STOCHASTIC SYSTEMS 
Stochastic control theory was first applied in this country at the 
Massachusetts Institute of Technology during World War II to synthesize 
fire control systems. In the 1960's it was applied to space navigation, 
guidance, and orbit determination in such well-known missions as Ranger, 
Mariner, and Apollo. Applications of the filtering theory, aspects of 
control theory include submarine navigation, fire control, aircraft 
navigation, practical schemes for detection theory, and numerical in- 
tegration. There have also been industrial applications; one example 
involved the problem of basic weight control in the manufacture of 
paper. 
The filtering and prediction theory developed by Wiener and Kolmogorov 
forms the cornerstone of stochastic control theory. It provides an 
estimate of the signal or the state of a process on the basis of observa- 
tion of the signal additively corrupted by noise. Unfortunately, the 
Wiener-Kolmogorov theory cannot be applied extensively because it requires 
the solution of the Wiener-Hopf integral equation. It is difficult to 
obtain closed form solutions to this equation, and it is not an easy 
equation to solve numerically. 
Kalman and Buey- give a solution to the filtering problem under 
weaker assumptions than those of the original Wiener problem. Their 
solution makes it possible to solve prediction and filtering problems 
recursively and is ideally suited for digital computers. Basically, it 
can be viewed as an algorithm which, given the observation process, 
sequentially computes in real time the conditional distribution of the 
signal process. The estimated state of the process is given as the 
output of a linear dynamical system driven by the observations. One 
determines the coefficients for the dynamical system by solving an 
initial value problem for a differential equation. This differential 
equation is easier to solve than the Wiener-Hopf equation. 
Our attention here will be limited to linear systems with quadratic 
cost functions. In this case the solution of the optimal control 
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