Remarks on the Applicability of Algebra and Analysis

There are virtually no areas of applied mathematics, the sciences and engineering which do not rely on the subject matter of the present book. This is especially true of abstract and linear algebra, real analysis, and functional analysis at the intermediate level (the level of the present book). Accordingly, the literature on the applicability of algebra and analysis is huge and any attempts at characterizing this literature, even at a most general and superficial level would be futile. In the following, we will identify samples from the literature of a specific area which the reader may want to consult for examples of applications of the type of mathematics used in our book. Generally speaking, this literature concerns the qualitative analysis of dynamical systems, under which we primarily mean well posedness of† dynamical systems, stability properties of invariant sets (resp. equilibria) and boundedness properties of motions. By necessity, our discussion will involve only representative citations of the various areas which we will touch upon. Furthermore, many sources that we cite may require familiarity with other specialized areas in mathematics for a complete understanding of the subject on hand.†

Dynamical Systems

A dynamical system is a four-tuple {T, X, A, S} †where T R denotes time set (R = (−∞,∞)); X is the state space (a metric space (X,d) †with metric d); A X †is the set of initial states; and S denotes a family ofmotions. For any fixed a  A, t0  T, †a mapping p,a,t0): Ta,t0  → X †is called a motion if p(t0,a,t0) = a †where Ta,t0 = [t0,t1)∩T, t1 > t0, †and t1 †is finite or infinite. In general, for a given pair (a,t0) A×T, †more than one motion may exist (see, e.g. [1]-[3]).

When T = R+ = [0,∞) we speak of a continuous-time dynamical system and when T = N = {0,1,2,….} we speak of a discrete-time dynamical system. When all motions in a continuous-time dynamical system are continuous with respect to time t (t Ta,t0), we speak of a continuous dynamical system and when one or more of the motions in such systems are not continuous with respect to t, we speak of a discontinuous dynamical system (DDS). When in a dynamical system parts of the motions evolve along continuous time while other parts evolve along discrete time, one speaks of a hybrid dynamical system (HDS) [4]. It turns out that every HDS (as defined above) can be embedded into a DDS, having identical qualitative properties (such as stability of an invariant set and boundedness of motions) as the HDS. Accordingly, the qualitative analysis of a given HDS can always be reduced to the analysis of an associated DDS.

When in the above, X is a finite dimensional normed linear space, one speaks of a finite dimensional dynamical system, and otherwise, of an infinite dimensional dynamical system. The motions of finite dimensional dynamical systems are determined by the solutions of ordinary differential equations and inequalities (see, e.g. [1]-[10]) and by ordinary difference equations and inequalities (see, e.g. [11]), while the motions of infinite dimensional dynamical systems may be characterized by the solutions of differential-difference equations (see, e.g. [12]), retarded and neutral functional differential equations (see, e.g. [13],[14]), Volterra integro-differential equations ( see, e.g. [15]-[17]), differential equations and inclusions defined on Banach (resp. Hilbert) spaces (see, e.g. [18]-[21]), certain classes of initial-value and initial-value and boundary-value problems determined by partial differential equations (see, e.g. [22]-[26]), as well as linear and nonlinear semigroups (see, e.g. [27]-[30]).

Discontinuous dynamical systems (DDS) arise in the modeling process of a variety of systems, including impulsive dynamical systems, switched systems, dynamical systems endowed with logic elements, discrete event systems, and so forth (see, e.g. [3]). Of course many (but not all) of these systems constitute special cases of† HDS (see, e.g. [4]).

The classical Lyapunov stability and boundedness results for continuous dynamical systems and discrete-time dynamical systems (as defined above) constitute a mature subject, spanning over a century [1]-[3]. On the other hand, the Lyapunov stability and boundedness results for DDS and HDS have been established more recently [3],[4].

The qualitative theory of dynamical systems described above is strictly concerned with topological aspects of mathematical systems (refer to chapter 5 on metric spaces in our book). This makes possible tidy definitions and classifications of dynamical systems. In the case of control systems where inputs, outputs and other structures play crucial roles, frequently there are algebraic constraints that need to be superimposed in the qualitative analysis (refer to chapters 3-4 and 6,7 in our book). In such cases, the definition of dynamical system and the classification of dynamical systems are not entirely straightforward. For example, in the case of control systems there does not seem to exist a universally accepted definition for hybrid dynamical system. However, there are no disagreements on the meaning of finite dimensional linear continuous time and discrete time control systems (see, e.g. [31]-[41]). Also, references for finite dimensional nonlinear control systems include [42],[43].


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