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AuthorZelikin, M. I. author
TitleControl Theory and Optimization I [electronic resource] : Homogeneous Spaces and the Riccati Equation in the Calculus of Variations / by M. I. Zelikin
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000
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Descript XII, 284 p. online resource


This book is devoted to the development of geometrie methods for studying and revealing geometrie aspects of the theory of differential equations with quadratie right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. The book contains the following three parts, to each of which aseparate book could be devoted: 1. the classieal calculus of variations and the geometrie theory of the Riccati equation (Chaps. 1-5), 2. complex Riccati equations as flows on Cartan-Siegel homogeneity daยญ mains (Chap. 6), and 3. the minimization problem for multiple integrals and Riccati partial difยญ ferential equations (Chaps. 7 and 8). Chapters 1-4 are mainly auxiliary. To make the presentation complete and self-contained, I here review the standard facts (needed in what folIows) from the calculus of variations, Lie groups and algebras, and the geometry of Grassยญ mann and Lagrange-Grassmann manifolds. When choosing these facts, I preยญ fer to present not the most general but the simplest assertions. Moreover, I try to organize the presentation so that it is not obscured by formal and technical details and, at the same time, is sufficiently precise. Other chapters contain my results concerning the matrix double ratio, comยญ plex Riccati equations, and also the Riccati partial differential equation, whieh the minimization problem for a multiple integral. arises in The book is based on a course of lectures given in the Department of Meยญ and Mathematics of Moscow State University during several years


1. Classical Calculus of Variations -- 2. Riccati Equation in the Classical Calculus of Variations -- 3. Lie Groups and Lie Algebras -- 4. Grassmann Manifolds -- 5. Matrix Double Ratio -- 6. Complex Riccati Equations -- 7. Higher-Dimensional Calculus of Variations -- 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral -- Epilogue -- Appendix to the English Edition -- References

Mathematics Differential geometry Calculus of variations Mathematics Differential Geometry Calculus of Variations and Optimal Control; Optimization


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