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New Relases Riccati Equations
Egorov, AI
Egorov, AI 2007. Riccati Equations Russian Academic Monographs 5, ISBN 978-954-642-296-5, Pensoft Publishers, Sofia-Moscow, 165x240, formulas, equations, comments, references, 384pp., in English, hardback.Price €URO 89.00.

Table of contents

PREFACE
INTRODUCTION
CHAPTER 1. MATRICES, OPERATORS, AND GROUPS
1.1. Matrix polynomials
1. Equivalent matrices and invariant factors
2. Jordan canonical form
3. Transformation of matrices
4. Commuting matrices
1.2. Functions of matrices
1. Majorant matrices. Series of matrices
2. Representation of functions of matrices
3. Analytic continuation of a function of matrix
4. Some properties of functions of matrices
1.3. Operators and abstract functions
1. Linear operator
2. The space of linear operators
3. Inverse operators
4. Abstract functions of a number argument
5. Resolvent operator R_(A)
6. Nonlinear operators and their di_erentiation
7. Higher derivatives and di_erentials. The Taylor formula
1.4. Banach algebras
1. Main de_nitions and examples
2. Main properties of the spectrum
3. Functional calculus on Banach algebras
4. Di_erentiation
1.5. Lie groups
1. Main de_nitions and examples
2. Extension theory
3. Extension of the group and in_nitesimal operator
4. Commutators
1.6. Multiparametrical groups in _nite-dimensional spaces
1. Groups, vector _elds, and Lie algebras
2. Determining equation and the Lie derivative
3. Extended groups and their invariants
4. Calculation of the main group
5. Group classi_cation and invariant solutions
CHAPTER 2. MATRIX ALGEBRAIC AND DIFFERENTIAL RICCATI EQUATIONS
2.1. Matrix polynomial equations
1. Equation AX - XB = 
2. Commuting matrices
3. Solution of the linear inhomogeneous equation
4. Scalar equation
5. Polynomial equations
2.2. Square root of a matrix
1. Equation with Jordan matrix
2. Equation with a singular matrix
2.3. Algebraic Riccati equation
1. General analysis. Examples
2. Equation of the form Y 2 + AY + Y B + P = 
3. Relation between Riccati equations and linear equations
2.4. Linear di_erential equations
1. Homogeneous equation
2. Inhomogeneous equation
3. Particular solution to the inhomogeneous equation. The Cauchy formula
4. Bernoulli equation
2.5. Scalar equation
1. General properties of solutions
2. Examples of integrable Riccati equations
3. Properties of solutions
4. Existence of solutions
5. Some additional properties of the Riccati equation
2.6. Matrix di_erential Riccati equation
1. The simplest properties of the equation
2. Equation with constant coe_cients
3. Existence of a solution
2.7. Group analysis of the Riccati equation
1. Groups admissible for equations. Fundamental solutions
2. Scalar Riccati equation
3. Riccati equation on the plane
2.8. Group analysis of the matrix Riccati equation
1. One-parametrical groups of transformations and their operators
2. Multiparametrical groups and their operators
3. Determining equation. Lie algebra
4. Integration of the Riccati equation by the change of variables
5. Invariant solutions
2.9. Linearization of the matrix Riccati equation
1. Linearizability conditions
2. Anharmonic ratio of solutions to equation (1)
2.10. The Riccati equation in the sweep method
1. Boundary-value problem for a scalar di_erential equation
2. Boundary-value problem for the vector di_erential equation
2.11. Riccati equation in control theory
1. Problems on analytical construction of controllers and on optimal stabilization [27]
2. Optimal Kalman{Bucy _lter
2.12. Approximate solution of matrix Riccati equations
1. Solution of the algebraic equation by the Schur method
2. Method of block reduction of Hamiltonian matrices
to the Schur form
3. Iterative method for solving the di_erential Riccati equation
CHAPTER 3. SOLVABILITY PROBLEMS FOR MATRIX DIFFERENTIAL RICCATI EQUATIONS
3.1. Connection between Riccati equations and linear systems
1. Auxiliary propositions
2. Variation of solutions
3. Transformation of Eq. (1.1) and system (1.4M)
3.2. Properties of solutions
1. Associated di_erential Riccati equations
2. Normality and anormality of solutions
3. Singular solutions to the matrix di_erential Riccati equation
3.3. Generalized systems of di_erential equations and matrix integral
Riccati equations
CHAPTER 4. RICCATI EQUATIONS IN CONTROL PROBLEMS FOR SYSTEMS WITH DISTRIBUTED PARAMETERS
4.1. Riccati equations in mathematical physics
1. Boundary-value problems and operators
2. Operator Riccati equation in mathematical physics
4.2. Boundary-value Riccati problem in control systems with distributed parameters
1. Problems on optimal distributed control
2. Intergo{di_erential boundary-value Riccati problem
3. Optimization problem with a control function dependent only on time
4. In_nite systems of di_erential Riccati equations
5. Construction of a formal solution to the boundary-value Riccati problem
6. Controlling a system with uncontrollable disturbances
4.3. Semigroups of linear operators
1. De_nition of semigroups and their main properties
2. Semigroups over a Hilbert space. Dissipative semigroups
3. Compact semigroups and operators
4. Extension of operators
4.4. Di_erential equations in functional spaces
1. Stationary equation
2. Problems with inhomogeneous boundary conditions
3. Evolution equation
4.5. General problem on analytical construction of controllers
1. Statement of the problem and its preliminary analysis
2. Synthesis of optimal control
3. Problem of optimal stabilization
REFERENCES
INDEX


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