Why study fluid mechanics? The primary reason is not even technical, it is cultural: a physicist is defined as one who looks around and understands at least part of the material world. One of the goals of this book is to let you understand how the wind blows and the water flows so that flying or swimming you may appreciate what is actually going on. The secondary reason is to do with applications: whether you are to engage with astrophysics or biophysics theory or build an apparatus for condensed matter research, you need the ability to make correct fluid-mechanics estimates; some of the art of doing this will be taught in the book. Yet another reason is conceptual: mechanics is the basis of the whole of physics in terms of intuition and mathematical methods. Concepts introduced in the mechanics of particles were subsequently applied to optics, electromagnetism, quantum mechanics, etc.; here you will see the ideas and methods developed for the mechanics of fluids, which are used to analyze other systems with many degrees of freedom in statistical physics and quantum field theory. And last but not least: at present, fluid mechanics is one of the most actively developing fields of physics, mathematics and engineering, so you may wish to participate in this exciting development.

Even for physicists who are not using fluid mechanics in their work, taking a one-semester course on the subject would be well worth the effort.

Now that we have learnt basic mechanisms and elementary interplay between non-linearity, dissipation and dispersion in fluid mechanics, where can we go from here? It is important to recognize that this book describes only a few basic types of flow and leaves whole sets of physical phenomena outside of its scope. It is impossible to fit all of fluid mechanics into the format of a single story with a few memorable protagonists. Here is a brief guide to further reading, more details can be found in the endnotes.

A comparable elementary textbook (which is about twice as big) is that of Acheson [2]; it provides extra material and some alternative explanations on the subjects described in Chapters 1 and 2. On the subjects of Chapter 3, a timeless classic is the book by Lighthill [15]. For a deep and comprehensive study of fluid mechanics as a branch of theoretical physics one cannot do better than use another timeless classic, volume VI of the Landau–Lifshitz course [14]. Apart from a more detailed treatment of the subjects covered here, it contains a variety of different flows, a detailed presentation of the boundary layer theory, the theory of diffusion and thermal conductivity in fluids, relativistic and superfluid hydrodynamics, etc. In addition to reading about fluids, it is worth looking at flows, which is as appealing aesthetically as it is instructive and helpful in developing a physicist's intuition.

Determining asymptotic properties of dynamical systems, including the formulation of a qualitative picture of the system's trajectories over large intervals of time, is one of the central questions of modern theory for adaptive systems. This is not surprising, for the very reason for adaptation is the lack of available measurement information. If such information is not available a priori, and carrying out numerical or physical experiments is not a feasible option, assessment of the qualitative properties of the system's behavior is often the only way to characterize the system. What are these qualitative properties? Informally, from these properties we should be able to tell, for example, how a system might respond to external perturbations, or how the system's variables behave over long intervals of time. Formally, we may wish to know whether the system is stable in some sense, whether its trajectories are bounded, and to what sets these trajectories will be confined with time.

In this chapter we shall provide a brief summary and necessary background about these qualitative properties of dynamical systems. We do not wish, however, to present an exhaustive review of all concepts. There are many excellent texts devoted to detailed analysis of every single issue mentioned above. Here we will rather review these concepts with a level of detail and generality just sufficient for developing a qualitative understanding of the problem of adaptation and the basics of methods of adaptive regulation.

Adaptation is amongst the most familiar and wide spread phenomena in nature. Since the early days of the nineteenth century it has puzzled researchers in broad areas of science. Since it had often been observed in responsive behaviors of biological systems, adaptation was initially understood as a regulatory mechanism that helps an animal to survive in a changing environment. Later the notion of adaptation was adopted in wider fields of science and engineering.

As a theoretical discipline it began to emerge as a branch of control theory during the first half of the twentieth century. Its beginning was marked by publications discussing basic principles of adaptation and its merits for engineering. Imprecise technology and mechanisms were, perhaps, amongst the strongest practical motivations for such a theory at that time. Various notions of adaptation were adopted by engineers and theoreticians in order to grasp, understand, and implement relevant features of this phenomenon in practice. The first applications of the new theory were simple schemes for extremal control of mechanical systems; these systems could be described by just a few linear ordinary differential equations. Since then adaptive controllers have evolved to encompass substantially more complex devices. The controlling devices themselves can now be viewed as nonlinear dynamical systems with specific input–output properties. Methods for the design and analysis of such systems are currently recognized by many in terms of the theory of adaptive control and systems identification.

In this chapter we discuss a range of synthesis problems in the domain of adaptive control and regulation for dynamical systems with nonlinear parametrization and, possibly, unstable target dynamics. Results presented in the previous chapters, such as e.g. the bottle-neck principle and (non-uniform) small-gain theorems from Chapter 4, will play important roles in the development of suitable formal statements of these problems. In particular, when specifying the target dynamics of an adapting system, we will exploit input–output characterizations such as input–state and input–output margins, and majorizing of mappings and functions. No stability requirements will be imposed on the target motions in the adapting system a priori. This will offer us greater flexibility and thus will create opportunities to overcome certain limitations of standard approaches (see Chapter 3) with regard to the target dynamics and nonlinear parametrization.

We begin by stating the general problem of adaptation and adaptive regulation and providing a set of solutions to this problem. Having developed these solutions, we will proceed by considering several specific problems of adaptation. These problems are

1. (1) adaptive regulation to invariant sets;

2. (2) adaptive control of interconnected nonlinear systems;

3. (3) parametric identification of systems of ordinary differential equations with monotone nonlinear parametrization;

4. (4) non-dominating adaptive control and identification for systems with general nonlinear parametrization of uncertainties.

In order to provide particular solutions to the general problem of adaptation and also to the specific problems (1)–(4) we introduce a synthesis method – the method of the virtual adaptation algorithm.

Consider spatiotemporal pattern representation in the framework of template matching, the oldest and most common method for detecting an object in an image. According to this method the image is searched for items that match a template. A template consists of one or more local arrays of values representing the object, e.g. intensity, color, or texture. A similarity value between these templates and certain domains of the image is calculated, and a domain is associated with the template once their similarity exceeds a given threshold.

Despite the simple and straightforward character of this method, its implementation requires us to consider two fundamental problems. The first relates to what features should be compared between the image S0(x, y) and the template Si(x, y), i ∈ I. The second problem is how this comparison should be done.

The normative answer to the question of what features should be compared invokes solving the issue of optimal image representation, ensuring the most effective utilization of available resources and, at the same time, minimal vulnerability to uncertainties. Solutions in principle to this problem are well known from the literature and can be characterized as spatial sampling. For example, when the resource is the frequency bandwidth of a single measurement mechanism, the optimality of spatially sampled representations is proven in Gabor's seminal work (Gabor 1946). In classification problems, the advantage of spatially sampled image representations is demonstrated in Ullman et al. (2002).

Consider a living organism or an artificial mechanism, which we shall refer to for the moment as a system, aiming to perform optimally in an uncertain environment. Despite the fact that the environment may be uncertain, we will suppose that we know the structure of the physical laws of the environment determining plausible motions of the system. Suppose that we even know what the system's action might be and assume that criteria of optimality according to which the system must determine its actions are available. Would we be able to decide a priori which particular action a system must execute or how it should adjust itself in order to maintain its behavior at the optimum?

Depending on the language describing the system's behavior, environment, and uncertainties a number of theoretical frameworks can be employed to find an answer to this non-trivial question. If the available information about the system is limited to a statistical description of the events and their likelihoods are known, then a good methodological candidate is the theory of statistical decision making. On the other hand, if the more sophisticated and involved apparatus of stochastic calculus is used to formalize the behavior of a system in an uncertain environment then a reasonable way to approach the analysis of such an object is to employ the theory of stochastic control and regulation. Despite these differences in how the behavior of a system may be described in various settings, there is a fundamental similarity in the corresponding theoretical frameworks.

In this chapter we provide analysis tools for dynamical systems described as input–output and input–state mappings (or simply operators) in the corresponding spaces. Such a description is advantageous and natural when mathematical models of the systems are vaguely known and uncertain. We will see that the basic properties of these input–output and input–state mappings (such as boundedness and continuity) constitute important information for our understanding of the various ways in which an adaptation can be organized in these systems.

In particular, we will see that some basic stability notions (Lyapunov stability of invariant sets (LaSalle and Lefschetz 1961), stability of solutions in the sense of Lyapunov, and input-to-state, input-to-output, output-to-state, and input–output stability (Zames 1966)) are equivalent to continuity of a certain mapping characterizing the dynamics of the system (Theorems 4.1 and 4.3).

As we have said earlier, real physical systems, however, are not always stable and hence their input–output and input–state characterizations are not always continuous. Moreover, the target dynamics of these systems should not necessarily admit continuous input–output or input–state description. Indeed, continuity of a mapping S at a given point u0 in essence reflects the fact that the value of the mapping S(u), can be made arbitrarily close to S(u0), provided that u remains sufficiently close to u0. In reality this is a quite idealistic picture, and in most cases such infinitesimal closeness is not needed.

In this chapter we provide an overview of mathematical formulations of the problem of adaptation in dynamical systems. Adaptation is considered here as a special regulatory process that emerges as a response of a physical system to changes in the environment. In order to understand the notion of adaptation the terms “regulation,” “response,” and “environment” need to be given some physical sense and precise mathematical definitions. Instead of inventing a new language of our own, we adopt these terms from the language of mathematical control theory. It is clear that this adoption of terms will create a certain bias in our approach. On the other hand, it will allow us to operate with rigorously defined and established objects that have already passed the test of time.

We start with a retrospective overview and analysis of main ideas in the existing literature on adaptation in the domain of control. These ideas gave rise to distinct mathematical statements of the problem of adaptation. We will discuss their strengths and limitations with respect to the demands of new real-world applications in biology, neuroscience, and engineering. As a result of the analysis, we will formulate a list of features that a modern theory of adaptation must inherit, and propose a program that will allow us to contribute to the development of such a theory in the following chapters.

Logical principles of adaptation

The theory of adaptive regulation, as a set of notions, methods, and tools for systematic study of systems adjusting their properties in response to changes in the environment, has a long history of development.