Continuity of the input response for p = ∞ (Corollary 1.6)

Corollary 1.6 (cor:p-infty-solution-continuous). Let X, U be Banach spaces and (𝕋, Φ) an abstract linear control system with p = ∞. For all u ∈ L^∞(ℝ₊; U), t ↦ Φ_t u is continuous on ℝ₊.

This answers positively Problem 2.4 of [Weiss 1989]. The proof (§ 2.4 of the paper) only uses Theorem 1.5 (ALCS.zero_class), the composition property and the continuity of semigroup orbits (Proposition 1.2, C0Semigroup.continuousOn_orbit):

• at t₀ = 0, ‖Φ_t u‖ ≤ κ(t) ‖u‖ → 0;
• at t₀ > 0, with t₁ = t₀ - δ and u₁ = u (t₁ + ·), the composition property gives Φ_{t₁ + r} u = T_r (Φ_{t₁} u) + Φ_r u₁, and ‖Φ_r u₁‖ ≤ κ(2δ) ‖u‖ for r ≤ 2δ.

The main statement is ALCS.continuousOn_Phi.

theorem AbstractLinearControl.ALCS.small_Phi {𝕜 : Type u_1} [RCLike 𝕜] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace 𝕜 X] {U : Type u_3} [NormedAddCommGroup U] [NormedSpace 𝕜 U] [CompleteSpace X] (S : ALCS 𝕜 X U ⊤) (u : ↥(MeasureTheory.Lp U ⊤ muPlus)) (ε : ℝ) (hε : 0 < ε) : ∃ η > 0, ∀ (r : ℝ), 0 ≤ r → r < η → ∀ (v : ↥(MeasureTheory.Lp U ⊤ muPlus)), ‖v‖ ≤ ‖u‖ → ‖(S.Φ r) v‖ < ε

Consequence of Theorem 1.5: for ε > 0 there is η > 0 such that ‖Φ r v‖ < ε for all r ∈ [0, η) and all inputs v with ‖v‖ ≤ ‖u‖.

theorem AbstractLinearControl.ALCS.continuousOn_Phi {𝕜 : Type u_1} [RCLike 𝕜] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace 𝕜 X] {U : Type u_3} [NormedAddCommGroup U] [NormedSpace 𝕜 U] [CompleteSpace X] (S : ALCS 𝕜 X U ⊤) (u : ↥(MeasureTheory.Lp U ⊤ muPlus)) : ContinuousOn (fun (t : ℝ) => (S.Φ t) u) (Set.Ici 0)

Corollary 1.6 (cor:p-infty-solution-continuous). Let X, U be real or complex Banach spaces and S = (𝕋, Φ) an abstract linear control system with p = ∞. For every input u ∈ L^∞(ℝ₊; U), the map t ↦ Φ t u is continuous on ℝ₊ = [0, ∞).