In this chapter we introduce various transformations on a system. An anti-causal system evolves in the backward time direction. To get the flow-inverted system we interchange the roles of the input and the output. Time-inversion means that we reverse the direction of time. To get a time-flow-inverted system we perform both of these transformations at the same time. Both well-posed and non-well-posed versions of these transformations are given.

Anti-causal systems

Up to now we have only considered causal systems which are well-posed in the forward time direction, i.e., we have always chosen the initial time to be smaller than the final time. It is possible to develop a completely analogous theory for anti-causal systems which are well-posed in the backward time direction. These systems appear naturally, e.g., when we want to pass from a system to its dual.

To get an anti-causal system it suffices to take a causal system and reverse the time direction as follows.

Definition 6.1.1 Let U, X, and Y be Banach spaces, and let 1 ≤ p ≤ ∞. An anti-causal Lp∣Reg-well-posed linear system Σ on (Y, X, U) consists of a quadruple Σ = satisfying the following conditions:

1. operator family = ચ-t, t ≥ 0, is a C0 semigroup on X;

2. B: Lp∣Regc(U) → X satisfies ચsBu = Bτsπ+u for all u ∈ Lp∣Regc(U) and all s ≤ 0;

3. ℭ: X → Lp∣Regloc(ℝ-; Y) satisfies ℭચsx = π-τsℭx for all x ∈ X and all s ≤ 0;

4. D: Lp∣Regloc, c(ℝ; U) → lp…