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Time-delay systems. Analysis, optimization and applications. (English) Zbl 0658.93001
North-Holland Systems and Control Series, Vol. 9. Amsterdam etc.: North- Holland (Elsevier Science Publishers B.V.) XVI, 504 p.; \$ 89.00; Dfl. 200.00 (1987).
This book is a good introduction to the qualitative theory of dynamic systems with after-effect. It starts with several mathematical models of actual time-delay processes. The first part deals with the mathematical description of time-delay systems. Large-scale time-delay systems and a linearization of nonlinear time-delay systems as well as state space and frequency domain representations are extensively considered here.
The second part gives an analysis of linear time-delay systems. It considers various solution properties, the fundamental matrix, adjoint state equations and other properties. Special attention is given to the problem of stability and controllability of time-delay systems. Such important problems as uniform asymptotic stability, the Lyapunov method, stability of stochastic time-delay systems, stability of time-delay systems with variable delays, Pontryagin’s stability method, several controllability and observability criteria are described in this part. Unfortunately, some interesting results concerning criteria of multipoint controllability, modal controllability (control) by using both integral and difference linear regulators, rank conditions for stabilization and others are not presented here.
Part III: Optimization (Chapter 6. Optimization of time-delay systems. Chapter 7. Suboptimal control of time-delay systems, Chapter 8. Near- optimum design on large-scale time-delay systems) includes, in particular, the following areas: the maximum principle, a generalized Riccati method, the dynamic programming method, a sensitivity approach, a hierarchical control and a computational algorithm. The next part is concerned with various applications of time-delay systems to gold rolling mills, water resources and others.
There are two appendices containing reviews on linear algebra, on the Laplace transform and modified z-transforms at the end of the book.
Reviewer: V.Marcenko

##### MSC:
 93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory 34K35 Control problems for functional-differential equations 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K20 Stability theory of functional-differential equations 49K99 Optimality conditions 93B03 Attainable sets, reachability 93B05 Controllability 93B07 Observability 93B35 Sensitivity (robustness) 93B40 Computational methods in systems theory (MSC2010) 93C05 Linear systems in control theory 93C10 Nonlinear systems in control theory 93A15 Large-scale systems 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D20 Asymptotic stability in control theory 93A13 Hierarchical systems 93E03 Stochastic systems in control theory (general) 93E15 Stochastic stability in control theory 93E20 Optimal stochastic control 93C95 Application models in control theory