observation operators as they arise in the context of various abstract frameworks that are motivated by partial differential equations with boundary/point control. Relevant classes of PDEs include: parabolic or parabolic-like equations, hyperbolic and Petrowski-type equations (such as plate equations and the Schrödinger equation), and hybrid systems of coupled PDEs of the type that arise in modern thermo-elastic and smart material applications. Purely PDE dynamical properties are critical in motivating the various abstract settings and in applying the corresponding theories to concrete PDEs arising in mathematical physics and in other recent technological applications.

Volume I covers the abstract parabolic theory, including both the finite and infinite horizon optimal control problems, as well as the corresponding min-max theory with non-definite quadratic cost. A lengthy chapter presents many multidimensional PDE illustrations with boundary/point control and observation. These include not only the traditional parabolic equations. such us the heat equation, but also second-order equations with structural ("high") damping, as well as thermo-elastic plate equations. Recently discovered, critical dynamical properties are provided in detail. Many of these new results are appearing here in print for the first time.

Volume II is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems including second-order hyperbolic equations with Dirichlet boundaries, plate equations and the Schrödinger equation under a variety of boundary controls. and structural acoustic models coupling two hyperbolic equations.

Volume III is in preparation.

Irena Lasiecka is Professor of Mathematics at the University of Virginia, Charlottesville. She has held positions at the Control Theory Institute of the Polish Academy of Sciences, the University of California, Los Angeles. and the University of Florida, Gainesville. She has authored or co-authored over 150 research papers and one other book in the area of POEs and their control theoretic properties. She serves on the editorial boards of Applied Mathematics and Optimisation, Journal of Mathematical Analysis and Applications, and the IEEE Transactions on Automatic Control, among others, and she holds, or has held, numerous offices in the professional societies SIAM, IFIP, and the AMS.

Roberto Triggiani is Professor of Mathematics at the University of Virginia at Charlottesville. He has held regular and visiting academic positions at several institutions in the United States and Europe. including Ohio State University, Ames, and the University of Florida, Gainesville. He has authored or co-authored over 140 research papers and one other book in PDEs and their control theoretic properties. He currently serves on the editorial boards of Applied Mathematics and Optimisation, Abstract and Applied Analysis, Systems and Control Letters.

2.3. Review by: Luciano Pandolfi.
Mathematical Reviews MR1745475 (2001m:93002).

The book under review is the first volume of a treatise whose aim is clearly stated by the first sentence of the introduction: "This three-volume treatise presents, in a unified framework, an up-to-date treatment of the quadratic optimal control theory for (linear) partial differential equations over a finite or infinite time horizon and related differential (integral) and algebraic Riccati equations." In this treatise the authors present most of their results on the quadratic regulator problem and on the Riccati equation for PDE. The results are obtained using the variational approach. This means that optimal controls are studied first and the Riccati operator is subsequently defined. After that it is proved that the Riccati operator solves the Riccati equation in a suitable sense. The converse method, known as the "direct" approach, which proves directly the existence of solutions of the Riccati equation, is discussed, too.

The authors follow an "abstract" presentation in the sense that two (not disjoint) classes of systems are defined in terms of the properties of the operators which describe (abstractly) the differential equation of the system; and the theory is deduced from such "abstract" properties. However, these abstract properties capture the essentials of many PDE models without artificial assumptions, as shown by a wealth of examples.
...
This book documents the impressive results obtained by the authors both on the theory and the applications of the regulator problem for parabolic PDE. It is a unique book which will become a standard reference. It should be found in every mathematical library.

3.1. From the Publisher of the 2011 reprint.

Originally published in 2000, this is the second volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

3.2. From the Preface.

This is the first volume of a comprehensive and up-to-date three-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon and related differential (integral) anti algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. An abstract space, operator theoretic treatment is provided, which is based on semigroup methods, and which is unifying across a few basic classes of evolution. A key feature of this treatise is the wealth of concrete multidimensional PDE illustrations, which fit naturally into the abstract theory, with no artificial assumptions imposed, at both the continuous and numerical level.

[Note. Preface continues identical to the Preface of Volume 1]

3.3. Review by: Luciano Pandolfi.
Mathematical Reviews MR1745476 (2001m:93003).

The volume under review is the second of a three-volume monograph (the third volume is still to be published). See the review of Volume 1 for general comments [Control theory for partial differential equations: continuous and approximation theories. I (2000)]. We repeat only that while Volume 1 is devoted to "parabolic" type systems, Volumes 2 and 3 deal with the "hyperbolic" case, and that the authors present the theories of the quadratic regulator problem (with cost functional coercive in the control variable) and of the Riccati differential equation for the classes of systems under study.

Volume 2 presents the theory of the finite horizon quadratic regulator problem, while the infinite horizon case and the numerical analysis of the regulator problem will be considered in Volume 3.
...
As in the case of Volume 1, the "abstract" theory emphasised in this monograph is in fact supported by a wealth of examples (which fill almost one half of the book). In particular, Chapter 9 presents several examples of systems with point control action (wave and Kirchhoff equations and two examples of coupled PDE, one acting on the boundary of the region where the second is defined, suggested by recent applications to noise reduction and smart materials).

The examples in Chapter 10 are examples of systems with boundary controls and include wave, Kirchhoff, Euler-Bernoulli and Schrödinger equations.

The treatment of the quadratic regulator problem presented in Volume 2 consistently follows the variational approach, but also the "direct" approach is presented when appropriate.

This book documents the impressive results obtained by the authors and their co-workers both on the theory and the applications of the regulator problem for hyperbolic PDE. It is a unique book which will become a standard reference and should be found in every mathematical library.

4.1. From the Abstract.

Mathematical control theory for a single partial differential equation (PDE) has dominated the research literature for quite a while, new, complex, and challenging issues have recently arisen in the context of coupled, or interconnected, PDE systems. This has led to a rapidly growing interest, and many unanswered questions, within the PDE community. By concentrating on systems of hyperbolic and parabolic coupled PDEs that are nonlinear, Mathematical Control Theory of Coupled PDEs seeks to provide a mathematical theory for the solution of three main problems: well-posedness and regularity of the controlled dynamics; stabilisation and stability; and optimal control for both finite and infinite horizon problems along with existence/uniqueness issues of the associated Riccati equations. These lectures are an outgrowth of recent research in the area of control theory for systems governed by coupled partial differential equations (PDEs). While mathematical control theory for a single PDE has long occupied a central place within a broader discipline often referred to as control of infinite-dimensional systems, more complex and challenging issues have arisen recently in the context of coupled or interconnected PDE systems. This is in response to modern technological demands where the interaction (coupling) between various components of the model is a central mechanism in controlling the system. On the other hand, mathematical tools developed in the context of single equations more often than not are inadequate for dealing with and accounting for the more complex interactive nature of the structures. This has propelled a rapidly growing interest within the PDE community, which has posed an array of new questions and problems, including, How can we take advantage of the coupling within the model in order to improve system's performance? This question is clearly tightly connected to the issue of (controlled) propagation of certain properties from one component of a system into another, more generally, creating of new phenomena via coupling. The purpose of these lectures is to describe several classes of coupled PDE models that display the above properties and to present mathematical tools and a cogent framework by which to successfully analyse the associated control problems. We concentrate on systems of coupled PDEs of hyperbolic and parabolic type, which, moreover, are nonlinear and may include thermal effects. (The latter further contribute to the mixing of various dynamical components of the overall dynamics.) The controls are rough and as such naturally arise in the context of smart material technology. Particular emphasis is paid to boundary and point controls whose mathematical treatment is more involved and leads to the theory of semigroups with unbounded inputs.

4.2. From the Preface.

Control theory of dynamical systems. What is control theory of a dynamical system? Traditionally, the classical viewpoint taken in the study of differential equations consisted in the (passive) analysis of the evolution properties displayed by a specific equation, or a class of equations, in response to given data.

By contrast, control theory injects an active mode of synthesis in the study of differential equations: it seeks to influence their dynamical evolution by selecting and synthesising suitable data (input functions, or control functions) from within a preassigned class, as to achieve a predetermined desired outcome or performance. Within the deterministic (as opposed to stochastic) setting, canonical examples of sought-after goals include:

(i) The optimal control problem: this consists in the minimisation, over a preassigned time horizon, of a preassigned performance cost functional, which penalises both the control function as well as its corresponding solution, within the preassigned class of admissible controls. Such examples include: reaching a target in minimal time or with minimal 'fuel' (energy) of both control and solution.

(ii) The problem of controllability: this consists, depending upon the dynamical properties of the differential equation at hand, in selecting a suitable control function, perhaps of minimal norm within a preassigned class, which steers a preassigned initial state exactly to (or to an arbitrarily small neighbourhood of ) a preassigned target state over a preassigned time interval, preferably 'the smallest' one. Thus stated, this is an example of 'open loop' control problem.

(iii) Stabilisation problem: This consists in forcing an original free (i.e., with no control) non-stable system to become asymptotically stable as time goes to infinity, by selecting, introducing and synthesising a suitable damping or dissipative term. The latter may be viewed as 'a control in feedback form,' and thus the resulting problem is an example of 'closed loop' control.

The aforementioned examples are just a few canonical illustrations of the target-oriented philosophy of synthesis, which is intrinsic to control theory. Moreover, the three sample control problems just mentioned are typically inter-dependent: for instance, the controllability or stabilisation problem are generally a necessary prerequisite for the study of some optimal control problems over an infinite time horizon. Since the outgrowth, in the late fifties, of control theory as a discipline in itself, within the fields of differential equations and optimisation theory, the above problems have been recognised as being at the heart of control theory. They, in turn, produce a wealth of ramifications and variations.

Finite versus infinite-dimensional control theory. Depending on the type of differential (or discrete) equations, the synthesis or implementation via 'actuators' of the sought-after control function - and, more generally, the resulting control theory - takes on diverse, dynamics-dependent features. Models described by ordinary differential equations (ODE) are usually referred to as being part of 'finite-dimensional' control theory, as the state of such a model is finite-dimensional, and so is the control function. By contrast, model described by (linear and non-linear) partial differential equations may be viewed as being (the most significant) part of 'infinite-dimensional' control theory.

Naturally, the specific and precise formulation of a control problem, such as the three samples above, need to be tuned to the particular class of dynamical systems at hand. These will then determine the techniques to be used and the nature of the results to be obtained. Within infinite-dimensional control theory, say in the linear case, it is possible to invoke the unifying formulation provided by semigroup theory. This, however, works best in the case of 'nice' classes of controls such as distributed controls, whose action upon the abstract differential equation is exercised by virtue of a bounded control operator. In this case, regularity theory provided by semigroup theory is adequate to formulate and solve, with the help of operator theory, a number of control-theoretic questions, or to ascertain their limitations in the context of infinite-dimensional spaces, where new pathological phenomena arise, of which there is no counterpart in the finite-dimensional case. All this, by and large, became available by the seventies.

Boundary/point control problems for single PDE's. By contrast, the situation is drastically different when the control action is "rough," such as it occurs when the control acts either as a 'boundary control' on the boundary (or part thereof) of the spatial domain; or else as a point control through a Dirac measure (or its derivative) supported at, say, an interior point of the spatial domain. In these latter cases - which are mathematically much more challenging and physically much more appealing and relevant - the control operator has its range in a space definitively larger than the state space, and with weaker topology.

Naturally, in order to extract best possible results and tune the technical tools to the problem at hand for a single PDE (as opposed to a system of PDE's), it is necessary to distinguish at the outset between different types of PDE classes: primarily, parabolic-like dynamics versus hyperbolic-like dynamics, with further sub-distinctions in the latter class. This is due to well-known, intrinsically different dynamical properties between these two classes. As a consequence, they lead to two drastically different basic abstract models, whose defining, characterising features set them apart. Accordingly, these two abstract models need, therefore, to be investigated by corresponding different technical strategies and tools. As a consequence, different types of distinctive results are achieved to characterise the two classes. All this dictates that, when dealing with a single equation, the abstract theory needs to bifurcate at the very outset into a parabolic-like model and hyperbolic-like basic models. Moreover, in the latter class, a further distinction into finite and infinite time horizon is called for when studying optimal control problems, to account for different, critical properties between these two cases.

In the case of classes of single PDE's - parabolic-like versus hyperbolic-like - such as they arise in boundary/point control, a rather comprehensive, up-to-date, abstract treatment of quadratic optimal control problems over a finite or infinite time horizon, and related Differential or Algebraic Riccati Equations, is given in [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)]. Both continuous and approximation theories are included.

These volumes present a far-reaching, technical extension of the deterministic optimal control problem, aimed at accommodating and encompassing multidimensional PDE's with boundary/point control and/or observation, in a natural way. Thus, throughout these volumes, emphasis is placed on unbounded control operators and/or, possibly, on unbounded observation operators as well, as they arise in the context of various abstract frameworks that are motivated by, and ultimately directed to, PDE with boundary/point control and observation. A key feature of the entire treatise is then a wealth throughout of concrete, multidimensional PDE's illustrations, which naturally fit into the abstract theory, with no artificial assumptions imposed, at both the continuous and numerical level.

In a definite sense, the present lecture notes are a successor work of [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)].

Boundary/point control problems for systems of coupled PDE's (possibly of different type). The present lectures deal with boundary/point control problems for systems of highly coupled PDE's, possibly of different type, such as they arise in modern technological applications. A motivating driving force is the so-called structural acoustic problem, which, accordingly, receives much attention in these lectures.

In one canonical setting, the system to be studied couples a hyperbolic (wave) equation [within the acoustic chamber] with an elastic parabolic-like equation (because of the so-called 'structural damping') defined on a flat elastic wall of the chamber, with coupling taking place at the boundary. Thus, the overall system is more 'complex' than just a single PDE: indeed, the overall system mixes a hyperbolic dynamical component with a parabolic dynamical component. The control action is 'rough.' Then, in regard of the quadratic optimal control problem, it is clear that the results on single PDE equations in [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)] cannot be applied directly. While the single PDE treatment [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)] serves as a foundational basis for both results to be expected, and techniques to be employed, some critical features of the problem are unpredictable, and new phenomena arise which require additional technical treatment. Would the parabolic component succeed in propagating its smoothing properties across, and onto, the whole coupled system? Would the resulting system behave globally more akin to a parabolic or to a hyperbolic structure? The theory included in these lectures will give a precise answer to these, and similar, questions.

The topic of systems of coupled PDE and their control-theoretic properties, like many classes defined by exclusion, is simply too vast to be treated in a systematic way. Here, we shall narrow the focus to classes of systems of coupled PDE's which are driven by recent technological applications. Accordingly, as to the scope, we shall restrict, mostly but not exclusively, to optimal control problems defined for coupled hyperbolic/parabolic, such as waves, plates and heat equations. For these, we shall then attempt to bring forward several approaches, techniques, methods which have proved successful in recent years in the resulting extension of the single PDE case of [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)]. As in the structural acoustic prototype model, we shall consider those situations where coupling takes place on the interface of two regions. At any rate, while we refer to established engineering literature for the modelling aspects, their range of validity and related practical aspects of implementation, our focus here is on the mathematical analysis of the problem. In particular, we seek to unveil new mathematical questions, or difficulties, inherent to the more complex coupled model, and then present suitable approaches for their solution.

The goal of these lectures. We shall concentrate on systems of coupled PDE's of hyperbolic and parabolic type, which, moreover, are non-linear and may include thermoelastic effects (the latter further contributes to the 'mixing' of various dynamical components of the overall dynamics). The controls are "rough" and such as they naturally arise in the context of smart material technology. For the above model, the ultimate goal of these lectures is to provide a mathematical theory for the solution of the following three main problems: (i) well-posedness and regularity; (ii) stabilisation and stability; and (iii) optimal control for finite or infinite horizon problems along with the associated Riccati Equations

When stabilisation is sought, emphasis is placed on boundary feedback controls (or damping terms on the boundary). As noted before, a positive solution of the stabilisation problem serves as a necessary prerequisite for the optimal control problem over an infinite time horizon. Moreover, a preliminary task for either one of the three problems consists in providing the appropriate functional analytic or operator-theoretic setting, which will include suitable unbounded operators to model actuators and boundary feedback controls.

Detailed list of topics. In order to study a control problem (which, in some sense , can be seen as being an inverse- problem), it is necessary to understand well the properties of the forward problem. This entails an analysis of the well-posedness and regularity of the underlying PDE system. Thus, Chapter 2 provides the rudiments of such an analysis as specialised to the systems of interest, which are predominantly of second order in time and with "rough" forcing terms. Particular models include: nonlinear wave equations, nonlinear plate equations (with and without thermoelasticity) and full von Karman systems, which couple elastodynamic waves with nonlinear plates. The analysis for waves and plates is a preliminary step toward the study of a well-posedness theory in structural acoustic interactions. Particular emphasis is given to the effects of nonlinear boundary conditions. These play a major role in the analysis to follow.

The study of asymptotic behaviour and stabilisability is taken up in Chapters 3 and 4. Chapter 3 starts with a presentation of some preliminary abstract stability inequalities which are used throughout the lectures. Detailed analysis of the stabilisation problem formulated for a benchmark case modelled by a semilinear wave equation is given in Section 3.2. Here, particular emphasis is given to the issue of low regularity of the solutions, the nonlinearity of boundary conditions, as well as the potential non-uniqueness of the solutions. Section 3.4 provides a collection of relevant stability results pertinent to asymptotic stability of nonlinear plate dynamics. Building upon the stability results for waves and plates, Chapter 4 deals entirely with stability issues pertinent to the nonlinear structural acoustic interactions. All three major damping mechanisms affecting flexible walls are discussed in details: structural, mechanical and thermal damping.

The remaining three Chapters of the Lectures deal with optimal control problems formulated in the context of structural acoustic models, where the control actuation is modelled by highly "unbounded" operators (as they arise in smart material technology. Here, the dynamics considered is predominantly linear. Moreover, as noted before, coupling on the interface and propagation of analyticity-hyperbolicity play major role in the arguments. In Chapter 4 we provide a detailed description of various models which includes both PDE's as well as abstract (semigroup) formulations for various structures considered.

Chapter 6 provides a complete solution to the optimal control problem defined on a finite time horizon. Here the main result is a derivation of an underlying Riccati equation, which culminates with a regularity analysis of the gain feedbacks. This, in turn, is critical for proper justification and formulation of the quadratic term in the Differential Riccati Equation. The subtlety of the issue is due to the unboundedness of the control operator, which enters quadratically into the Riccati equation. Thus, a priori, such equation may not be even properly formulated and the meaning of its solution is not clear in advance. On the other hand, in the case of dynamics generated by analytic semigroups, the theory of such Riccati equations is complete and well understood [I Lasiecka and R Triggiani, Control Theory for Partial Differential Equations (2000)]. This is due to regularising effects of the analytic dynamics which permits a proper definition of the gain operators. The main contribution of Chapters 6 and 7 is in showing that for interactive structures which arise in structural acoustic problems (which are not analytic ), we still have complete solvability of the Riccati equations. The corresponding Riccati equations do have well defined solutions with meaningful gain operators which lead to a feedback solution of the optimisation problem. Chapter 6 deals with the infinite horizon problem where issues addressed already in the finite horizon problem are compiled with an intrinsic stability/stabilisation property for the overall system. It is here where we use the stability results developed in Chapters 3 and 4. The ultimate gaol of this chapter is to provide a rigorous analysis of the existence and uniqueness of the resulting nonstandard Algebraic Riccati Equations.

As a final note we mention that no much space is devoted in these lectures to the issue of exact controllability. The main reason for this is that point controls ( as they typically arise in smart materials applications) offer very little in the context of exact controllability in the desired spaces of finite energy. In fact, most of the results here are of negative nature. An additional difficulty of the problem is the hybrid connection of the system which is well known, even for much simpler models, to be not exactly controllable in such spaces. Thus, the main difficulties with controlling exactly structural acoustic models are due to (i) hybrid nature of connections, and (ii) well known and documented limited effectiveness of point controls in the context of infinite dimensional (particularly, hyperbolic) systems.

4.3. Review by: Angelo Favini.
Mathematical Reviews MR1879543 (2003a:93002).

The present book is an outgrowth of lectures given by the author at the University of Nebraska in 1999 and contains many important results obtained by the author and her collaborators in recent years. In particular, the main results in Chapters 5-7 are based on research papers by the author and G Avalos.

The lectures deal with boundary or point control problems for systems of highly coupled nonlinear PDEs, of parabolic and hyperbolic type, arising in technological applications. In particular, much attention is directed to the structural acoustic problem, where the overall system mixes a hyperbolic dynamical component with a parabolic dynamical component and the control action is rough.

The book contains 7 chapters, the first of which is introductory. Chapter 2 furnishes an analysis of the well-posedness and regularity of the underlying PDE system. Particular models, such as nonlinear wave equations, nonlinear plate equations and full von Kármán equations, are considered.

Chapters 3 and 4 are devoted to the study of asymptotic behaviour and stabilisability. In Chapter 5 there is provided a detailed description of the structural acoustic control problems by means of the semigroup and PDE models.

Chapters 6 and 7 provide a complete solution to the optimal control problem on a finite-time horizon and on an infinite-time horizon for such models, respectively. The main contribution in these chapters is in showing that, for the not-analytic interactive structures arising in acoustic problems, the relative Riccati equations have well-defined solutions with meaningful gain operators leading to a feedback solution of the optimisation problem.

Each chapter finishes with interesting comments and a list of open problems.

5.1. From the Publisher.

This book consists of five introductory contributions by leading mathematicians on the functional analytic treatment of evolutions equations. In particular the contributions deal with Markov semigroups, maximal $L^{p}$-regularity, optimal control problems for boundary and point control systems, parabolic moving boundary problems and parabolic non-autonomous evolution equations. The book is addressed to PhD students, young researchers and mathematicians doing research in one of the above topics.

5.2. From the Preface.

Evolution equations describe time dependent processes as they occur in physics, biology, economy or other sciences. Mathematically, they appear in quite different forms, e.g., as parabolic or hyperbolic partial differential equations, as integro-differential equations, as delay or difference differential equations or more general functional differential equations. While each class of equations has its own well established theory with specific and sophisticated methods, the need for a unifying view becomes more and more urgent. To this purpose functional analytic methods have been applied in recent years with increasing success. In particular the concepts of Abstract Cauchy Problems and of Operator Semigroups on Banach spaces allow a systematic treatment of general evolution equations preparing the ground for a better theory even for special equations.

It is the aim of this volume to make this evident. Five contributions by leading experts present recent research on functional analytic aspects of evolution equations.
...
Control theoretic aspects of evolution equations in finite dimensions have been studied for a long time. Thanks to functional analytic tools there is now a well established infinite dimensional theory described in some recent monographs. However, this theory does not cover so-called boundary and point control problems. Irena Lasiecka from the University of Virginia (USA) developed (mostly with Roberto Triggiani) a systematic approach to these problems using a beautiful combination of abstract semigroups methods and sharp PDE estimates. In her contribution she explains this approach and discusses illustrating examples such as systems of coupled wave, plate and heat equations.

5.3. Review by: Angelo Favini.
Mathematical Reviews MR2108960 (2005m:49038).

Irena Lasiecka's article presents some recent results from the author and some co-authors on optimal control problems and related Riccati equations in the context of partially analytic dynamics with boundary and point control.
...
In Part I of this lecture the theory relevant to this class of singular systems is explained. Part II provides various concrete examples of control systems, illustrating applicability of the theory and motivating the abstract theory, too.

The first example is a system from thermoelasticity whose dynamics consist of a natural coupling between parabolic and hyperbolic equations. The second one involves composite materials and this model, too, provides a hyperbolic (Kirchhoff plate) / parabolic (heat equation) configuration. The third example involves structural acoustic interaction in a 3-dimensional acoustic chamber.

6.1. From the Publisher.

The steady-state solutions to Navier-Stokes equations on a bounded domain $\Omega \subset \mathbb{R}^{d}, d = 2, 3$, are locally exponentially stabilisable by a boundary closed-loop feedback controller, acting tangentially on the boundary $\partial \Omega$, in the Dirichlet boundary conditions. The greatest challenge arises from a combination between the control as acting on the boundary and the dimensionality $d=3$. If $d=3$, the non-linearity imposes and dictates the requirement that stabilisation must occur in the space $(H^{3/2+\epsilon}(\Omega ))^{3}, \epsilon > 0$, a high topological level. A first implication thereof is that, due to compatibility conditions that now come into play, for $d = 3$, the boundary feedback stabilising controller must be infinite dimensional. Moreover, it generally acts on the entire boundary ∂ Ω. Instead, for $d = 2$, where the topological level for stabilisation is $(H^{3/2-\epsilon}(\Omega ))^{2}$, the boundary feedback stabilising controller can be chosen to act on an arbitrarily small portion of the boundary. Moreover, still for $d = 2$, it may even be finite dimensional, and this occurs if the linearised operator is diagonalisable over its finite-dimensional unstable subspace. In order to inject dissipation as to force local exponential stabilisation of the steady-state solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite time-horizon is introduced for the linearised N-S equations. As a result, the same Riccati-based, optimal boundary feedback controller which is obtained in the linearised OCP is then selected and implemented also on the full N-S system. For $d = 3$, the OCP falls definitely outside the boundaries of established optimal control theory for parabolic systems with boundary controls, in that the combined index of unboundedness - between the unboundedness of the boundary control operator and the unboundedness of the penalisation or observation operator - is strictly larger than $\large\frac{3}{2}\normalsize$, as expressed in terms of fractional powers of the free-dynamics operator. In contrast, established (and rich) optimal control theory [L-T.2 ] of boundary control parabolic problems and corresponding algebraic Riccati theory requires a combined index of unboundedness strictly less than 1. An additional preliminary serious difficulty to overcome lies at the outset of the program, in establishing that the present highly non-standard OCP - with the aforementioned high level of unboundedness in control and observation operators and subject, moreover, to the additional constraint that the controllers be point-wise tangential - be non-empty; that is, it satisfies the so-called Finite Cost Condition [L-T.2].

6.2. Contents.

Chapters.

1. Introduction.

2. Main results.

3. Proof of Theorems 2.1 and 2.2 on the linearised system (2.4): $𝑑 = 3$.

4. Boundary feedback uniform stabilisation of the linearised system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $𝑑 = 3$.

5. Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati based boundary feedback control. Case $𝑑 = 3$.

6. Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control.

7. A PDE-interpretation of the abstract results in Sections 5 and 6.

6.3. Review by: Angelo Favini.
Mathematical Reviews MR2215059 (2008g:93093).

This research contains, and improves on, a number of previous results of the authors on the subject.
...
Techniques and arguments are strictly related to previous results by Barbu and Triggiani, who considered the interior stabilisation problem of the Navier-Stokes equations.

7.1. From the Publisher.

The authors consider abstract nonlinear second order evolution equations with a nonlinear damping. Questions related to long time behaviour, existence and structure of global attractors are studied. Particular emphasis is put on dynamics which - in addition to nonlinear dissipation - have noncompact semilinear terms and whose energy may not be necessarily decreasing. For such systems the authors first develop a general theory at the abstract level. They then apply the general theory to nonlinear wave and plate equations exhibiting the aforementioned characteristics and are able to provide new results pertaining to several open problems in the area of structure and properties of global attractors arising in this class of PDE dynamics.

7.2. Contents.

Chapters.

1. Introduction.

2. Abstract results on global attractors.

3. Existence of compact global attractors for evolutions of the second order in time.

4. Properties of global attractors for evolutions of the second order in time.

5. Semilinear wave equation with a nonlinear dissipation.

6. Von Karman evolutions with a nonlinear dissipation.

7. Other models from continuum mechanics.

7.3. Review by: Alain Haraux.
Mathematical Reviews MR2438025 (2009i:37200).

This monograph, divided into seven chapters, is devoted mainly to the existence and properties of compact global attractors for second-order evolution equations with nonlinear damping. The main applications which are derived concern the wave equation and several models of plate equations, including the von Kármán model. The authors consider various kinds of damping terms and nonlinear conservative forces. In the case of critical growth for the conservative terms, they need either some structural conditions on the nonlinearity or strong damping terms to insure compactness of the attractor. In addition to compactness, they also cover the study of regularity of the elements of the attractor as well as finite-dimensionality results. Some results on the rate of decay to equilibrium are also given. After four chapters on abstract equations, Chapter 5 is devoted to applications concerning nonlinear wave equations. The von Kármán model with or without rotational forces is studied in detail in Chapter 6. Finally, Chapter 7 considers a number of other problems of the fourth order with respect to space, such as Berger's plate model, Mindlin-Timoshenko beams and plates, Kirchhoff-Boussinesq plate equations and at the end some second-order problems with strong damping which enjoy specific smoothing properties similar to the regularising effect of analytic linear semigroups.

8.1. From the Publisher.

In the study of mathematical models that arise in the context of concrete applications, the following two questions are of fundamental importance: (i) well-posedness of the model, including existence and uniqueness of solutions; and (ii) qualitative properties of solutions. A positive answer to the first question, being of prime interest on purely mathematical grounds, also provides an important test of the viability of the model as a description of a given physical phenomenon. An answer or insight to the second question provides a wealth of information about the model, hence about the process it describes. Of particular interest are questions related to long-time behaviour of solutions. Such an evolution property cannot be verified empirically, thus any in a-priori information about the long-time asymptotics can be used in predicting an ultimate long-time response and dynamical behaviour of solutions. In recent years, this set of investigations has attracted a great deal of attention. Consequent efforts have then resulted in the creation and infusion of new methods and new tools that have been responsible for carrying out a successful analysis of long-time behaviour of several classes of nonlinear PDEs.

8.2. From the Preface.

The main goal of this book is to present recently developed mathematical methods of interest in the study of models that arise in continuum (nonlinear) mechanics of elastic bodies and are subject to different external influences and loads. In order to focus and streamline the exposition, we use the well-known von Karman model for the dynamics of plates and shells with large deflections as a benchmark prototype for the description of nonlinear oscillations. The theory of well-posedness and long-time behaviour associated with von Karman evolutions has reached a high level of maturity, if not completeness. Thus, the time seems ripe to collect relevant results and to present them in a complete and also self-contained manner. In addition. the von Karman model epitomises many important features and mathematical difficulties that arise in the study of attractors for various nonlinear PDEs. These include: (i) predominantly hyperbolic or hyperbolic-like type of dynamics combined with nonlinearity of the dissipation: (ii) critical (or potentially supercritical, with respect to Sobolev's embeddings) level of nonlinear terms which, in turn, results in the loss of compactness: (iii) loss of gradient flow structure due to the presence of external forces that are non-conservative. In order to cope with these difficulties, we develop and present an array of new methods that are capable of handling some of the issues discussed above. The novelty of the methods proposed is twofold, at two levels: (i) an abstract level within the realm of dynamical systems and (ii) a more concrete PDE level, within the context of establishing the validity of certain inverse-type inequalities. The book contains a number of new original results that appear in print for the first time.

Although our final results on well-posedness and long-time behaviour of solutions to the problem of nonlinear oscillations of plates and shells are presented in the context of von Karman evolutions, the techniques developed transcend this particular model and are applicable to a broad variety of nonlinear models in mechanics that share similar features and obstacles in their mathematical treatment.

We hope that the methods presented and the ideas developed in this book will be useful not only to the interested mathematical community working with differential equations and dynamical systems. but also to physicists and engineers interested in both the mathematical background and the asymptotic analysis of infinite-dimensional dissipative systems that arise in continuum mechanics. Much of the analysis in this book is devoted to a rigorous reduction of infinite-dimensional (PDE) systems to some finite-dimensional structures, which arc described only by finitely many degrees of freedom. These finite-dimensional structures should be of interest to application-oriented scientists, who pursue the actual design and implementation of algorithmic schemes, aiming at an accurate reconstruction and simulation of real infinite-dimensional phenomena.

8.3. Contents

Introduction.

1. Preliminaries.

2. Evolutionary Equations.

3. Von Karman Models with Rotational Forces.

4. Von Karman Equations without Rotational Inertia.

5. Thermoelastic Plates.

6. Structural acoustic problems and plates in a potential flow of gas.

7. Attractors for evolutionary equations.

8. Long-time behaviour of second-order abstract equations.

9. Plates with internal damping.

10. Plates with boundary damping.

11. Thermoelasticity.

12. Composite wave-plate systems.

13. Inertial manifolds for von Karman plate equations.

8.4. Review by: Alexander Figotin.
SIAM Review 53 (3) (2011), 586-589.

This almost 800-page monograph by Igor Chueshov and Irena Lasiecka is probably the most detailed treatise ever written on the von Karman evolution equations originally introduced in 1910 by von Karman.
...

The goals pursued by the authors in this monograph go far beyond a systematic presentation of the essentially complete mathematical theory of von Karman evolution equations. Their monograph presents a number of mathematical methods, including the authors' original contributions, which are essential for continuum nonlinear mechanics of elastic bodies subjected to external loads and related studies of nonlinear oscillations and wave propagation. The choice of von Karman theory as the focal point of the exposition was motivated by its richness and high level of maturity. Also, as the authors point out, "the von Karman model epitomizes many important features and mathematical difficulties that arise in the study of attractors for various nonlinear PDEs." These include (i) hyperbolic or hyperbolic-like dynamics involving a nonlinearity of the dissipation; (ii) a critical level of nonlinear terms which, in turn, results in the loss of compactness; (iii) a loss of gradient flow structure due to the presence of external forces that are nonconservative. To handle these difficulties the authors develop and present an array of new methods. The proposed methods are novel at two levels: an abstract level and a more concrete PDE level. The main results presented in this book are related to the following subjects.

- Well-posedness of solutions in a finite energy space, including Hadamard well-posedness.

- Regularity of solutions corresponding to more regular initial data.

- Long-time behaviour of solutions when $t \rightarrow ∞$, including (i) existence of global attractors; (ii) properties of attractors such as structure, dimensionality, and smoothness; (iii) rate of convergence of solutions to points of equilibria; (iv) inertial manifolds; (v) determining functionals.

The presentation of many ideas is based on von Karman evolution subject to different damping mechanisms. In addition to a single von Karman evolution the authors consider various interactive coupled systems (structure-acoustic interaction; structure gas-flow interactions), where the von Karman evolution constitutes just one component of such a system. In this latter case, the emphasis is on propagating smoothness properties from one component to another and on transferring stability properties.

The book consists of two parts. Part I of the book (Chapters 1-6) deals with the issues of well-posedness and regularity of solutions, and Part II (Chapters 7-13) presents results on long-time behaviour. Subjects covered in these chapters are as follows.

Chapter 1 provides the mathematical background needed for the analysis of von Karman evolution. It presents proper ties of the relevant Sobolev-Hardy-Lizorkin spaces, elements of nonlinear monotone operator theory, theory of biharmonic equations, and properties of von Karman brackets defining the Airy stress functions.

Chapter 2 provides an abstract treatment of second-order nonlinear evolution equations with nonlinear damping within the framework of nonlinear semigroup theory. The treatment depends heavily on the boundary conditions. A detailed analysis of various boundary conditions and their impact on well-posedness and regularity theory is presented at the end this chapter.

Chapter 3 deals with the well-posedness issues related to von Karman evolution ac counting for rotational inertia parameter a. This corresponds to hyperbolic-like dynamics with finite speed of propagation. Nonlinear dissipation both in the interior and on the boundary is considered.

Chapter 4 repeats the same programme for von Karman evolution without the rotational inertia term ($a = 0$). In that case the dynamics has infinite speed of propagation. Because of the lack of regularising effect on the velocity caused by rotational terms, the well-posedness of the model requires more subtle tools that involve some elements of harmonic analysis.

Chapter 5 provides the analysis of coupled structures involving von Karman evolution. In this chapter thermoelastic coupling with a heat equation is considered. Well-posedness and regularity theory along with analyticity of dynamics in the non-rotational case are presented.

Chapter 6 deals with two other coupled interactive structures that involve von Karman evolutions as one of the components.

Chapter 7 presents a treatment of long time behaviour of dynamical systems and provides a foundation for a general abstract theory developed for autonomous dynamical systems. This includes a discussion of compactness, fractal dimensionality of compact sets, gradient flows, and inertial manifolds.

Chapter 8 specialises and further develops the treatment of second-order systems. Various methods and concrete criteria for compactness, finite dimensionality, and smoothness of attractors are presented. New criteria for long-time behaviour are related to recent inverse-type estimates developed in the context of control theory of PDEs.

Chapter 9 presents long-time behaviour results for von Karman evolution with nonlinear internal damping. Existence of at tractors, and properties of attractors such as smoothness and fractal dimension, are studied. For models without rotational inertia new more sophisticated approaches are developed since the effect of the source is no longer compact with respect to the dynamics.

Chapter 10 takes up the same problem within the context of boundary damping. In this case the propagation of damping from the boundary into the interior plays a major role. Models both with and without rotational inertia are considered. To deal with criticality of the source in the case of rotational free models new approaches are developed allowing the combination of multiplier techniques with some specialised arguments in dynamical systems.

Chapter 11 describes long-time behaviour of thermoelastic plates with von Karman nonlinearity. There is an interesting phenomenon associated with the fact that rotational terms change the linearised dynamics from hyperbolic to parabolic. The challenge is to characterise long-time behaviour of parameters (i.e., fractal dimension of the attractor) that is uniform with respect to parabolicity or hyperbolicity encoded in the model. This is achieved following some novel criteria.

Chapter 12 presents three examples of interactive dynamical systems involving von Karman evolution. The first two involve acoustic-structure interaction (isothermal and heat generating), whereas the third one deals with structure-gas-flow interaction. Interface coupling between the von Karman plate and the acoustic wave equation describes a model of acoustic structure interactions. For the gas flow case, the von Karman plate is coupled on the boundary with a linearised potential flow equation. For all models, long-time dynamics is studied.

Chapter 13 deals with existence of inertial manifolds. The results depend strongly on the spectral properties of the operator describing the model, and the geometry of the domain plays a dominant role. Existence of inertial manifolds for von Karman evolution is established under various types of damping such as viscous damping, strong (structural damping), and thermal damping.

The appendix provides the necessary background and preliminary material used throughout the book.

The book contains a number of original results that appear in print for the first time.

The book's presentation requires from the reader a fairly high level of mathematical sophistication, but if this barrier is passed the reader interested in applied problems will be rewarded. All the mathematical methods and asymptotic models discussed in the book were developed with real physical and engineering problems in mind. In particular, a substantial part of the monograph is devoted to a rigorous reduction of infinite-dimensional PDE systems to finite-dimensional structures which can be a solid basis for further finite element numerical analysis.

9.1. From the Abstract.

These lectures present the analysis of stability and control of long time behaviour of PDE models described by nonlinear evolutions of hyperbolic type. Specific examples of the models under consideration include: (i) nonlinear systems of dynamic elasticity: von Karman systems, Berger's equations, Kirchhoff-Boussinesq equations, nonlinear waves (ii) nonlinear flow-structure and fluid-structure interactions, (iii) and nonlinear thermo-elasticity. A characteristic feature of the models under consideration is criticality or super-criticality of sources (with respect to Sobolev's embeddings) along with super-criticality of damping mechanisms which, in addition, may be also geometrically constrained.

Our aim is to present several methods relying on cancelations, harmonic analysis and geometric analysis, which enable to handle criticality and also super-criticality in both sources and the damping of the underlined nonlinear PDE. It turns out that if carefully analysed the nonlinearity can be taken "advantage of" in order to produce implementable damping mechanism.

Another goal of these lectures is the understanding of control mechanisms which are geometrically constrained. The final task boils down to showing that appropriately damped system is "quasi-stable" in the sense that any two trajectories approach each other exponentially fast up to a compact term which can grow in time. Showing this property - formulated as quasi-stability estimate - is the key and technically demanding issue that requires suitable tools. These include: weighted energy inequalities, compensated compactness, Carleman's estimates and some elements of microlocal analysis.

9.2. From the Introduction.

These lectures are devoted to the analysis of stability and control of long time behaviour of PDE models described by nonlinear evolutions of hyperbolic type. Specific examples of the models under consideration include: (i) nonlinear systems of dynamic elasticity: von Karman systems, Berger's equations, Kirchhoff-Boussinesq equations, nonlinear waves (ii) nonlinear flow - structure and fluid-structure interactions, (iii) and nonlinear thermo-elasticity. A goal to accomplish is to reduce the asymptotic behaviour of the dynamics to a tractable finite dimensional and possibly smooth sets. This type of results beside having interest in its own within the realm of dynamical systems, are fundamental for control theory where finite dimensional control theory can be used in order to forge a desired outcome for the dynamics evolving in the attractor.

A characteristic feature of the models under consideration is criticality or super-criticality of sources (with respect to Sobolev's embeddings) along with super-criticality of damping mechanisms which may be also geometrically constrained. This means the actuation takes place on a "small" sub-region only. Super-criticality of the damping is often a consequence of the "rough" behaviour of nonlinear sources in the equation. Controlling supercritical potential energy may require a calibrated nonlinear damping that is also supercritical. On the other hand super-linearity of the potential energy provides beneficial effect on the long time boundedness of semigroups. From this point of view, the nonlinearity does help controlling the system but, at the same time, it also does raise a long list of mathematical issues starting with a fundamental question of uniqueness and continuous dependence of solutions with respect to the given (finite energy) data. It is known that solutions to these problems can not be handled by standard nonlinear analysis-PDE techniques.

The aim of these lectures is to present several methods of nonlinear PDE which include cancelations, harmonic analysis and geometric methods which enable to handle criticality and also super-criticality in both sources and the damping. It turns out that if carefully analysed the nonlinearity can be taken "advantage of" in order to produce implementable control algorithms.

Another aspects that will be considered is the understanding of control mechanisms which are geometrically constrained. Here one would like to use minimal sensing and minimal actuating (geometrically) in order to achieve the prescribed goal. This is indeed possible, however analytical methods used are more subtle. The final task boils down to showing that appropriately damped system is "quasi-stable" in the sense that any two trajectory approach each other exponentially fast up to modulo a compact term which can grow in time. Showing this property- formulated as quasi-stability estimate -is the key and technically demanding issue that requires suitable tools. These include: weighted energy in- equalities, compensated compactness, Carleman's estimates and some elements of microlocal analysis.

9.3. Review by: Wim T van Horssen.
Mathematical Reviews MR3340991.

In these lecture notes the authors give a nice introduction to the field of stability properties and long-time behaviour of solutions of initial-boundary value problems for nonlinear, hyperbolic, evolution equations. These equations include problems for systems in dynamic elasticity, nonlinear flow-structure and fluid-structure interactions, and nonlinear thermo-elasticity. The lecture notes are organised in the following way. First, different PDE models (such as wave and plate equations) are described with interior and boundary damping. Then, general tools from functional analysis are introduced to describe the well-posedness of weak solutions and to prove the existence of attractors (including the structure, dimensionality, and smoothness of these attractors). Finally, the methods introduced are applied to the aforementioned problems of hyperbolic type.
Each model problem is studied as follows. First, the problem and the assumptions are formulated, and the results are presented. Possible extensions and open problems are discussed. Then, sketches of the proofs are given with references to the literature for more complete proofs.

10.1. From the Publisher.

This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions. The first chapter provides a general description of a fluid-structure interaction, which is formulated within a realistic framework, where the structure subject to a frictional damping moves within the fluid. The second chapter then offers a multifaceted description, with often surprising results, of the case of the static interface; a case that is argued in the literature to be a good model for small, rapid oscillations of the structure. The third chapter describes flow-structure interaction where the compressible Navier-Stokes equations are replaced by the linearised Euler equation, while the solid is taken as a nonlinear plate, which oscillates in the surrounding gas flow. The final chapter focuses on a the equations of nonlinear acoustics coupled with linear acoustics or elasticity, as they arise in the context of high intensity ultrasound applications.

10.2. Review by: Giuseppe Saccomandi.
Mathematical Reviews MR3822414.

Irena Lasiecka and Justin T Webster contribute the chapter Flow-plate interactions: well-posedness and long time behavior. This is a very interesting chapter about the mathematical well-posedness and long-time behaviour of the flow-plate interactions. The mathematical methods used by the authors are the theory of continuous semigroups and the classical theory of infinite-dimensional dynamical systems. The chapter is well written and not only the mathematics but also the mechanics of the various models considered are explained in detail. This is a great reference for researchers interested in this topic.