By Michael Schulz

This ebook presents an advent to the research and the regulate mechanism of actual, chemical, organic, technological and financial types and their nonequilibrium evolution dynamics. robust emphasis is put on the basis of variational rules, evolution and keep watch over equations, numerical tools, statistical ideas and strategies for fixing or estimation of stochastic keep an eye on difficulties for structures with a excessive measure of complexity. particularly, the valuable objective of this publication is constructing a synergetic connection among theoretical suggestions and actual purposes. This publication is a contemporary advent and a valuable software for researchers in addition to for graduate students interested in econophysics and comparable subject matters.

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**Example text**

4. After a brief discussion of several ﬁeld theories, we deﬁne generalized Euler–Lagrange equations describing the control of ﬁeld equations. Furthermore, the control of ﬁelds via controllable sources and boundary conditions is discussed. Chaos control, controllability, and observability are the key points of Chap. 5. This part of the book is essentially addressed to dynamic systems with a moderate number of degrees of freedoms and therefore a moderate degree of complexity. Such systems are often observed at mesoscopic spatial scales.

D. D. Farmer, R. Shaw: An Introduction to Dynamical Systems (Springer, Berlin Heidelberg New York, 1997) 4 3. R. Balescu: Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975) 4 4. L. Boltzmann: J. f. Math. 100, 201 (1887) 4 5. W. Gibbs: Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, CT, 1902) 4 6. I. Sinai: Russian Math. Surv. 25, 137 (1970) 5 7. I. Arnold, A. Avez: Ergodic Problems of Classical Mechanics, Foundations and Applications (Benjamin, New York, 1968) 6 8.

Such problems occur in a very natural way, if we wish to control weak deviations from a given nominal curve. But several tools and concepts estimating the stability of controlled systems and several linear regulator problems will also be discussed here. The control of ﬁelds is another, often physically motivated class of control problems which we will study in Chap. 4. After a brief discussion of several ﬁeld theories, we deﬁne generalized Euler–Lagrange equations describing the control of ﬁeld equations.