Noncausal Continuous Time Systems

In this chapter we study various types of noncausal continuous time systems. Contrary to the usual continuous time systems obeying the equations

(7.1a)

$$ x\left( t \right) = - iAx\left( t \right) + Bu\left( t \right), $$

(7.1b)

$$ y\left( t \right) = - iCx\left( t \right) + Du\left( t \right), $$

where

t

∈ ℝ

+

is time,

u(t)

is input,

y(t)

is output,

x(t)

is the state, and −

iA

generates a strongly continuous semigroup, we now consider

t

∈ ℝ and require −

iA

to be exponentially dichotomous. This amounts to dropping the causality assumption on the linear system. Various theories can be developed, parallelling existing theories for causal systems. In Section 7.1 we require −

iA

to be exponentially dichotomous and

B

and

C

to be bounded. This includes the direct generalization of finite-dimensional linear systems theory, where

A, B, C

, and

D

are all matrices and

A

does not have real eigenvalues. In Section 7.2 we pass to a formalism with two state spaces (one densely and continously imbedded into the other), where the exponentially dichotomous operator −

iA

on the larger state space extends that on the smaller state space, the input operator

B

is bounded from the input space

into

the larger state space, and the output operator

C

is bounded

from

the smaller state space into the output space. Also adopting a complex Hilbert space setting, we thus obtain the so-called extended Pritchard-Salamon realizations. At the same time we discuss left and right Pritchard-Salamon realizations, where only one state space is used at the time.