I The Simplest Problem in Calculus of Variations. - 1. Introduction. - 2. Minimum Problems on an Abstract SpaceElementary 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 StochasticIntegral. - 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. 15. 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. Language: English