Deterministic and stochastic optimal control

TLDR: In this paper, the authors considered the problem of optimal control of Markov diffusion processes in the context of calculus of variations, and proposed a solution to the problem by using the Euler Equation Extremals.

Abstract: 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.