Noncausal Continuous Time Systems

Abstract

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

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

((7.1a))

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

((7.1b))

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.