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The stability and control of discrete processes. (English) Zbl 0606.93001
Applied Mathematical Sciences, 62. New York etc.: Springer-Verlag. VII, 150 p. DM 49.80 (1986).
Professor J. P. LaSalle died in 1983 at the age of 67. The reviewed book is being published posthumously with the careful assitance of Kenneth R. Meyer, one of the students of Professor LaSalle. This book is concerned with the stability and controllability of a discrete dynamical system. For the difference equation $$x'=Tx$$, where the function $$T: R^ m\to R^ m$$ is continuous, $$x=x(n)$$, $$x'=x(n+1)$$, the well known theorems of boundedness, stability, asymptotic stability and unstability are presented and discussed, together with the invariance principle. For the discrete linear system $$x'=Ax$$ and the higher-order equation $$\Psi (z)y=y^{(m)}+a_{m-1}y^{(m-1)}+...+a_ 0y=0$$, where A is an $$m\times m$$ real or coplex matrix, $$y^{(m)}=y(m+n)$$, $$y=y(n)$$, and $$a_ i$$ $$(i=0,...,m-1)$$ are constants, in order to show how to obtain the general solutions of $$x'=Ax$$ and $$\Psi (z)y=0$$, two algorithms for computing $$A^ n$$ are given. Necessary and sufficient conditions for $$x'=Ax$$ to be equivalent to $$\Psi (z)y=0$$ are discussed. The readers also can see by an example how one can solve a system of higher-order difference equations. For the general nonhomogeneous linear system $$x'=Ax+f(n)$$, where $$f: J_ 0\to C^ m$$ is the input function or forcing term, the solution of the complete equation satisfying $$x(0)=x^ 0$$ is given by the variation of constant formula, and the method of undetermined coefficients may be used to find a particular solution with $$f(n)=p(n)\lambda^ n$$, where $$\lambda\in C$$ and $$p(n)=\sum^{r}_{j=0}n^ ja^ j$$ is a vector polynomial of degree r. The author also discusses stability by linear approximation and the problem of forced oscillations when the forcing term f is periodic. Finally, for the linear control theory, including many fundamentally important questions such as controllability, observability, reachability and stabilization by linear feedback are discussed in detail. (The control of linear systems; pole assignment; minimum energy control; minimal time-energy feedback control; observers; state estimation; stabilization by dynamic feedback, etc.)
Moreover, a large number of examples and exercises throughout the book is very useful for the reader’s better understanding of the subject. From the above description of the book its character is quite clear: the last publication of Professor LaSalle is on a subject which contains many interesting ideas, is very useful in applications and can be understood at an undergraduate level.
Reviewer: Ch.He

##### MSC:
 93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory 93C55 Discrete-time control/observation systems 93D15 Stabilization of systems by feedback 39A11 Stability of difference equations (MSC2000) 93B55 Pole and zero placement problems 93B05 Controllability 93C05 Linear systems in control theory 93B07 Observability 93D20 Asymptotic stability in control theory