随机优化问题是对受随机扰动影响的动力学系统的研究，该系统可以被控制以优化某些性能准则。在过去的几年中，控制理论的研究取得了长足的发展，特别是受到数学金融带来的随机优化问题的启发。涉及线性动力学和二次性能标准的问题通常称为线性调节器问题。通常的控制框架可能是研究最深入的控制问题，线性二次最优控制问题或线性调节器问题是用于处理一个由一组微分方程控制的系统的性能指标的最小化问题。本书是一部英文版的数学专著，深入研究了线性二次性最佳控制相关知识。

1 Introduction
1.1 Background
1.2 Motivation and objectives of the book
1.3 Layout plan of the book
1.4 Notations
2 Literature Survey
2.1 Introduction
2.2 Literatures on stochastic optimal control problems
2.3 Literature on Bellman's optimality principle or Dynamic program
ming principle
2.4 Works on the Hamilton-Jacobi-Bellman (HJB) equation or Dynamic
programming equation
2.5 Brief survey of literature on viscosity and classical solution of HJB
equation
2.6 Literatures on the existence and development of optimal policies with
reference to cost control
2.7 Concluding remarks
3 Stochastic Differential Equations relating to Stochastic Control The
ory
3.1 Introduction
3.2 Preliminaries
3.2.1 Some definitions
3.2.2 Stochastic integrals
3.2.3 Stochastic differential equations (SDEs)
3.3 Linear control systems
3.4 Optimal control problems
3.4.1 Linear regulator problem
3.4.2 Stochastic control problems in standard forms
3.4.3 The linear-quadratic regulator problem
3.5 Concluding remarks
4 Viscosity Solution of the Degenerate Bellman Equation of Linear
Regulator Control Problem
4.1 Introduction
4.2 Stochastic linear regulator control problem
4.2.1 Problem formulation
4.2.2 The Hamilton-Jacobi-Bellman Equation
4.2.3 Value function
4.3 Viscosity solutions of the Degenerate Bellman Equation
4.3.1 Definition of viscosity solution
4.3.2 Viscosity properties of the value function
4.3.3 Dymnamic programming princtiple
4.4 Convergence of the value function
4.4.1 The value function is a viscosity solution of degenerate Bell
man equation
4.5 Uniqueness of degenerate Bellman equation
4.6 Stability properties of viscosity solutions
4.6.1 The limiting value function is a viscosity solution of degenerate
Bellman equation
4.7 Concluding remarks
5 Existence of Classical Solution of the Degenerate Bellman Equation
and Optimal Control
5.1 Introduction
5.2 Classical or Smooth solution of the degenerate Bellman equation
5.2.1 Convexity of the value function
5.2.2 Smoothness of the value function
5.3 An application to control theory
5.3.1 Optimal control
5.4 Concluding remarks
6 Summary and Conclusions
Bibliography
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