Front Cover -- Advanced Mathematical Tools for Automatic Control Engineers -- Copyright Page -- Table of Contents -- Preface -- Notations and Symbols -- List of Figures -- Part I: Matrices and Related Topics -- Chapter 1. Determinants -- 1.1 Basic Definitions -- 1.2 Properties of Numerical Determinants, Minors and Cofactors -- 1.3 Linear Algebraic Equations and the Existence of Solutions -- Chapter 2. Matrices and Matrix Operations -- 2.1 Basic Definitions -- 2.2 Some Matrix Properties -- 2.3 Kronecker Product -- 2.4 Submatrices, Partitioning of Matrices and Schur's Formulas -- 2.5 Elementary Transformations on Matrices -- 2.6 Rank of a Matrix -- 2.7 Trace of a Quadratic Matrix -- Chapter 3. Eigenvalues and Eigenvectors -- 3.1 Vectors and Linear Subspaces -- 3.2 Eigenvalues and Eigenvectors -- 3.3 The Cayley-Hamilton Theorem -- 3.4 The Multiplicities and Generalized Eigenvectors -- Chapter 4. Matrix Transformations -- 4.1 Spectral Theorem for Hermitian Matrices -- 4.2 Matrix Transformation to the Jordan Form -- 4.3 Polar and Singular-Value Decompositions -- 4.4 Congruent Matrices and the Inertia of a Matrix -- 4.5 Cholesky Factorization -- Chapter 5. Matrix Functions -- 5.1 Projectors -- 5.2 Functions of a Matrix -- 5.3 The Resolvent for a Matrix -- 5.4 Matrix Norms -- Chapter 6. Moore-Penrose Pseudoinverse -- 6.1 Classical Least Squares Problem -- 6.2 Pseudoinverse Characterization -- 6.3 Criterion for Pseudoinverse Checking -- 6.4 Some Identities for Pseudoinverse Matrices -- 6.5 Solution of Least Squares Problem Using Pseudoinverse -- 6.6 Cline's Formulas -- 6.7 Pseudo-Ellipsoids -- Chapter 7. Hermitian and Quadratic Forms -- 7.1 Definitions -- 7.2 Nonnegative Definite Matrices -- 7.3 Sylvester Criterion -- 7.4 The Simultaneous Transformation of a Pair of Quadratic Forms -- 7.5 Simultaneous Reduction of more than Two Quadratic Forms.

7.6 A Related Maximum-Minimum Problem -- 7.7 The Ratio of Two Quadratic Forms -- Chapter 8. Linear Matrix Equations -- 8.1 General Type of Linear Matrix Equation -- 8.2 Sylvester Matrix Equation -- 8.3 Lyapunov Matrix Equation -- Chapter 9. Stable Matrices and Polynomials -- 9.1 Basic Definitions -- 9.2 Lyapunov Stability -- 9.3 Necessary Condition of the Matrix Stability -- 9.4 The Routh-Hurwitz Criterion -- 9.5 The Liénard-Chipart Criterion -- 9.6 Geometric Criteria -- 9.7 Polynomial Robust Stability -- 9.8 Controllable, Stabilizable, Observable and Detectable Pairs -- Chapter 10. Algebraic Riccati Equation -- 10.1 Hamiltonian Matrix -- 10.2 All Solutions of the Algebraic Riccati Equation -- 10.3 Hermitian and Symmetric Solutions -- 10.4 Nonnegative Solutions -- Chapter 11. Linear Matrix Inequalities -- 11.1 Matrices as Variables and LMI Problem -- 11.2 Nonlinear Matrix Inequalities Equivalent to LMI -- 11.3 Some Characteristics of Linear Stationary Systems (LSS) -- 11.4 Optimization Problems with LMI Constraints -- 11.5 Numerical Methods for LMI Resolution -- Chapter 12. Miscellaneous -- 12.1 Lambda-Matrix Inequalities -- 12.2 Matrix Abel Identities -- 12.3 S-Procedure and Finsler Lemma -- 12.4 Farkaš Lemma -- 12.5 Kantorovich Matrix Inequality -- Part II: Analysis -- Chapter 13. The Real and Complex Number Systems -- 13.1 Ordered Sets -- 13.2 Fields -- 13.3 The Real Field -- 13.4 Euclidean Spaces -- 13.5 The Complex Field -- 13.6 Some Simple Complex Functions -- Chapter 14. Sets, Functions and Metric Spaces -- 14.1 Functions and Sets -- 14.2 Metric Spaces -- 14.3 Summary -- Chapter 15. Integration -- 15.1 Naive Interpretation -- 15.2 The Riemann-Stieltjes Integral -- 15.3 The Lebesgue-Stieltjes Integral -- 15.4 Summary -- Chapter 16. Selected Topics of Real Analysis -- 16.1 Derivatives -- 16.2 On Riemann-Stieltjes Integrals.

16.3 On Lebesgue Integrals -- 16.4 Integral Inequalities -- 16.5 Numerical Sequences -- 16.6 Recurrent Inequalities -- Chapter 17. Complex Analysis -- 17.1 Differentiation -- 17.2 Integration -- 17.3 Series Expansions -- 17.4 Integral Transformations -- Chapter 18. Topics of Functional Analysis -- 18.1 Linear and Normed Spaces of Functions -- 18.2 Banach Spaces -- 18.3 Hilbert Spaces -- 18.4 Linear Operators and Functionals in Banach Spaces -- 18.5 Duality -- 18.6 Monotonic, Nonnegative and Coercive Operators -- 18.7 Differentiation of Nonlinear Operators -- 18.8 Fixed-Point Theorems -- Part III: Differential Equations and Optimization -- Chapter 19. Ordinary Differential Equations -- 19.1 Classes of ODE -- 19.2 Regular ODE -- 19.3 Carathéodory's Type ODE -- 19.4 ODE with DRHS -- Chapter 20. Elements of Stability Theory -- 20.1 Basic Definitions -- 20.2 Lyapunov Stability -- 20.3 Asymptotic Global Stability -- 20.4 Stability of Linear Systems -- 20.5 Absolute Stability -- Chapter 21. Finite-Dimensional Optimization -- 21.1 Some Properties of Smooth Functions -- 21.2 Unconstrained Optimization -- 21.3 Constrained Optimization -- Chapter 22. Variational Calculus and Optimal Control -- 22.1 Basic Lemmas of Variation Calculus -- 22.2 Functionals and their Variations -- 22.3 Extremum Conditions -- 22.4 Optimization of Integral Functionals -- 22.5 Optimal Control Problem -- 22.6 Maximum Principle -- 22.7 Dynamic Programing -- 22.8 Linear Quadratic Optimal Control -- 22.9 Linear-Time Optimization -- Chapter 23. H2 and H∞ Optimization -- 23.1 H2-Optimization -- 23.2 H∞-Optimization -- Bibliography -- Index.