1 Basic Properties of Hilbert Spaces. - 1. 0 Introduction. - 1. 1 Basic Definitions. - 1. 2 Examples of Hilbert Spaces. - 1. 3 Hilbert Spaces from Hilbert Spaces. - 1. 4 Convex Sets and Projections. - 1. 5 Orthogonality and Orthonormal Bases. - 1. 6 Continuous Linear Functionals. - 1. 7 Riesz Representation Theorem. - 1. 8 Weak Convergence. - 1. 9 Nonlinear Functionals and Generalized Curves. - 1. 10 The Hahn-Banach Theorem. - 2 Convex Sets and Convex Programming. - 2. 0 Introduction. - 2. 1 Elementary Notions. - 2. 2 Support Functional of a Convex Set. - 2. 3 Minkowski Functional. - 2. 4 The Support Mapping. - 2. 5 Separation Theorem. - 2. 6 Application to Convex Programming. - 2. 7 Generalization to Infinite Dimensional Inequalities. - 2. 8 A Fundamental Result of Game Theory: Minimax Theorem. - 2. 9 Application: Theorem of Farkas. - 3 Functions Transformations Operators. - 3. 0 Introduction. - 3. 1 Linear Operators and their Adjoints. - 3. 2 Spectral Theory of Operators. - 3. 3 Spectral Theory of Compact Operators. - 3. 4 Operators on Separable Hilbert Spaces. - 3. 5 L2 Spaces over Hilbert Spaces. - 3. 6 Multilinear Forms. - 3. 7 Nonlinear Volterra Operators. - 4 Semigroups of Linear Operators. - 4. 0 Introduction. - 4. 1 Definitions and General Properties of Semigroups. - 4. 2 Generation of Semigroups. - 4. 3 Semigroups over Hilbert Spaces: Dissipative Semigroups. - 4. 4 Compact Semigroups. - 4. 5 Analytic (Holomorphic) Semigroups. - 4. 6 Elementary Examples of Semigroups. - 4. 7 Extensions. - 4. 8 Differential Equations: Cauchy Problem. - 4. 9 Controllability. - 4. 10 State Reduction: Observability. - 4. 11 Stability and Stabilizability. - 4. 12 Boundary Input: An Example. - 4. 13 Evolution Equations. - 5 Optimal Control Theory. - 5. 0 Introduction. - 5. 1 Preliminaries. - 5. 2 Linear Quadratic Regulator Problem. - 5. 3 Linear Quadratic Regulator Problem: Infinite Time Interval. - 5. 4 Hard Constraints. - 5. 5 Final Value Control. - 5. 6 Time Optimal Control Problem. - 6 Stochastic Optimization Theory. - 6. 0 Introduction. - 6. 1 Preliminaries. - 6. 2 Measures on Cylinder Sets. - 6. 3 Characteristic Functions and Countable Additivity. - 6. 4 Weak Random Variables. - 6. 5 Random Variables. - 6. 6 White Noise. - 6. 7 Differential Systems. - 6. 8 The Filtering Problem. - 6. 9 Stochastic Control. - 6. 10 Physical Random Variables. - 6. 11 Radon-Nikodym Derivatives. - 6. 12 Nonlinear Stochastic Equations. Language: English