Abstract
This paper establishes necessary and sufficient conditions for subsystems within a coupled differential inclusion system on a generalized Hilbert space to remain within prescribed closed sets. By analyzing the behavior of vector fields on the boundary of each closure from a spectral theoretic perspective, we derive the "Finite Respect Condition," which ensures that the self-dissipation operator dominates the perturbations caused by interactions. The main theorem proves the Harmony Equivalence: the satisfaction of a budget inequality is equivalent to the global preservation of the viability kernel in the product space. This mathematical structure is implemented as a verifiable certificate based on $\Sigma_1$ arithmetic, providing a foundation for autonomous safety guarantees in distributed systems.
1. Introduction
Axiom (Principle over Patent)
The Finite Respect Principle is an ethical and mathematical axiom defining the coexistence of systems. We release the budget inequality condition $\Budget_i \le \kappa_i \ell_i$ and its $\Sigma_{1}$ verification algorithm as a universal public good. This openness does not negate the patentability of specific control mechanisms or implementation technologies but presents a meta-rule that they must adhere to.
Stability analysis of interconnected dynamical systems is a central issue in modern mathematical sciences. In particular, when each subsystem possesses a specific energy functional $V_i$ and is required to operate within a region defined by a certain threshold $R_i$ (a finite closure), the safety of the entire system depends critically on the balance between individual dissipation and mutual interference.
The purpose of this paper is to formulate a rigorous condition for multiple closures to coexist without infringing upon each other's boundaries within the framework of viability theory using Bouligand tangent cones—namely, the "Finite Respect Principle." Specifically, we extend the classical Gershgorin circle theorem to the context of nonlinear operators and show that the dominance of energy flux at the boundary is equivalent to the existence of a globally invariant set.
This paper follows the tradition of classical viability theory as developed by Aubin–Cellina (1984), Aubin–Frankowska (1990), and Cârjă–Necula–Vrabie (2007). Conversely, from the perspective of analyzing networked dissipative systems, it aligns with the small-gain theorem and dissipativity theory established by Dashkovskiy–Rüffer–Wirth (2010, 2011) and Arcak (2016). The primary objective of this work is to reconstruct these concepts under the unified geometric condition of the "Finite Respect Principle" and to elevate them into a form computationally implementable as a $\Sigma_1$ certificate.
2. Mathematical Formulation
Let $\H = \bigoplus_{i=1}^N \H_i$ be the direct sum of real Hilbert spaces, and let $x_i \in \H_i$ denote the state variable on each subspace $\H_i$.
For each subsystem $i$, we define a Lyapunov candidate function $V_i: \H_i \to \R_{\ge 0}$ and an admissible closed set
$$ \Omega_i := \set{ x_i \in \H_i \mid V_i(x_i) \le R_i }. $$
Let $\Omega := \prod_{i=1}^N \Omega_i$ be the product closure in the total state space.
For the sake of rigor, this paper primarily focuses on the case where
$$ \H_i \cong \R^{n_i} $$
(a finite-dimensional real vector space with the standard Euclidean norm).
In this setting, closed and bounded sets are compact with respect to the strong topology, and the level set $\Omega_i$ of the Lyapunov function $V_i$ is also compact under appropriate coercivity assumptions.
(To extend this to general infinite-dimensional Hilbert spaces, one may assume the weak compactness of the level sets and develop the subsequent arguments with respect to the weak topology.)
The time evolution of the system is described by the following differential inclusion:
$$ \dot{x}_i(t) \in F_i(x(t)) - \partial \phi_i(x_i(t)), \quad \text{a.e. } t \ge 0 $$
where $F_i$ represents the vector field incorporating interactions, and $\partial \phi_i$ represents the subdifferential of a dissipation potential. To ensure the existence and uniqueness of solutions, we impose the following regularity conditions.
Assume that the map $F: \H \to 2^{\H}$ is upper semicontinuous and takes non-empty, compact, convex values (Filippov regularity). Furthermore, assume that $V_i$ is of class $C^1$, and its gradient $\nabla V_i$ is Lipschitz continuous.
For the fundamental theory of differential inclusions and Filippov solutions, refer to classical works such as Aubin–Cellina (1984) and Filippov (1988). Although extensions to systems on metric spaces (Badreddine–Frankowska, 2022) and generalizations to Stieltjes-type differential inclusions (Satco–Smyrlis, 2023) have been actively studied in recent years, this paper focuses on the standard setting in finite-dimensional Euclidean spaces, prioritizing theoretical robustness and ease of implementation.
We introduce an observation operator $\T_i: \H_i \to \R_{\ge 0}$ that measures the "task intensity" of each subsystem (specifically, we assume a convolution form like $\T_i(x_i) = \norm{\K_i * x_i}$, but we treat it more generally here).
Along each trajectory $x(t)$, the time derivative of the energy $V_i$ satisfies the following estimate:
$$ \frac{\diff}{\diff t}V_i(x_i(t)) \le - \kappa_i \bigl( \T_i(x_i) \bigr)^2 + \T_i(x_i) \sum_{j \neq i} \beta_{ij} \T_j(x_j) $$
where $\kappa_i > 0$ is the self-dissipation constant, and $\beta_{ij} \ge 0$ represents the interference coefficient from system $j$ to system $i$.
Furthermore, to characterize the behavior of observables on the boundary $\partial \Omega_i = \set{ x_i \mid V_i(x_i) = R_i }$, we assume the following norm equivalence.
For any $i$, there exist constants $m_i, L_i, \alpha_i, \zeta_i > 0$ such that under the boundary condition $V_i(x_i) = R_i$, the following holds:
$$ \ell_i := m_i \sqrt{\frac{R_i}{\zeta_i}} \;\le\; \T_i(x_i) \;\le\; L_i \sqrt{\frac{R_i}{\alpha_i}} =: u_i^{\max} $$
For each $i$, consider the set
$$ \Omega_i = \set{ x_i \in \H_i \mid V_i(x_i) \le R_i }. $$
Assume $V_i \in C^1(\H_i)$ and that $\nabla V_i(x_i) \neq 0$ holds on the boundary $V_i(x_i)=R_i$.
Then, the Bouligand tangent cone $T_{\Omega_i}(x_i)$ is given by
$$ T_{\Omega_i}(x_i)
= \set{ v_i \in \H_i \mid \inner{\nabla V_i(x_i)}{v_i} \le 0 }. $$
Therefore, for the product set $\Omega = \prod_{i=1}^N \Omega_i$, for any $x \in \partial \Omega$ and $v = (v_1,\dots,v_N) \in \H$,
$$ v \in T_{\Omega}(x)
\quad\Longleftrightarrow\quad
\forall i \text{ with } V_i(x_i) = R_i,\ \inner{\nabla V_i(x_i)}{v_i} \le 0 $$
holds.
Describing invariance conditions using Bouligand tangent cones is a standard method dating back to Aubin (1990) and Aubin–Frankowska (1990). In recent years, applications of the Nagumo condition to specific sets such as polyhedra and ellipsoids (Song, 2022; Lv et al., 2023), and theoretical extensions to Wasserstein spaces (Bonnet-Weill et al., 2025) and generalized tangent cones (Eftekharinasab, 2023; Preprint 2025) have been advanced. The Finite Respect Principle of this paper significantly reduces verification costs by condensing these geometric conditions into the algebraic form of "budget inequalities."
3. Main Theorem: The Finite Respect Principle
We derive the condition for system viability, i.e., for any initial value $x(0) \in \Omega$, the solution $x(t)$ remains within $\Omega$ for all $t \ge 0$.
For each system $i$, the maximum interference inflow (Budget) is defined as follows:
$$ \Budget_i(\mathbf{R}) := \sum_{j \neq i} \beta_{ij} u_j^{\max} = \sum_{j \neq i} \beta_{ij} L_j \sqrt{\frac{R_j}{\alpha_j}} $$
Assume that for all $i \in \{1, \dots, N\}$, the following inequality (Respect Condition) holds:
$$ \Budget_i(\mathbf{R}) \le \kappa_i \ell_i $$
Then, the product closure $\Omega$ is Forward Invariant. That is,
$$ \forall x_0 \in \Omega, \;\forall t \ge 0 \implies x(t) \in \Omega. $$
Proof.
According to the modern formulation of Nagumo's Theorem, a necessary and sufficient condition for a closed set $\Omega$ to be invariant under the differential inclusion $\dot{x} \in F(x)$ is that at any boundary point $x \in \partial \Omega$, the intersection of the vector field $F(x)$ and the Bouligand tangent cone $T_{\Omega}(x)$ of $\Omega$ is non-empty:
$$ F(x) \cap T_{\Omega}(x) \neq \emptyset. $$
Since $\Omega$ is a product set, it suffices to verify the condition on the boundary $\partial \Omega_i$ for each component.
Consider $x \in \partial \Omega_i \times \prod_{j \neq i} \Omega_j$ for some $i$. At this point, $V_i(x_i) = R_i$, and $V_j(x_j) \le R_j$ ($j \neq i$).
The outward normal vector to the level set of $V_i$ is $\nabla V_i(x_i)$. The condition for invariance is $\dot{V}_i \le 0$, which means in terms of the inner product:
$$ \sup_{v \in F_i(x)} \inner{\nabla V_i(x_i)}{v} \le 0. $$
From Assumption 2.2,
$$ \dot{V}_i \le -\kappa_i \bigl( \T_i(x_i) \bigr)^2 + \T_i(x_i) \sum_{j \neq i} \beta_{ij} \T_j(x_j). $$
Using the bounds on the boundary (Assumption 2.3), on $V_i(x_i)=R_i$, we have $\T_i(x_i) \ge \ell_i$, and on $V_j(x_j) \le R_j$, we have $\T_j(x_j) \le u_j^{\max}$.
Furthermore, note that $f(t) = -\kappa_i t^2 + A t$ is a decreasing function for $t \ge A/2\kappa_i$ (the Respect Condition usually implies that the operating point is in this decreasing region).
Evaluating the worst case,
$$ \begin{aligned}
\dot{V}_i &\le -\kappa_i \ell_i^2 + \ell_i \sum_{j \neq i} \beta_{ij} u_j^{\max} \\
&= \ell_i \left( -\kappa_i \ell_i + \Budget_i(\mathbf{R}) \right).
\end{aligned} $$
Since $\Budget_i(\mathbf{R}) \le \kappa_i \ell_i$ by assumption,
$$ \dot{V}_i \le \ell_i \cdot 0 = 0. $$
Therefore, the vector field on the boundary points towards the interior of $\Omega$ or along the tangent plane, satisfying the Nagumo condition.
4. Harmony Equivalence and the Inverse Problem
Throughout this paper, $\F$ denotes the family of vector fields satisfying the following conditions:
- $F \colon \H \to 2^{\H}$ satisfies the regularity assumption of Definition 2.1.
That is, $F$ is upper semicontinuous and takes non-empty compact convex values (defining a differential inclusion in the sense of Filippov).
- Along any Filippov solution $x(\cdot)$ generated by $F$, the time derivative of the energy $V_i$ satisfies the estimate of Assumption 2.2:
\[
\frac{\diff}{\diff t} V_i(x_i(t))
\;\le\;
- \kappa_i \bigl(\T_i(x_i(t))\bigr)^2
+ \T_i(x_i(t)) \sum_{j\neq i} \beta_{ij} \T_j(x_j(t)).
\]
- Each potential $V_i$ and observation $\T_i$ satisfies the boundary evaluation of Assumption 2.3:
\(
\ell_i \le \T_i(x_i) \le u_i^{\max}.
\)
When $F\in\F$, we call $F$ an
"\emph{admissible vector field (in the sense of the Finite Respect Principle)}".
Theorem 3.2 gave a sufficient condition, but the following theorem demonstrates the geometric necessity of this condition.
Consider the class $\F$ of admissible vector fields introduced in Definition 4.1. Then, the following two propositions are equivalent:
- (Strong Invariance) For any $F \in \F$, $\Omega$ is forward invariant.
- (Respect Inequality) For all $i$, $\Budget_i(\mathbf{R}) \le \kappa_i \ell_i$ holds.
Suppose $x^\ast \in \partial\Omega$ and $v^\ast = (v_1^\ast,\dots,v_N^\ast)\in\H$ are given, such that for any $i$, $v_i^\ast \in T_{\Omega_i}(x_i^\ast)$ and
\[
\inner{\nabla V_i(x_i^\ast)}{v_i^\ast}
= - \kappa_i \bigl(\T_i(x_i^\ast)\bigr)^2
+ \T_i(x_i^\ast)\sum_{j\neq i}\beta_{ij}\T_j(x_j^\ast).
\]
Then, there exists a Filippov regular map $F^\ast \colon \H \to 2^{\H}$ satisfying Definition 2.1 and Assumption 2.2 such that
\[
v^\ast \in F^\ast(x^\ast).
\]
In particular, $F^\ast$ is upper semicontinuous and takes non-empty compact convex values, and $F^\ast \in \F$ holds in the sense of Definition 4.1.
Proof Sketch of Lemma 4.2.
We follow standard methods of Aubin--Cellina type regularization.
First, take a sufficiently small radius $r>0$ and define a single-valued map $\tilde F \colon \H \to \H$ on the neighborhood $B_r(x^\ast)$ of $x^\ast$ as
\[
\tilde F(x) := \chi(x)\,v^\ast
\]
where $\chi$ is a smooth cutoff function with $\chi(x^\ast)=1$ and $\chi(x)=0$ outside $B_r(x^\ast)$.
Due to the $C^1$ property of $V_i$ and the continuity of $\nabla V_i, \T_i$, by choosing $r$ sufficiently small, the deviation between the right-hand side of Assumption 2.2 and the target value along any trajectory staying in $B_r(x^\ast)$ can be kept arbitrarily small.
Next, taking the Filippov regularization of $\tilde F$,
\[
F^\ast(x)
:= \operatorname*{co}\bigl\{
\lim_{k\to\infty} \tilde F(x_k)
\mid x_k \to x,\; x_k \notin N
\bigr\}
\]
(the convex hull of the essential limits excluding a set of measure zero $N$), general theory by Aubin--Frankowska (e.g., Aubin--Frankowska (1990), Chap. 2) ensures that $F^\ast$ satisfies the Filippov regularity of Definition 2.1, and $F^\ast(x^\ast)=\{v^\ast\}$ holds at $x^\ast$.
Furthermore, from the choice of $r$, it can be constructed such that the estimate of Assumption 2.2 is maintained along trajectories generated by $F^\ast$.
Thus, we obtain $F^\ast\in\F$.
Proof of Theorem 4.1.
(2) $\implies$ (1) has already been shown by Theorem 3.2, so we show (1) $\implies$ (2).
We assume by way of contradiction that (2) does not hold.
Then, there exists an index $k$ such that
$$ \Budget_k(\mathbf{R}) > \kappa_k \ell_k $$
holds. Define
$$ \Phi_k(x)
:= -\kappa_k \bigl(\T_k(x_k)\bigr)^2
+ \T_k(x_k) \sum_{j\neq k} \beta_{kj}\, \T_j(x_j) $$
for each point on the boundary $\partial\Omega$. From Assumption 2.2, for any trajectory and any $x\in\partial\Omega$,
$$ \dot V_k(x_k) \le \Phi_k(x) $$
holds.
From the continuity of $\T_i$ and the compactness of $\partial\Omega$ (Remark 2.3'), $\Phi_k : \partial\Omega \to \R$ attains a maximum value. Let this maximum value be $\Phi_k(x^\ast)$.
On the other hand, considering the function
$$ \psi(t_0,t_1,\dots,t_N)
:= -\kappa_k t_0^2 + t_0\sum_{j\neq k}\beta_{kj} t_j $$
under the constraints
$$ \T_k(x_k) \in [\ell_k,u_k^{\max}], \quad
\T_j(x_j) \in [0,u_j^{\max}] \ (j\neq k),
$$
the right-hand side is a concave quadratic in $t_0$, and the Respect Condition is designed such that it is monotonically decreasing for $t_0 \ge \ell_k$.
Therefore,
$$ \max \psi(t_0,t_1,\dots,t_N)
= \psi(\ell_k,u_1^{\max},\dots,u_N^{\max})
= \ell_k\bigl(\Budget_k(\mathbf{R})-\kappa_k\ell_k\bigr) > 0. $$
Thus,
$$ \max_{x\in\partial\Omega}\Phi_k(x)
\;\ge\;
\ell_k\bigl(\Budget_k(\mathbf{R})-\kappa_k\ell_k\bigr) > 0 $$
follows, and in particular, there exists $x^\ast\in\partial\Omega$ satisfying $\Phi_k(x^\ast) > 0$.
Here, we choose one vector $v_k^\ast\in\H_k$ that
saturates the inequality
$$ \sup_{v\in F_k(x^\ast)}
\inner{\nabla V_k(x_k^\ast)}{v}
\le \Phi_k(x^\ast) $$
at $x^\ast$.
For example, setting
$$ v_k^\ast
:= \frac{\Phi_k(x^\ast)}{\|\nabla V_k(x_k^\ast)\|^2}\,
\nabla V_k(x_k^\ast) $$
yields $\inner{\nabla V_k(x_k^\ast)}{v_k^\ast} = \Phi_k(x^\ast)$.
For other components $i\neq k$, we take arbitrary $v_i^\ast\in T_{\Omega_i}(x_i^\ast)$ that do not disturb the tangent cone condition, and define
$$ v^\ast := (v_1^\ast,\dots,v_N^\ast) \in \H. $$
Next, applying Lemma 4.2 to the $x^\ast$ and $v^\ast$ constructed above, we fix one $F^\ast \in \F$ satisfying $v^\ast \in F^\ast(x^\ast)$.
From this construction,
$$ \sup_{v\in F_k^\ast(x^\ast)}
\inner{\nabla V_k(x_k^\ast)}{v}
= \inner{\nabla V_k(x_k^\ast)}{v_k^\ast}
= \Phi_k(x^\ast) > 0 $$
follows.
According to Lemma 2.4 and Remark 2.5, this is equivalent to
$$ F^\ast(x^\ast)\cap T_{\Omega}(x^\ast)=\emptyset, $$
meaning the Nagumo condition is violated.
Thus, $\Omega$ is not forward invariant for $F^\ast$.
However, assumption (1) states that "$\Omega$ is forward invariant for any $F\in\F$," which is a contradiction.
Therefore, if (1) holds, then (2) must also hold.
5. Connection to $\Sigma_{1}$ Implementation
We define a verification protocol belonging to $\Sigma_{1}$ (the arithmetic hierarchy with a single existential quantifier) to realize the aforementioned mathematical guarantees on computer systems.
Algorithm 5.1: Finite Respect Verification via Outward Rounding
Input: Parameter set $\Theta = \{ \alpha_i, \zeta_i, m_i, L_i, \kappa_i, \beta_{ij}, R_i \}_{i,j}$
Output: $\mathtt{Boolean}$
- For each $i$, compute the lower bound $\underline{\ell}_i \leftarrow \mathtt{round\_down}(m_i \sqrt{R_i/\zeta_i})$.
- For each $j$, compute the upper bound $\overline{u}_j \leftarrow \mathtt{round\_up}(L_j \sqrt{R_j/\alpha_j})$.
- For each $i$, compute the interference upper bound $\overline{\Budget}_i \leftarrow \sum_{j \neq i} \mathtt{round\_up}(\beta_{ij} \overline{u}_j)$.
- Verify: Determine if $\forall i, \overline{\Budget}_i \le \mathtt{round\_down}(\kappa_i \underline{\ell}_i)$ holds.
When this algorithm returns true, the hypothesis of Theorem 3.2 is mathematically strictly satisfied, including floating-point errors.
Here, we assume all parameters $\Theta$ are given as rational numbers, and each rational number $q$ is encoded via a pair of signed integers $(p, r) \in \Z \times \N$ as $q = p / 2^r$.
The rounding operations $\mathtt{round\_down}, \mathtt{round\_up}$ are fixed as primitive recursive functions for this encoding (e.g., floor/ceiling for binary fixed-point representation).
Fix an ascending sequence of prime numbers $(p_0, p_1, p_2, \dots)$.
For a given parameter sequence $\Theta$, represent the state transition of Algorithm 5.1 for one step by an abstract machine with finitely many registers.
Encode each state as a natural number $c \in \N$ such that register contents, program counter, flags, etc., can be recovered by finitely many primitive recursive projections.
Then, letting
$$ \mathrm{Step}(\Theta, c, c') $$
be the binary predicate "transition from state $c$ to state $c'$ in one step according to the rules of Algorithm 5.1," $\mathrm{Step}$ is primitive recursively definable, including rounding operations.
Furthermore, for a natural number $m$, consider the prime factorization of $m$:
$$ m = \prod_{t=0}^{\infty} p_t^{e_t(m)}. $$
Assume the exponent $e_t(m)$ can be primitive recursively computed from $m$ and $t$ via a prime factorization algorithm.
Define the state at each step as
$$ c_t(m) := e_t(m) - 1 $$
(if $e_t(m)=0$, we regard $c_t(m)=-1$, representing an "unused" state for sufficiently large $t$).
Define the length $k(m)$ as
$$ k(m) := \max\{ t \mid e_t(m) > 0 \} $$
(if the set is empty, $k(m) = -1$). This maximum operation is also given primitive recursively by bounded search.
Using the above, define the predicate
$$ \mathrm{Trace}(\Theta, m) $$
as follows:
- $c_0(m)$ is the initial state corresponding to input $\Theta$ (this can be determined primitive recursively from the encoding of $\Theta$),
- For all $t < k(m)$, $\mathrm{Step}(\Theta, c_t(m), c_{t+1}(m))$ holds,
- The final state $c_{k(m)}(m)$ is a "completed state returning output $\mathtt{true}$".
Since all quantifications appearing here are
bounded by $t \le k(m)$, $\mathrm{Trace}(\Theta, m)$ can be defined as an arithmetic formula consisting only of bounded quantifiers, i.e., a primitive recursive predicate.
Fix the above encoding and Algorithm 5.1. Let $\Theta$ be the encoded parameter sequence.
(1) For a natural number $m$, the predicate
$$ \mathrm{Trace}(\Theta, m) $$
representing "there exists a computational trace of length $m$ or less where Algorithm 5.1 returns output $\mathtt{true}$ for input $\Theta$" is primitive recursively definable.
(2) Therefore, the following predicate
$$ \mathrm{Respect\_Cert}(\Theta)
\;:\Longleftrightarrow\;
\exists m \in \N\ \mathrm{Trace}(\Theta, m) $$
can be described in $\SigOne$ form (an arithmetic formula with a single leading existential quantifier).
(3) Letting $\Theta^{\mathrm{real}}$ be the true values corresponding to the real-valued parameters, if Algorithm 5.1 returns $\mathtt{true}$, then due to the properties of outward rounding, the Finite Respect Condition of Theorem 3.2
$$ \forall i,\ \Budget_i(\mathbf{R}) \le \kappa_i \ell_i $$
holds strictly for $\Theta^{\mathrm{real}}$.
Consequently,
$$ \mathrm{Respect\_Cert}(\Theta) \Rightarrow
\text{"$\Omega$ is forward invariant for all admissible vector fields"} $$
holds.
References
- M. Arcak, "Diagonal Stability of a Class of Cyclic Systems and Its Connection with the Secant Criterion", Automatica 42, 1531–1537, 2006.
- M. Arcak, C. Meissen, and A. Packard, "Networks of Dissipative Systems: Compositional Certification of Stability, Performance, and Safety", Springer, 2016.
- J.-P. Aubin, "A Survey of Viability Theory", SIAM J. Control Optim. 28, 749–788, 1990.
- J.-P. Aubin and A. Cellina, "Differential Inclusions: Set-Valued Maps and Viability Theory", Springer, 1984.
- J.-P. Aubin and H. Frankowska, "Set-Valued Analysis", Birkhäuser, 1990 (Reprint 2008).
- Z. Badreddine and H. Frankowska, "Viability and Invariance of Systems on Metric Spaces", Nonlinear Analysis 220, 2022.
- A. Berman and D. Hershkowitz, "Matrix Diagonal Stability and Its Implications", SIAM J. Discrete Math. 4(3), 377–382, 1983.
- A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences", Academic Press, 1979 (SIAM Classics 1994).
- B. Bonnet-Weill, A. Domínguez Corella, and H. Frankowska, "Viability Theory in the 1-Wasserstein Space", arXiv:2511.19779, 2025.
- Q. Cai and M. Fu, "Input-to-State Stability and Gain Computation for Networked Control Systems", Control Theory and Technology 21, 292–300, 2023.
- O. Cârjă, M. Necula, and I. I. Vrabie, "Viability, Invariance and Applications", North-Holland Math. Studies 207, 2007.
- S. N. Dashkovskiy, B. Rüffer, and F. Wirth, "Small Gain Theorems for Large Scale Systems and Interconnections", SIAM J. Control Optim. 48, 2010.
- S. Dashkovskiy, H. Ito, and F. Wirth, "On a Small Gain Theorem for ISS Networks in Dissipative Lyapunov Form", Eur. J. Control 17(4), 357–365, 2011.
- X. Duan, J. Lam, et al., "Graph-Theoretic Stability Conditions for Metzler Matrices", SIAM J. Matrix Anal. Appl. 42, 2021.
- K. Eftekharinasab, "On a Generalization of the Nagumo–Brezis Theorem", Acta Comment. Univ. Tartu. Math., 2023.
- A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides", Kluwer, 1988.
- P. Hájek and P. Pudlák, "Metamathematics of First-Order Arithmetic", Springer, 1998.
- H.-S. Hong et al., "A Small-Gain Approach for Consensus of Heterogeneous Multi-Agent Systems", ISA Transactions, 2025.
- R. Kaye, "Models of Peano Arithmetic", Oxford University Press, 1991.
- D. Krokavec et al., "Linear Matrix Inequalities in Fault Detection Filter Design for Metzler-Type Systems", Machines 13(1), 2025.
- J. Lv et al., "Determining the Viability of an Unbounded Polyhedron for Differential Inclusions", Math. Problems in Engineering, 2023.
- D. E. Marelli et al., "Generalization of Small-Gain Theorem and Distributed Testing", Automatica 152, 2023.
- R. Ofir, "A Networked Small-Gain Theorem Based on Discrete-Time Diagonal Stability", arXiv:2411.03380, 2024.
- B. Satco and G. Smyrlis, "Viability and Filippov-type Lemma for Stieltjes Differential Inclusions", Set-Valued Var. Anal., 2023.
- Y. Song, "Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem", arXiv:2207.05429, 2022.
- Z. Sun, "A Gallery of Diagonal Stability Conditions with Structured Matrices and Review Papers", Preprint, 2023.