内容简介：　　　

　　The classical Pontryag in maximum principle (addressed to deterministic finite dimensional control systems) is one of the three milestones in modern control theory. The corresponding theory is by now well-developed in the deterministic infinite dimensional setting and for the stochastic differential equations. However, very little is known about the same problem but for controlled stochastic (infinite dimensional) evolution equations when the diffusion term contains the control variables and the control domains are allowed to be non-convex. Indeed, it is one of the longstanding unsolved problems in stochastic control theory to establish the Pontryagin type maximum principle for this kind of general control systems: this book aims to give a solution to this problem. This book will be useful for both beginners and experts who are interested in optimal control theory for stochastic evolution equations.

英文目录：
1 The Mathematical Formulation of Fully Developed Turbulence
　　 1.1 Introduction to Turbulence
　　 1.2 The Navier–Stokes Equation for Fluid Flow
　　　　 1.2.1 Energy and Dissipation
　　 1.3 Laminar Versus Turbulent Flow
　　 1.4 Two Examples of Fluid Instability Creating Large Noise
　　　　 1.4.1 Stability
　　 1.5 The Central Limit Theorem and the Large Deviation Principle ,in Probability Theory
　　　　 1.5.1Cram´er’sTheorem
　　　　 1.5.2StochasticProcessesandTimeChange
　　 1.6 Poisson Processes and Brownian Motion
　　　　 1.6.1 Finite-Dimensional Brownian Motion
　　　　 1.6.2 The Wiener Process
　　 1.7 The Noise in Fully Developed Turbulence
　　　　 1.7.1 The Generic Noise
　　 1.8 The Stochastic Navier–Stokes Equation for Fully Developed Turbulence
2 Probability and the Statistical Theory of Turbulence
　　 2.1 Ito Processes and Ito’s Calculus
　　 2.2 The Generatorofan Ito Diffusionand Kolmogorov’s Equation
　　　　 2.2.1 The Feynman–Kac Formula
　　　　 2.2.2 Girsanov’s Theorem and Cameron–Martin
　　 2.3 Jumps and L´evy Processes
　　 2.4 Spectral Theory for the Operator K
　　 2.5 The Feynman–Kac Formula and the Log-Poissonian Processes
　　 2.6 The Kolmogorov–Obukhov–She–Leveque Theory
　　 2.7 Estimates of the Structure Functions
　　 2.8 The Solution of the Stochastic Linearized Navier–Stokes Equation
3 The Invariant Measure and the Probability Density Function
　　 3.1 The Invariant Measure of the Stochastic Navier–Stokes Equation
　　　　 3.1.1 The Invariant Measure of Turbulence
　　 3.2 The Invariant Measure for the Velocity Differences
　　 3.3 The Differential Equation for the Probability Density Function
　　 3.4 The PDF for the Turbulent Velocity Differences
　　 3.5 Comparison with Simulations and Experiments
　　 3.6 Description of Simulations and Experiments
　　 3.7 The Invariant Measure of the Stochastic Vorticity Equation
　　　　 3.7.1 The Invariant Measure of Turbulent Vorticity
4 Existence Theory of Swirling Flow
　　 4.1 Leray’s Theory
　　 4.2 The A Priori Estimate of the Turbulent Solutions
　　 4.3 Existence Theory of the Stochastic Navier–Stokes Equation
Appendix A The Bound for a Swirling Flow
Appendix B Detailed Estimates of S2 and S3
Appendix C The Generalized Hyperbolic Distributions
Reference
Index