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Partially Observable Linear Systems Under Dependent Noises (Systems & Control

Noise is a rich concept playing an underlying role in human activity. Consideration of the noise phenomenon in arts and sciences, respectively, makes the distinction between both domains more obvious. Artists create "deliberate noise"; the masterpieces of literature, music, modern fine art etc. are those where a clear idea, traditionally related to such concepts as love, is presented under a skilful veil of "deliberate noise." On the contrary, sciences fight against noise; a scientific discovery is a law of nature extracted from a noisy medium and refined.

This book discusses the methods of fighting against noise. It can be regarded as a mathematical view of specific engineering problems with known and new methods of control and estimation in noisy media.

The main feature of this book is the investigation of stochastic optimal control and estimation problems with the noise processes acting dependently on the state (or signal) and observation systems. While multiple early and recent findings on the subject have been obtained and challenging problems remain to be solved, this subject has not yet been dealt with systematically nor properly investigated. The discussion is given for infinite dimensional systems, but within the linear quadratic framework for continuous and finite time horizon. In order to make this book self-contained, some background material is provided.

Consequently, the target readers of this book are both applied mathematicians and theoretically oriented engineers who are designing new technology, as well as students of the related branches. The book may also be used as a reference manual in that part of functional analysis that is needed for problems ofinfinite dimensional linear systems theory.

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Table of Contents

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.


"The book is very well and carefully written. It is an excellent reference on the complete sets of equations for the optimal controls and for the optimal filters under wide band noises and shifted white noises and their possible application to navigation of spacecraft. Independently, it can serve as a useful reference on the part of functional analysis that is needed for problems of infinite-dimensional linear systems theory. The book is written for both applied mathematicians and theoretically oriented engineers as well as for students at a graduate level. The control community will find this reference an important contribution to the modern control and estimation theory literature." --Mathematical Reviews "This book deals with key issues in control theory, namely the interaction between optimal control and observation and/or estimation issues!. The book!will be of interest to researchers in optimal control and estimation." --Zentralblatt Math

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