There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

http://control.disp.uniroma2.it/galeani/CO/materiale/automatica2000.pdf

Survey Constrained model predictive control: Stability and optimality

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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

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Controlled Markov processes and viscosity solutions

I The Simplest Problem in Calculus of Variations.- 1. Introduction.- 2. Minimum Problems on an Abstract Space-Elementary Theory.- 3. The Euler Equation Extremals.- 4. Examples.- 5. The Jacobi Necessary Condition.- 6. The Simplest Problem in n Dimensions.- II The Optimal Control Problem.- 1. Introduction.- 2. Examples.- 3. Statement of the Optimal Control Problem.- 4. Equivalent Problems.- 5. Statement of Pontryagin's Principle.- 6. Extremals for the Moon Landing Problem.- 7. Extremals for the Linear Regulator Problem.- 8. Extremals for the Simplest Problem in Calculus of Variations.- 9. General Features of the Moon Landing Problem.- 10. Summary of Preliminary Results.- 11. The Free Terminal Point Problem.- 12. Preliminary Discussion of the Proof of Pontryagin's Principle.- 13. A Multiplier Rule for an Abstract Nonlinear Programming Problem.- 14. A Cone of Variations for the Problem of Optimal Control.- 15. Verification of Pontryagin's Principle.- III Existence and Continuity Properties of Optimal Controls.- 1. The Existence Problem.- 2. An Existence Theorem (Mayer Problem U Compact).- 3. Proof of Theorem 2.1.- 4. More Existence Theorems.- 5. Proof of Theorem 4.1.- 6. Continuity Properties of Optimal Controls.- IV Dynamic Programming.- 1. Introduction.- 2. The Problem.- 3. The Value Function.- 4. The Partial Differential Equation of Dynamic Programming.- 5. The Linear Regulator Problem.- 6. Equations of Motion with Discontinuous Feedback Controls.- 7. Sufficient Conditions for Optimality.- 8. The Relationship between the Equation of Dynamic Programming and Pontryagin's Principle.- V Stochastic Differential Equations and Markov Diffusion Processes.- 1. Introduction.- 2. Continuous Stochastic Processes Brownian Motion Processes.- 3. Ito's Stochastic Integral.- 4. Stochastic Differential Equations.- 5. Markov Diffusion Processes.- 6. Backward Equations.- 7. Boundary Value Problems.- 8. Forward Equations.- 9. Linear System Equations the Kalman-Bucy Filter.- 10. Absolutely Continuous Substitution of Probability Measures.- 11. An Extension of Theorems 5.1,5.2.- VI Optimal Control of Markov Diffusion Processes.- 1. Introduction.- 2. The Dynamic Programming Equation for Controlled Markov Processes.- 3. Controlled Diffusion Processes.- 4. The Dynamic Programming Equation for Controlled Diffusions a Verification Theorem.- 5. The Linear Regulator Problem (Complete Observations of System States).- 6. Existence Theorems.- 7. Dependence of Optimal Performance on y and ?.- 8. Generalized Solutions of the Dynamic Programming Equation.- 9. Stochastic Approximation to the Deterministic Control Problem.- 10. Problems with Partial Observations.- 11. The Separation Principle.- Appendices.- A. Gronwall-Bellman Inequality.- B. Selecting a Measurable Function.- C. Convex Sets and Convex Functions.- D. Review of Basic Probability.- E. Results about Parabolic Equations.- F. A General Position Lemma.

Deterministic and stochastic optimal control

Normal and Integral Currents

1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4.8 The multiplier rule.- 5 Integration.- 5.1 Intervals.- 5.2 Measure.- 5.3 Integrals over En.- 5.4 Integrals over bounded sets.- 5.5 Iterated integrals.- 5.6 Integrals of continuous functions.- 5.7 Change of measure under affine transformations.- 5.8 Transformation of integrals.- 5.9 Coordinate systems in En.- 5.10 Measurable sets and functions further properties.- 5.11 Integrals: general definition, convergence theorems.- 5.12 Differentiation under the integral sign.- 5.13 Lp-spaces.- 6 Curves and line integrals.- 6.1 Derivatives.- 6.2 Curves in En.- 6.3 Differential 1-forms.- 6.4 Line integrals.- *6.5 Gradient method.- *6.6 Integrating factors thermal systems.- 7 Exterior algebra and differential calculus.- 7.1 Covectors and differential forms of degree 2.- 7.2 Alternating multilinear functions.- 7.3 Multicovectors.- 7.4 Differential forms.- 7.5 Multivectors.- 7.6 Induced linear transformations.- 7.7 Transformation law for differential forms.- 7.8 The adjoint and codifferential.- *7.9 Special results for n = 3.- *7.10 Integrating factors (continued).- 8 Integration on manifolds.- 8.1 Regular transformations.- 8.2 Coordinate systems on manifolds.- 8.3 Measure and integration on manifolds.- 8.4 The divergence theorem.- *8.5 Fluid flow.- 8.6 Orientations.- 8.7 Integrals of r-forms.- 8.8 Stokes's formula.- 8.9 Regular transformations on submanifolds.- 8.10 Closed and exact differential forms.- 8.11 Motion of a particle.- 8.12 Motion of several particles.- Axioms for a vector space.- Mean value theorem Taylor's theorem.- Review of Riemann integration.- Monotone functions.- References.- Answers to problems.

Functions of Several Variables

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Wendell H. Fleming

Papers

Controlled Markov processes and viscosity solutions

Deterministic and stochastic optimal control

Normal and Integral Currents

Functions of Several Variables

Book Review: Discrete-time Markov control processes: Basic optimality criteria