Infinite-dimensional input-to-state stability and Orlicz spaces

1. Introduction

• Unlike the bounded-control setting, where ISS and iISS are known to be equivalent, unbounded control operators arise in applications such as boundary control systems.The introduction notes that this distinction does not arise for linear systems in finite dimensions.Source: §1 · p. 2
• For linear infinite-dimensional systems with unbounded control operators, the paper relates ISS and iISS through admissibility and general input spaces.For linear systems, ISS is equivalent to admissibility together with exponential stability of the semigroup.Source: §1 · p. 2
• For L∞ inputs, iISS is equivalent to ISS with respect to some Orlicz space EΦ, and Orlicz-space ISS generalizes Lp-ISS by interpolating L1- and L∞-ISS.The introduction also notes that L∞-iISS implies zero-class admissibility, which is slightly stronger than admissibility.Source: §1 · p. 3
• For parabolic diagonal systems with scalar input, L∞-ISS implies ISS with respect to some Orlicz space, yielding equivalence of L∞-ISS and L∞-iISS.In this setting, every linear bounded operator from U to X−1 is L∞-admissible, which gives a characterization of ISS.Source: §1 · p. 3
• The paper gives a parabolic diagonal example that is L∞-ISS but not Lp-ISS for any finite p, and connects the general L∞-ISS-to-iISS question to a problem due to G. Weiss.The introduction describes the relation between L∞-ISS and L∞-iISS across settings schematically in Table 1.Source: §1 · p. 3, Table 1 · p. 3

2.1. The setting and definitions.

• The system is modeled with A generating a C0-semigroup on X and B bounded into the extrapolation space X−1, where it may be unbounded into X.The mild solution is represented through the variation-of-parameters formula on X−1.Source: §2.1 · pp. 3–4
• Admissibility ensures that input-driven states lie in X; infinite-time admissibility bounds the input-to-state map uniformly, while zero-class admissibility makes its bound vanish as time tends to zero.The paper formulates ISS, iISS, and UBEBS relative to general input spaces Z.Source: §2.1 · pp. 4–6
• Zero-class admissibility guarantees continuous X-valued mild solutions, and L∞-iISS implies L∞-zero-class admissibility.The latter result gives a sufficient condition for the continuity conclusion in the L∞ case.Source: §2.1 · p. 5, §2.2 · p. 8

2.2. Relations between the stability notions.

• For 1 ≤ p < ∞, Lp-ISS, Lp-iISS, and Lp-UBEBS together with exponential stability are equivalent.For general Z, Z-ISS is equivalent to Z-admissibility plus exponential stability; Z-iISS implies Z-admissibility and exponential stability.Source: §2.2 · pp. 7–8
• For bounded B, the stability equivalences in Proposition 2.13 reduce to exponential stability, with Assumption (B) supporting the additional equivalences.The paper notes that the assertions are independent of Z in this bounded-control setting.Source: §2.2 · pp. 8–9
• When B is unbounded, whether a system is Lp-ISS or Lp-iISS depends crucially on p.Even the scalar system with X = U = C, A = −1, and B = 1 is not L1-zero-class admissible.Source: §2.2 · pp. 9–10

• Orlicz spaces provide an input-space framework for relating L∞-ISS and L1-ISS to stability notions that use integral input bounds.The section motivates this framework by treating the integral in the iISS definition as a type of norm.Source: §3 · p. 10

Theorem 3.1

• L∞-iISS holds exactly when the system is ISS with respect to EΦ for some Young function Φ.The theorem also characterizes this condition by exponential stability of the semigroup together with EΦ-admissibility.Source: §3 · p. 10

Theorem 3.2

• For a Young function Φ satisfying the Δ2-condition, EΦ-ISS, EΦ-iISS, and EΦ-UBEBS with an exponentially stable semigroup are equivalent.The result generalizes the corresponding equivalence for Lp spaces, which are Orlicz spaces satisfying the Δ2-condition.Source: §3 · p. 10

Theorem 3.4

• L1-ISS, L1-iISS, and EΦ-ISS for every Young function Φ are equivalent.Source: §3 · p. 10

• The section answers positively, for the stated parabolic diagonal class, whether L∞-ISS implies L∞-iISS.The preceding result established the reverse implication for infinite-dimensional systems more generally.Source: §4 · pp. 14–15

Theorem 4.1

• For scalar-input parabolic diagonal systems with a q-Riesz eigenbasis and eigenvalues in a left-half-plane sector, L∞-iISS and L∞-ISS are equivalent.The theorem states L∞-iISS under these assumptions for B in L(C, X−1), and hence also L∞-zero-class admissibility.Source: §4 · pp. 14–15

Remark 4.3

• The EΦ-admissibility space supplied by the general result need not satisfy the Δ2-condition.The example is not Lp-admissible for any finite p, which rules out EΦ-admissibility for every Young function Φ satisfying Δ2.Source: §4 · p. 16

Theorem 4.5

• For the diagonal setting, L∞-admissibility is equivalent to a resolvent bound and to membership of s ↦ 1/s in Lq(C+, μ).The measure μ is specified in the theorem by the eigenvalues and coefficients of B.Source: §4 · p. 17

Remark 4.6

• Without exponential stability, the equality between L∞-admissible control operators and X−1 does not hold in general.A counterexample is given on ℓ2.Source: §4 · p. 17

5. Some Examples

• The one-dimensional Dirichlet heat equation with boundary control at 1 is formulated as a parabolic diagonal system and shown to be L∞-iISS.The control operator is represented using the sine basis; the section also gives L∞-ISS and L∞-iISS estimates.Source: §5 · pp. 17–18
• For the constructed parabolic diagonal system, the admissibility criterion rules out Lp-admissibility for every finite p.The passage gives the conclusion for p > 2 and extends it to all p ≥ 1.Source: §5 · p. 19
• An example is L∞-iISS and infinite-time L∞-admissible, yet not Lp-admissible for any finite p; its EΦ-admissibility therefore requires a Young function failing the Δ2-condition.The example’s sequence satisfies the stated summability condition, while its failure of finite-p admissibility excludes every Δ2 Young function.Source: §5 · p. 19

6. Conclusions and Outlook

• For parabolic diagonal systems with scalar input, L∞-iISS and L∞-ISS coincide.The paper also relates iISS with respect to L1 and L∞ to ISS with respect to Orlicz spaces, and extends the ISS/iISS equivalence to Orlicz spaces satisfying Δ2.Source: §6 · p. 19
• For linear infinite-dimensional systems with possibly unbounded control operators, ISS is equivalent to admissibility together with exponential stability of the semigroup.The paper studies inputs in general function spaces.Source: §6 · p. 19
• Future directions include nonanalytic diagonal systems, general analytic systems, zero-class admissibility, and strongly stable semigroups.The paper notes that results for parabolic diagonal systems have recently been adapted to more general analytic semigroups using holomorphic functional calculus.Source: §6 · p. 19
• The open question of whether L∞-admissibility ensures mild-solution continuity is linked to a disjunction: universal positive continuity or an exponentially stable system where it fails.The paper does not answer the question; Proposition 6.1 states that at least one of the two assertions holds.Source: §6 · pp. 19–20

Appendix A. Orlicz Spaces

• The Orlicz class consists of measurable functions satisfying the defining integral condition; unlike the Orlicz space, it need not be a vector space.The Orlicz class is convex and embeds in L1, and each L1 function belongs to some such class.Source: Appendix A · p. 20
• A Young function is generated by a right-continuous, nondecreasing function that vanishes at zero, is positive for positive arguments, and tends to infinity.The appendix also defines complementary Young functions through a complementary-function construction.Source: Appendix A · p. 20
• Complementary Young functions support Young’s inequality and its integrated form, while Orlicz Hölder inequality gives a continuous embedding of LΦ into L1.The appendix also records the corresponding relation between the spaces EΦ, the Orlicz class, and LΦ.Source: Appendix A · pp. 20–21
• The Orlicz space LΦ is equipped here with the Luxemburg norm, and is a Banach space.For Φ(t) = t^p with 1 < p < ∞, it coincides with vector-valued Lp up to equivalent norms.Source: Appendix A · p. 21
• When Φ satisfies the Δ2-condition, EΦ, the Orlicz class, and LΦ coincide.The appendix defines Φ-mean convergence and an essential-growth comparison between Young functions, with a theorem relating such convergence to norm convergence.Source: Appendix A · pp. 21–22