Deterministic and Stochastic Optimal Control / Edition 1

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The first part of this book presents the essential topics for an introduction to deterministic optimal control theory. The second part introduces shastic optimal control for Markov diffusion processes. It also inlcudes two other topics important for applications, namely, the solution to the shastic linear regulator and the separation principle.

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Product Details

  • ISBN-13: 9780387901558
  • Publisher: Springer New York
  • Publication date: 11/17/1975
  • Series: Stochastic Modelling and Applied Probability Series, #1
  • Edition description: 1st ed. 1975. Corr. 2nd printing 1982
  • Edition number: 1
  • Pages: 222
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.63 (d)

Table of Contents

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 Shastic Differential Equations and Markov Diffusion Processes.- 1. Introduction.- 2. Continuous Shastic Processes; Brownian Motion Processes.- 3. Ito’s Shastic Integral.- 4. Shastic 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. Shastic 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.

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