Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory

The spectral or canonical factorization of matrix- or operator-valued function

F

defined on the imaginary axis is defined as

F

=

Y *X

, where

Y

±1

,

X

±1

are H

−

(bounded and holomorphic on Re

z

> 0), or, more generally,

Y

±1

,

X

±1

belong to some weighted strong H

2

space.

It is well known that the invertibility of the corresponding Toeplitz operator

P

+

FP

0+

is necessary for this factorization to exist, where

P

+

: L

2

→ H

2

is the orthogonal projection. When

F

is positive, this condition is also sufficient for the factors to be H

∞

. In the general (indefinite) case, this is not so. However, if

F

is smooth enough, then the H

∞

canonical factorization does exist even in the indefinite case; we give a solution assuming that

F

is the Fourier transform of a measure with no singular continuous part.

If the (Popov function determined by the) transfer function of a control system has a canonical factorization, then a well-posed optimal state feedback exists for the corresponding control problem. Conversely, a well-posed optimal state feedback determines a canonical factorization of the transfer function. We generalize this to unstable systems, i.e., to transfer functions that are holomorphic and bounded on some right half-plane | Re

z

>

r

.

Then we show that if the generalized Popov Toeplitz operator is uniformly positive, then the canonical factorization exists (the stable case is well known). However, the results on the regularity of the factors and in the nonpositive case remain very few — we explain them and the remaining open problems.