# Noncausal Continuous Time Systems

Chapter

## Abstract

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

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

(7.1a)

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

(7.1b)

*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.## Keywords

Transfer Function Operator Function Bounded Linear Operator Continuous Semigroup Complex Hilbert Space
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© Birkhäuser Verlag AG 2008