文档介绍：Control Theory: From Classical to Quantum
Optimal, Stochastic, and Robust Control
Notes for Quantum Control Summer School, Caltech, August 2005
. James∗
Department of Engineering
Australian National University
Matthew.******@
Contents
1 Introduction 3
2 Deterministic Dynamic Programming and Viscosity Solutions 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Value Functions are Viscosity Solutions . . . . . . . . . . . . . . . . . . . . 12
The Distance Function is a Viscosity Solution . . . . . . . . . . . . 12
The Optimal Control Value Function is a Viscosity Solution . . . . 14
and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Stochastic Control 22
Some Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . 23
Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
∗This work was supported by the Australian Research Council.
1
Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Observation Processes . . . . . . . . . . .