A Vector Small-Gain Theorem for General Nonlinear Control Systems

1. Introduction

• The vector small-gain approach uses a weak semigroup property and vector Lyapunov functions and functionals to address finite- and infinite-dimensional systems.The paper targets uniform and non-uniform input-to-output and input-to-state stability, including ODEs, retarded functional differential equations, and sampled-data systems.Source: §1 · p. 2
• Earlier nonlinear small-gain results commonly required a semigroup property, while the cited ISS small-gain framework considered internal and external stability together.The paper describes its weak semigroup property as more relaxed and notes that it permits study of a wide class of feedback systems, including hybrid and switched systems.Source: §1 · pp. 1–2

2. Global Asymptotic Stability for Monotone Discrete-Time Systems

• A MAX-preserving map can be represented coordinatewise as the maximum of non-decreasing scalar gain functions.This characterizes the map through component functions γ_ij, with each output coordinate given by a maximum over input coordinates.Source: §1 · p. 4
• For a continuous MAX-preserving map, global asymptotic stability is equivalent to each scalar gain function satisfying γ_ii(s) < s for every s > 0.When the dimension exceeds one, the proposition also gives small-gain conditions for compositions of the component gain functions.Source: §1 · p. 4

3. A Vector Small-Gain Theorem for a Wide Class of Systems

• Under hypotheses H1–H3, including global asymptotic stability of the associated discrete-time system, the theorem gives a bound on the vector Lyapunov function in terms of initial state and input.The theorem assumes a vector Lyapunov function and a continuous MAX-preserving map.Source: §3 · p. 7
• If the output is bounded by a function of the vector Lyapunov values, the system satisfies input-to-output stability with the theorem's specified gain.The stated gain is formed from the mapping G defined in equation (3.5).Source: §3 · pp. 7–8
• When c and β are bounded, the theorem also gives uniform input-to-output stability with the same stated gain.The gain is expressed using the components G_i.Source: §3 · p. 8

4.A. Systems Described by ODEs

• The passages report that system (4.1) satisfies IOS with the stated gain; if β∈K+ is bounded, system (4.1) with output Y=H(t,x) also satisfies UIOS with that gain.Source: 4.A. Systems of ODEs · p. 9
• Corollary 4.2 gives sufficient vector Lyapunov conditions for ODE ISS under the same small-gain conditions, and for UISS when β is bounded.The stated gain is explicitly constructed from the functions defined in (4.7)–(4.8).Source: 4.A. Systems of ODEs · p. 10
• Corollary 4.2 includes Theorem 3.4 in as a special case; cited -based characterizations require Lipschitz right-hand sides and provide no overall gain formula.The text also states that Corollary 4.2 uses the same cyclic small-gain conditions as.Source: 4.A. Systems of ODEs · p. 10
• For large-scale interconnected systems, subsystem ISS-Lyapunov functions and gains satisfying the small-gain conditions guarantee UISS for the overall system.The subsystem gain functions are the actual gains in the max-formulation of each subsystem’s UISS property.Source: 4.A. Systems of ODEs · p. 10

4.B. Systems Described by RFDEs

• For RFDEs, Theorem 4.4 uses almost-Lipschitz-on-bounded-sets functionals in sufficient Lyapunov-like conditions for (U)IOS and states that IOS gains are explicit.The theorem’s conditions are formulated for functionals on histories of length r_i, with 0 ≤ r_i ≤ r.Source: 4.A. Systems of ODEs · p. 12

Vector Lyapunov Functions and Functionals

• Under the small-gain conditions, system (4.10) satisfies IOS, and also UIOS when β is bounded, with the stated input gain.The gain is θ^-1∘γ^-1∘a, where θ is defined by (4.7)–(4.8).Source: 4.A. Systems of ODEs · p. 12
• Under the same small-gain conditions, system (4.10) satisfies ISS, and also UISS when β is bounded, with the stated input gain.The gain is θ^-1∘γ^-1∘a, where θ is defined by (4.7)–(4.8).Source: 4.A. Systems of ODEs · p. 13
• Theorem 4.4 and Corollary 4.5 allow Razumikhin functions; for ODEs, they also allow locally Lipschitz functions and time-discontinuous right-hand sides.In the time-delay case, the diagonal gain functions play a significant role.Source: 4.A. Systems of ODEs · p. 13

4.C. Sampled-data Systems

• For sampled-data system (4.21), Theorem 4.8 gives sufficient vector Lyapunov conditions for UIOS, with an explicit gain from u and zero gain from d~.The gain is θ^-1∘γ^-1∘a, with θ defined by (4.7)–(4.8).Source: 4.A. Systems of ODEs · p. 15
• Under the small-gain conditions, sampled-data system (4.21) satisfies UISS with the stated gain from u and zero gain from d~.This follows from Corollary 4.9 using the Lyapunov conditions specified for the ISS case.Source: 4.A. Systems of ODEs · p. 15
• Theorem 4.8 and Corollary 4.9 include Theorem 3.1 and Corollary 3.3 in as special cases, without requiring ’s additional growth hypothesis on H.That hypothesis requires R ≥ 0 and a function p ∈ K such that H(x) ≤ R + p(|x|).Source: 4.A. Systems of ODEs · p. 15

Example 5.1: Time-delay system

• The time-delay system is robustly globally asymptotically stable when the component constants satisfy c_i < a_i for every i.The example also notes that conditions on diagonal terms cannot generally be avoided.Source: §5 · p. 16
• For the time-delay example, the stated small-gain conditions hold when the parameter λ is sufficiently close to 1.This follows from the component inequalities and precedes the robust stability conclusion.Source: §5 · p. 16

Example 5.2: Biochemical control circuit model

• Under hypothesis (H), every solution starting in the positive interior converges to the non-trivial equilibrium X* of the biochemical circuit.The result allows g to be neither bounded nor strictly increasing, and establishes almost-global convergence even when two equilibria exist.Source: §5 · p. 17

Conclusions

• The theorem yields vector Lyapunov characterizations of uniform and non-uniform IOS and permits explicit computation of the overall system gain.The results generalize many recent small-gain results.Source: §6 · p. 19
• Allowing diagonal terms in the gain map gives the results direct applications to time-delay systems, with examples demonstrating the methodology on large-scale systems including biotechnology applications.Source: §6 · p. 19
• Coupled-system internal and external stability involving iISS subsystems is identified as a future research topic; preliminary results for two ISS and/or iISS subsystems are cited as.Source: §6 · p. 19

Proposition 2.7

• Proposition 2.7 is completed by proving (iv) ⇒ (i), (iii) ⇒ (ii), and (ii) ⇒ (iv), with (i) ⇒ (iii) following from Proposition 2.1.The proof of (ii) ⇒ (iv) uses a contradiction: a nonzero vector satisfying x ≥ Γ(x) would imply a closed cycle contradicting (ii).Source: Appendix-Proofs · pp. 19–20

Theorem 3.1

• An induction on k proves estimate (A.2) after a finite transient, uniformly over initial times in [0,T] and initial states with norm at most R.When β and c are bounded, the transient can be chosen independently of T.Source: Appendix-Proofs · pp. 20–21
• Theorem 3.1’s proof establishes that Σ is RFC from the input and that estimate (A.1) holds for all t ≥ 0.The proof obtains RFC by contradiction using the BIC property and bounds on the transition map.Source: Appendix-Proofs · p. 21

• Bounded c and β with T-independent kτ allow f to be chosen independently of T, and the ε adjustment establishes (A.2) for k+1.The proof notes that k+1 is also independent of T under these conditions.Source: Appendix-Proofs · pp. 22–23

Proof of Theorem 4.1

• For Theorem 4.1, the proof uses the stated inequalities to establish (3.1), hypothesis (H2), and (3.2), thereby verifying the hypotheses of Theorem 3.1.The component estimates use Lemma 3.5 in to obtain class-KL bounds.Source: Appendix-Proofs · pp. 23–24

Proof of Theorem 4.4

• The proof obtains (H3) and (H4) as direct consequences of stated inequalities and related definitions, then verifies (H1) and (H2) in two steps for the specified initial-condition ranges.Source: Appendix-Proofs · p. 25

Proof for system (4.10)

• The text states that inequalities (A.39) show (3.1) holds, and separately states that if β is bounded, then c is bounded as well.Source: Appendix-Proofs · p. 26
• For system (4.10), the proof establishes RFC and uses a contradiction argument with continuity to conclude that (3.1) and (3.2) hold for the considered initial states and inputs.The argument first derives RFC from the preceding solution estimate, then rules out a violation by approximating the initial history using continuity.Source: Appendix-Proofs · p. 27

Proof of Theorem 4.8

• For system (4.21), the proof defines vector Lyapunov functions V_i and W and uses the stated inequalities to establish hypotheses (H3) and (H4) of Theorem 3.1.The construction uses W(t,x) = exp(-μt)(1 + Q(x)); the proof then concludes that all hypotheses of Theorem 3.1 hold.Source: Appendix-Proofs · pp. 27–29
• For system (4.21), the sampled-data argument derives a derivative implication for each V_i and uses it to obtain the comparison estimates needed for (3.1).The proof considers sampling times and held inputs, then applies a KL comparison estimate to the V_i functions.Source: Appendix-Proofs · p. 28
• For system (4.21), estimates for W and the V_i functions are combined to establish (3.2), completing the verification of Theorem 3.1’s hypotheses.The proof derives bounds from (A.51)–(A.57) before concluding that all hypotheses hold.Source: Appendix-Proofs · pp. 28–29