Partially Observable Linear Systems Under Dependent Noises

1 Basic Elements of Functional Analysis.- 1.1 Sets and Functions.- 1.1.1 Sets and Quotient Sets.- 1.1.2 Systems of Numbers and Cardinality.- 1.1.3 Systems of Sets.- 1.1.4 Functions and Sequences.- 1.2 Abstract Spaces.- 1.2.1 Linear Spaces.- 1.2.2 Metric Spaces.- 1.2.3 Banach Spaces.- 1.2.4 Hilbert and Euclidean Spaces.- 1.2.5 Measurable and Borel Spaces.- 1.2.6 Measure and Probability Spaces.- 1.2.7 Product of Spaces.- 1.3 Linear Operators.- 1.3.1 Bounded Operators.- 1.3.2 Inverse Operators.- 1.3.3 Closed Operators.- 1.3.4 Adjoint Operators.- 1.3.5 Projection Operators.- 1.3.6 Self-Adjoint, Nonnegative and Coercive Operators.- 1.3.7 Compact, Hilbert—Schmidt and Nuclear Operators.- 1.4 Weak Convergence.- 1.4.1 Strong and Weak Forms of Convergence.- 1.4.2 Weak Convergence and Convexity.- 1.4.3 Convergence of Operators.- 2 Basic Concepts of Analysis in Abstract Spaces.- 2.1 Continuity.- 2.1.1 Continuity of Vector-Valued Functions.- 2.1.2 Weak Lower Semicontinuity.- 2.1.3 Continuity of Operator-Valued Functions.- 2.2 Differentiability.- 2.2.1 Differentiability of Nonlinear Operators.- 2.2.2 Differentiability of Operator-Valued Functions.- 2.3 Measurability.- 2.3.1 Measurability of Vector-Valued Functions.- 2.3.2 Measurability of Operator-Valued Functions.- 2.3.3 Measurability of G1- and G2-Valued Functions.- 2.4 Integrability.- 2.4.1 Bochner Integral.- 2.4.2 Fubini’s Property.- 2.4.3 Change of Variable.- 2.4.4 Strong Bochner Integral.- 2.4.5 Bochner Integral of G1- and G2-Valued Functions.- 2.5 Integral and Differential Operators.- 2.5.1 Integral Operators.- 2.5.2 Integral Hilbert-Schmidt Operators.- 2.5.3 Differential Operators.- 2.5.4 Gronwall’s Inequality and Contraction Mappings.- 3 Evolution Operators.- 3.1 Main Classes of Evolution Operators.- 3.1.1 Strongly Continuous Semigroups.- 3.1.2 Examples.- 3.1.3 Mild Evolution Operators.- 3.2 Transformations of Evolution Operators.- 3.2.1 Bounded Perturbations.- 3.2.2 Some Other Transformations.- 3.3 Operator Riccati Equations.- 3.3.1 Existence and Uniqueness of Solution.- 3.3.2 Dual Riccati Equation.- 3.3.3 Riccati Equations in Differential Form.- 3.4 Unbounded Perturbation.- 3.4.1 Preliminaries.- 3.4.2 ?*-Perturbation.- 3.4.3?-Perturbation.- 3.4.4 Examples.- 4 Partially Observable Linear Systems.- 4.1 Random Variables and Processes.- 4.1.1 Random Variables.- 4.1.2 Conditional Expectation and Independence.- 4.1.3 Gaussian Systems.- 4.1.4 Random Processes.- 4.2 Stochastic Modelling of Real Processes.- 4.2.1 Brownian Motion.- 4.2.2 Wiener Process Model of Brownian Motion.- 4.2.3 Diffusion Processes.- 4.3 Stochastic Integration in Hilbert Spaces.- 4.3.1 Stochastic Integral.- 4.3.2 Martingale Property.- 4.3.3 Fubini’s Property.- 4.3.4 Stochastic Integration with Respect to Wiener Processes.- 4.4 Partially Observable Linear Systems.- 4.4.1 Solution Concepts.- 4.4.2 Linear Stochastic Evolution Systems.- 4.4.3 Partially Observable Linear Systems.- 4.5 Basic Estimation in Hilbert Spaces.- 4.5.1 Estimation of Random Variables.- 4.5.2 Estimation of Random Processes.- 4.6 Improving the Brownian Motion Model.- 4.6.1 White, Colored and Wide Band Noise Processes.- 4.6.2 Integral Representation of Wide Band Noises.- 5 Separation Principle.- 5.1 Setting of Control Problem.- 5.1.1 State-Observation System.- 5.1.2 Set of Admissible Controls.- 5.1.3 Quadratic Cost Functional.- 5.2 Separation Principle.- 5.2.1 Properties of Admissible Controls.- 5.2.2 Extended Separation Principle.- 5.2.3 Classical Separation Principle.- 5.2.4 Proof of Lemma 5.15.- 5.3 Generalization to a Game Problem.- 5.3.1 Setting of Game Problem.- 5.3.2 Case 1: The First Player Has Worse Observations.- 5.3.3 Case 2: The Players Have the Same Observations.- 5.4 Minimizing Sequence.- 5.4.1 Properties of Cost Functional.- 5.4.2 Minimizing Sequence.- 5.5 Linear Regulator Problem.- 5.5.1 Setting of Linear Regulator Problem.- 5.5.2 Optimal Regulator.- 5.6 Existence of Optimal Control.- 5.6.1 Controls in Linear Feedback Form.- 5.6.2 Existence of Optimal Control.- 5.6.3 Application to Existence of Saddle Points.- 5.7 Concluding Remarks.- 6 ntrol and Estimation under Correlated White Noises.- 6.1 Estimation: Preliminaries.- 6.1.1 Setting of Estimation Problems.- 6.1.2 Wiener—Hopf Equation.- 6.2 Filtering.- 6.2.1 Dual Linear Regulator Problem.- 6.2.2 Optimal Linear Feedback Filter.- 6.2.3 Error Process.- 6.2.4 Innovation Process.- 6.3 Prediction.- 6.3.1 Dual Linear Regulator Problem.- 6.3.2 Optimal Linear Feedback Predictor.- 6.4 Smoothing.- 6.4.1 Dual Linear Regulator Problem.- 6.4.2 Optimal Linear Feedback Smoother.- 6.5 Stochastic Regulator Problem.- 6.5.1 Setting of the Problem.- 6.5.2 Optimal Stochastic Regulator.- 7 Control and Estimation under Colored Noises.- 7.1 Estimation.- 7.1.1 Setting of Estimation Problems.- 7.1.2 Reduction.- 7.1.3 Optimal Linear Feedback Estimators.- 7.1.4 About the Riccati Equation (7.15).- 7.1.5 Example: Optimal Filter in Differential Form.- 7.2 Stochastic Regulator Problem.- 7.2.1 Setting of the Problem.- 7.2.2 Reduction.- 7.2.3 Optimal Stochastic Regulator.- 7.2.4 About the Riccati Equation (7.48).- 7.2.5 Example: Optimal Stochastic Regulator in Differential Form.- 8 Control and Estimation under Wide Band Noises.- 8.1 Estimation.- 8.1.1 Setting of Estimation Problems.- 8.1.2 The First Reduction.- 8.1.3 The Second Reduction.- 8.1.4 Optimal Linear Feedback Estimators.- 8.1.5 About the Riccati Equation (8.40).- 8.1.6 Example: Optimal Filter in Differential Form.- 8.2 More About the Optimal Filter.- 8.2.1 More About the Riccati Equation (8.40).- 8.2.2 Equations for the Optimal Filter.- 8.3 Stochastic Regulator Problem.- 8.3.1 Setting of the Problem.- 8.3.2 Reduction.- 8.3.3 Optimal Stochastic Regulator.- 8.3.4 About the Riccati Equation (8.81).- 8.3.5 Example: Optimal Stochastic Regulator in Differential Form.- 8.4 Concluding Remarks.- 9 Control and Estimation under Shifted White Noises.- 9.1 Preliminaries.- 9.2 State Noise Delaying Observation Noise: Filtering.- 9.2.1 Setting of the Problem.- 9.2.2 Dual Linear Regulator Problem.- 9.2.3 Optimal Linear Feedback Filter.- 9.2.4 About the Riccati Equation (9.27).- 9.2.5 About the Optimal Filter.- 9.3 State Noise Delaying Observation Noise: Prediction.- 9.4 State Noise Delaying Observation Noise: Smoothing.- 9.5 State Noise Delaying Observation Noise: Stochastic Regulator Prob-lem.- 9.6 Concluding Remarks.- 10 Control and Estimation under Shifted White Noises (Revised).- 10.1 Preliminaries.- 10.2 Shifted White Noises and Boundary Noises.- 10.3 Convergence of Wide Band Noise Processes.- 10.3.1 Approximation of White Noises.- 10.3.2 Approximation of Shifted White Noises.- 10.4 State Noise Delaying Observation Noise.- 10.4.1 Setting of the Problem.- 10.4.2 Approximating Problems.- 10.4.3 Optimal Control and Optimal Filter.- 10.4.4 Application to Space Navigation and Guidance.- 10.5 State Noise Anticipating Observation Noise.- 10.5.1 Setting of the Problem.- 10.5.2 Approximating Problems.- 10.5.3 Optimal Control and Optimal Filter.- 11 Duality.- 11.1 Classical Separation Principle and Duality.- 11.2 Extended Separation Principle and Duality.- 11.3 Innovation Process for Control Actions.- 12 Controllability.- 12.1 Preliminaries.- 12.1.1 Definitions.- 12.1.2 Description of the System.- 12.2 Controllability: Deterministic Systems.- 12.2.1 CCC, ACC and Rank Condition.- 12.2.2 Resolvent Conditions.- 12.2.3 Applications of Resolvent Conditions.- 12.3 Controllability: Stochastic Systems.- 12.3.1 ST-Controllability.- 12.3.2 CT-Controllability.- 12.3.3 ST-Controllability.- Comments.- Bibiography.- Index of Notation.