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Analysis and control of polynomial dynamic models with biological applications. (English) Zbl 1404.92005

Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-815495-3/pbk; 978-0-12-815496-0/ebook). xvii, 165 p. (2018).
If one thinks of classical autonomous dynamical systems of relevance to applications outside physics, chemical models (following mass action or Michaelis-Menten kinetics) and population models (like Lotka-Volterra) come to one’s mind. Similar types of models have been studied over the years, and the authors here present a common generalization of many of these types, thereby including much of their own work. Quasipolynomial (QP) models have linear combinations of quasipolynomials as right hand sides, where quasipolynomials are of the form \(x_1^{a_1}\cdot \dots\cdot x_n^{a_n}\) with real numbers \(a_k\). They are written as generalized Lotka-Volterra (LV) systems. It is investigated how chemical reaction networks (CRN) can be subsumed under QP models. Kinetic models with mass action rates can be realized as LV models and vice-versa. So-called bio-CNRs (such as Michaelis-Menten) can be realized in a similar form, but with rational right hand sides. These can be embedded in models obeying mass action kinetics. In a similar way, more general smooth nonlinear models can be embedded in QP models. This is achieved by introducing new variables. A chapter studies transformations systematically, another one is dedicated to the existence of stationary points and their stability. Here, natural Lyapunov functions and the famous deficiency zero and deficiency one theorems find their places. In a special chapter, the authors turn their attention to feedback control. First, QP control systems are stabilized by optimizing feedback gain for the system linearized at the equilibrium point. The bilinear matrix inequality arising is attacked iteratively by a linear matrix inequality. Properly manipulating the feedback gain allows to apply kinetic theory to closed-loop systems with zero deficiency. The theory described is applied in a final chapter on case studies. An appendix presents the necessary mathematical basics, especially on linear programming and control theory. This book has the character of a guide through the maze of primary literature. It provides a good survey of the structure of the theory. For the harder aspects of important results cited, the authors refer to other published sources. In this respect, the part on control is particularly demanding.

MSC:

92-02 Research exposition (monographs, survey articles) pertaining to biology
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C42 Systems biology, networks
93D09 Robust stability
93D15 Stabilization of systems by feedback
34D23 Global stability of solutions to ordinary differential equations
49N10 Linear-quadratic optimal control problems
34K35 Control problems for functional-differential equations
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