The use of operator factorization for linear control and estimation

The linear filtering, prediction and smoothing problems as well as the linear quadratic control problems can very generally be formulated as operator equations using basic linear algebra.

The equations are of Fredholm type II, and they are difficult to solve directly.

It is shown how the operator can be factorized into two Volterra operators using a matrix Riccati equation. Recursive solution of these triangular operator equations is then obtained by two initial value differential equations.

The proofs of smoothing and optimal control under known disturbances are in this way especially clear and simple.