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• catalog abstract ""This book is devoted to geometric methods in the theory of differential equations with quadratic right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. Connections of the calculus of variations and the Riccati equation with the geometry of Lagrange-Grassmann manifolds and classical Cartan-Siegel homogeneity domains in a space of several complex variables are considered. The book is addressed to undergraduate and graduate students, scientific researchers and all specialists interested in the problems of geometry, the calculus of variations, and differential equations."--BOOK JACKET.".
• catalog alternative "Homogeneous spaces and the Riccati equation in the calculus of variations".
• catalog alternative "Odnorodnye prostranstva i uravnenie rikkati v variatsionnom ischislenii 1. English.".
• catalog contributor b12294657.
• catalog created "c2000.".
• catalog date "2000".
• catalog date "c2000.".
• catalog dateCopyrighted "c2000.".
• catalog description ""This book is devoted to geometric methods in the theory of differential equations with quadratic right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. Connections of the calculus of variations and the Riccati equation with the geometry of Lagrange-Grassmann manifolds and classical Cartan-Siegel homogeneity domains in a space of several complex variables are considered. The book is addressed to undergraduate and graduate students, scientific researchers and all specialists interested in the problems of geometry, the calculus of variations, and differential equations."--BOOK JACKET.".
• catalog description "Ch. 1. Classical Calculus of Variations. 1. Euler Equation. 2. Hamiltonian Formalism. 3. Theory of the Second Variation. 4. Riccati Equation. 5. Morse Index. 6. Jacobi Envelope Theorem. 7. Strong Minimum. 8. Poincare-Cartan Integral Invariant. 9. Fields of Extremals -- Ch. 2. Riccati Equation in the Classical Calculus of Variations. 1. Riccati Equation as a Sufficient Condition for Positivity of the Second Variation. 2. Riccati Equation for a Problem with Differential Constraints. 3. Riccati Equation and the Grassmann Manifold. 4. Grassmann Manifolds of Lower Dimension -- Ch. 3. Lie Groups and Lie Algebras. 1. Lie Groups: Definition and Examples. 2. Lie Algebras. 3. Lie Groups of Lower Dimension. 4. Adjoint Representation and Killing Form. 5. Semisimple Lie Groups. 6. Homogeneous and Symmetrical Spaces. 7. Totally Geodesic Submanifolds -- ".
• catalog description "Ch. 4. Grassmann Manifolds. 1. Three Approaches to the Description of the Grassmann Manifolds. 2. Lagrange-Grassmann Manifolds. 3. Riccati Equation as a Flow on the Manifold G[subscript n](R[superscript 2n]). 4. Systems Associated with a Linear System of Differential Equations -- Ch. 5. Matrix Double Ratio. 1. Matrix Double Ratio on the Grassmann Manifold. 2. Clifford Algebras. 3. Totally Geodesic Submanifolds of Grassmann Manifolds. 4. Curves with a Scalar Double Ratio. 5. Fourth Harmonic as a Geodesic Symmetry. 6. Clifford Parallels. 7. Connection Between Clifford Parallels and Isoclinic Planes. 8. Matrix Double Ratio on the Lagrange-Grassmann Manifold. 9. Morse-Maslov-Arnol'd Index in the Leray-Kashivara Form. 10. Fourth Harmonic as an Isometry of the Lagrange-Grassmann Manifold. 11. Application of the Matrix Double Ratio to the Study of the Riccati Equation -- Ch. 6. Complex Riccati Equations. 1. Cartan-Siegel Domains. 2. Klein -- ".
• catalog description "Ch. 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral. 1. Riccati Equation in the Case of the Degenerate Legendre Condition. 2. Reducing the Dirichlet Integral to the Integral of Its Principal Part. 3. Relation of the Riccati Partial Differential Equation to the Euler Equation. 4. Connection Defined by a Solution to the Riccati Partial Differential Equation.".
• catalog description "Includes bibliographical references and indexes.".
• catalog description "Poincare Upper Half-Plane and Generalized Siegel Upper Half-Plane. 3. Complexified Riccati Equation as a Flow on the Generalized Siegel Upper Half-Plane. 4. Flow on Cartan-Siegel Homogeneity Domains. 5. Matrix Analog of the Schwarz Differential Operator -- Ch. 7. Higher-Dimensional Calculus of Variations. 1. Minimal Surfaces. 2. Necessary Optimality Conditions for a Multiple Integral. 3. Vector Bundles. 4. Distributions and the Frobenius Theorem. 5. Connection in a Linear Bundle. 6. Levi-Civita Connection. 7. Nonnegativity Conditions of the Second Variation. 8. Field Theory in the Weyl Form. 9. Caratheodory Transformation. 10. Field Theory in the Caratheodory Form -- ".
• catalog extent "xii, 284 p. :".
• catalog identifier "3540667415".
• catalog isPartOf "Encyclopaedia of mathematical sciences, 0938-0396 ; v. 86".
• catalog issued "2000".
• catalog issued "c2000.".
• catalog language "eng rus".
• catalog language "eng".
• catalog publisher "Berlin ; New York : Springer,".
• catalog subject "Calculus of variations.".
• catalog subject "Commande, Théorie de la. ram".
• catalog subject "Control theory.".
• catalog subject "Mathematical optimization.".
• catalog subject "Optimisation mathématique. ram".
• catalog subject "QA402.3 .O4613 2000".
• catalog subject "Riccati equation.".
• catalog tableOfContents "Ch. 1. Classical Calculus of Variations. 1. Euler Equation. 2. Hamiltonian Formalism. 3. Theory of the Second Variation. 4. Riccati Equation. 5. Morse Index. 6. Jacobi Envelope Theorem. 7. Strong Minimum. 8. Poincare-Cartan Integral Invariant. 9. Fields of Extremals -- Ch. 2. Riccati Equation in the Classical Calculus of Variations. 1. Riccati Equation as a Sufficient Condition for Positivity of the Second Variation. 2. Riccati Equation for a Problem with Differential Constraints. 3. Riccati Equation and the Grassmann Manifold. 4. Grassmann Manifolds of Lower Dimension -- Ch. 3. Lie Groups and Lie Algebras. 1. Lie Groups: Definition and Examples. 2. Lie Algebras. 3. Lie Groups of Lower Dimension. 4. Adjoint Representation and Killing Form. 5. Semisimple Lie Groups. 6. Homogeneous and Symmetrical Spaces. 7. Totally Geodesic Submanifolds -- ".
• catalog tableOfContents "Ch. 4. Grassmann Manifolds. 1. Three Approaches to the Description of the Grassmann Manifolds. 2. Lagrange-Grassmann Manifolds. 3. Riccati Equation as a Flow on the Manifold G[subscript n](R[superscript 2n]). 4. Systems Associated with a Linear System of Differential Equations -- Ch. 5. Matrix Double Ratio. 1. Matrix Double Ratio on the Grassmann Manifold. 2. Clifford Algebras. 3. Totally Geodesic Submanifolds of Grassmann Manifolds. 4. Curves with a Scalar Double Ratio. 5. Fourth Harmonic as a Geodesic Symmetry. 6. Clifford Parallels. 7. Connection Between Clifford Parallels and Isoclinic Planes. 8. Matrix Double Ratio on the Lagrange-Grassmann Manifold. 9. Morse-Maslov-Arnol'd Index in the Leray-Kashivara Form. 10. Fourth Harmonic as an Isometry of the Lagrange-Grassmann Manifold. 11. Application of the Matrix Double Ratio to the Study of the Riccati Equation -- Ch. 6. Complex Riccati Equations. 1. Cartan-Siegel Domains. 2. Klein -- ".
• catalog tableOfContents "Ch. 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral. 1. Riccati Equation in the Case of the Degenerate Legendre Condition. 2. Reducing the Dirichlet Integral to the Integral of Its Principal Part. 3. Relation of the Riccati Partial Differential Equation to the Euler Equation. 4. Connection Defined by a Solution to the Riccati Partial Differential Equation.".
• catalog tableOfContents "Poincare Upper Half-Plane and Generalized Siegel Upper Half-Plane. 3. Complexified Riccati Equation as a Flow on the Generalized Siegel Upper Half-Plane. 4. Flow on Cartan-Siegel Homogeneity Domains. 5. Matrix Analog of the Schwarz Differential Operator -- Ch. 7. Higher-Dimensional Calculus of Variations. 1. Minimal Surfaces. 2. Necessary Optimality Conditions for a Multiple Integral. 3. Vector Bundles. 4. Distributions and the Frobenius Theorem. 5. Connection in a Linear Bundle. 6. Levi-Civita Connection. 7. Nonnegativity Conditions of the Second Variation. 8. Field Theory in the Weyl Form. 9. Caratheodory Transformation. 10. Field Theory in the Caratheodory Form -- ".
• catalog title "Control theory and optimization I : homogeneous spaces and the Riccati equation in the calculus of variations / M.I. Zelikin.".
• catalog title "Homogeneous spaces and the Riccati equation in the calculus of variations".
• catalog type "text".