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Semiglobal Input-Delay Tolerance Algorithm for Distributed Nonconvex Optimization of Networked Nonlinear Systems
Jing-Zhe Xu, Zhi-Wei Liu, Ming-Feng Ge, Yan-Wu Wang, Dinxin He
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 6/20/2026, 5:00:19 AM
Summary
This paper proposes a Semiglobal Input-Delay Tolerant (SIDT) algorithm designed for distributed optimization in Networked Nonlinear Systems (NNSs) subject to input delays and consensus constraints. The algorithm achieves Input-Delay Tolerant Semiglobal Convergence (IDTSC), ensuring that for any prescribed compact initial set, there exists an admissible delay bound under which the system converges to the optimal solution. The research extends the applicability of distributed optimization from strictly convex problems to nonconvex problems by relaxing the strong convexity requirement to the Polyak-Łojasiewicz (P-Ł) condition. The proposed method utilizes a hierarchical design and Input-to-State Stability (ISS) analysis to handle the coupling between nonlinear dynamics and input delays.
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Relation Signals (4)
Semiglobal Input-Delay Tolerance Algorithm (SIDT) → achieves → Input-Delay Tolerant Semiglobal Convergence (IDTSC)
confidence 100% · a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC
Semiglobal Input-Delay Tolerance Algorithm (SIDT) → appliedto → Networked Nonlinear Systems (NNSs)
confidence 100% · the SIDT algorithm broadens its applicability to nonconvex optimization [in NNSs]
Semiglobal Input-Delay Tolerance Algorithm (SIDT) → uses → Input-to-State Stability (ISS)
confidence 95% · Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed
Polyak-Łojasiewicz (P-Ł) condition → enables → nonconvex optimization
confidence 90% · by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization.
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Abstract
Abstract:This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays.
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- Source: https://arxiv.org/abs/2606.19871v1
- Canonical: https://arxiv.org/abs/2606.19871v1
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Semiglobal Input-Delay Tolerance Algorithm for Distributed Nonconvex Optimization of Networked Nonlinear Systems†thanks: This manuscript is an extended version of our paper published in IEEE Transactions on Automatic Control, 2026, doi:10.1109/TAC.2026.3702930. © 2026 IEEE. Personal use is permitted. For other uses, permission must be obtained from IEEE. Jing-Zhe Xu1 Zhi-Wei Liu1 Ming-Feng Ge2 Yan-Wu Wang1 and Dinxin He1 Abstract This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays. 11footnotetext: School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: jzxu@hust.edu.cn; zwliu@hust.edu.cn; wangyw@hust.edu.cn; hedingxin@hust.edu.cn).22footnotetext: School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China (e-mail: gemf@cug.edu.cn).33footnotetext: Acknowledgment: This work was supported by the National Natural Science Foundation of China under Grants U24A20268, 624B2055, 62373162, and 62473349. Index Terms Input delay tolerance, distributed optimization, nonconvex optimization, networked nonlinear systems, semiglobal asymptotic convergence. 1 Introduction Distributed optimization has attracted sustained attention in networked systems because of its decomposability, data locality, and robustness to communication imperfections [1, 2, 3, 4, 5, 6, 7, 8]. Most of these works now exist along two main lines: discrete-time strategies [3, 4, 5] and continuous-time strategies [6, 7, 8], to find the solution of the optimization problem. In most continuous-time studies, the “state” typically represents an algorithmic iterate rather than a physical plant state, and node dynamics are either omitted or idealized as single-integrator models (effectively x˙i=ui x_i=u_i) [6, 7, 8]. As a consequence, these results only focus on solving the optimization computation problem itself essentially, not the joint problem of regulating a dynamical network while solving the distributed optimization task. As the field has matured, distributed optimization has moved from purely algorithmic studies to practical deployments in sensor networks [9, 10], machine learning [11], smart grids [12, 13] and robotics networks [14, 15]. In these settings the decision variables are the physical states of nodes rather than abstract iterates, and the dynamics of the node are richer than a single integrator and often nonlinear [9, 10, 11, 12, 13, 14, 15]. Thus, algorithms that merely compute an optimizer [6, 7, 8] are not directly applicable since the design must both solve the distributed optimization problem and regulate the nonlinear dynamics so that the states converge to the optimizer. To this end, recent works therefore blend control with optimization [13, 14, 15, 16, 17]. For example, hierarchical distributed controllers have been developed for fleets with Euler-Lagrange dynamics, including unmanned surface vehicles [15], and small gain methods are used to achieve optimal output consensus in uncertain nonlinear multi-agent systems [16]. However, most of these optimization control designs are developed under delay free assumptions [14, 15, 9, 10, 11, 12, 13, 16, 17]. In above application domains (e.g., robotics networks) that apply these methods, delays are difficult to avoid [18, 19, 20]. Specifically, hardware limits and environmental disturbances introduce latency in sensing, computation, communication, and actuation [18]. When delays interact with nonlinear dynamics, they corrupt gradient feedback and create residual terms that can destroy forward invariance and, even when arbitrarily small, undermine stability and prevent convergence to the optimal solution [21]. Previous delay handling optimization methods use Lyapunov Krasovskii functionals and are effective only for linear systems with delays [22, 23]. Moreover, optimization methods based on switching communication graphs face related limits [24], as they offer no general stability guarantee for nonlinear plants. Therefore, distributed optimization control for networked nonlinear systems with delays remains largely gap. Furthermore, many practical applications of distributed optimization, including multirobot motion planning [25, 15] and economic dispatch in power grids [26], necessitate strict adherence to state constraints [25, 27, 26, 28, 29, 30, 15]. These state constraints are intrinsic to real systems, often arising from physical and safety limits, task-level coordination that enforces state consistency, and conservation requirements at the network scale [25, 26]. Nevertheless, designing algorithms that minimize global cost functions within the required constraints is a formidable challenge. To address this, consensus-based optimization approaches utilizing nonsmooth Lyapunov functions have been proposed [15, 27, 28, 29, 30]. Note that these methods remain inadequate for systems with input delays, as they rely on optimization in delay-free environments that consider only current states and control inputs. The presence of input delays necessitates the incorporation of historical state information, thereby complicating the enforcement of real-time constraints. This fundamental alteration in the optimization problem’s structure underscores the need for more robust algorithms capable of handling input delays in constrained distributed optimization. Building on the insights from previous discussions, this paper investigates a class of distributed convex and nonconvex optimizations for networked nonlinear systems (NNSs) subject to input delays and consensus constraints. To this end, a novel optimization control algorithm is proposed to compute the optimal solution within consensus constraints and, at the same time, regulate the node states to approach the computed optimal solution. The key contributions of this paper are summarized as follows: 1. A novel concept of input-delay tolerant semiglobal convergence (IDTSC) is provided to cope with the fact that unknown input delays together with nonlinear node dynamics can break the solvability of the distributed optimization problem and prevent the network state from converging. Compared with linear-delay optimization schemes [24, 22, 23], IDTSC concept follows a semiglobal principle that links the radius of the initial condition set to an admissible delay margin, thereby offering implementable guarantees for distributed optimization over nonlinear nodes. 2. Unlike existing distributed constrained optimization methods without considering time delays [27, 28, 29, 30], the proposed algorithm couples a hierarchical structure with input-to-state stability (ISS) theory guaranteeing that the states return to and stay in the constraint set and are driven toward the constrained optimizer under the input delay. Importantly, the algorithm is delay-independent, implying that its gains do not depend on the delay and require no delay-based retuning. 3. The strong-convexity requirement is relaxed to the Polyak–Łojasiewicz (P–Ł) condition, under which the proposed design still guarantees input-delay tolerant practical semiglobal convergence for distributed optimization of NNSs with input delays and consensus constraints. The paper is organized as follows: Section I introduces preliminaries and formulates the problem. Section I presents the SIDT algorithm and the theoretical results. Section IV provides numerical validation, and section V draws a conclusion with future directions. 2 Problem Formulation And Preliminary 2.1 Notations and Definitions Notations: Let ℝnR^n denote the n-dimensional real vector space, and ℝn×nR^n× n denotes the space of n×n× n real matrices. The absolute value of a variable x is denoted by |x||x|, and the function sig(x)a sig(x)^a is defined as sign(x)|x|a sign(x)|x|^a. Additionally, InI_n represents n×n× n diagonal matrix with ones on the main diagonal. n1_n and n0_n denote the n-dimensional all-ones and all-zero column vector, respectively. ∇f(x)∇ f(x) and ∇2f(x)∇^2f(x) represent the gradient vector and Hessian matrix of a scalar field f(x)f(x), respectively. Let ⊗ denote the Kronecker product. The following stability concept is foundational for in this paper. Definition 1. (Global asymptotic and local exponential stability) Consider a nonlinear system: x˙=f(x,u), x=f(x,u), (1) where x∈ℝnx ^n is the state, and u∈ℝmu ^m is the feedback control law. This system (1) is said to be globally asymptotically and locally exponentially (GALE) stabilizable through state feedback if there exists a smooth feedback control law u=ξ(x)u=ξ(x), where ξ:ℝn→ℝmξ:R^n ^m is a smooth function satisfying ξ(0)=0ξ(0)=0, such that the closed-loop system: x˙=f(x,ξ(x))≜f¯(x), x=f(x,ξ(x)) f(x), exhibits global asymptotic stability and local exponential stability at the equilibrium point x=0x=0. 2.2 Communication graph Consider an undirected graph =(,ℰ)G=(V,E), where V represents a set of N nodes, and ℰ⊆×E V×V denotes the edges between them. For each node i, the neighborhood set is defined as i=j∈∣(i,j)∈ℰN_i=\j∈V (i,j)∈E\. The connectivity of the graph is represented by the adjacency matrix A, where aij>0a_ij>0 if nodes i and j are connected (i.e., (i,j)∈ℰ(i,j)∈E), and aij=0a_ij=0 otherwise. The degree matrix D is a diagonal matrix, with the i-th diagonal entry given by di=∑j≠iaijd_i= _j≠ ia_ij, representing the degree of node i. The Laplacian matrix ℒL of the graph is then defined as ℒ=−L=D-A. 2.3 Problem Description Consider a NNS of n nodes interacting over a graph =(,ℰ)G=(V,E). Each node i∈i has scalar state xi(t)∈ℝmx_i(t) ^m and the input is ui(t)∈ℝmu_i(t) ^m. The dynamics are governed by: x˙i(t)=fi(xi(t))+gi(xi(t))ui(t−d),i=1,…,n, x_i(t)=f_i(x_i(t))+g_i(x_i(t))u_i(t-d), i=1,…,n, (2) where d denotes input delay, smooth maps fi:ℝm→ℝmf_i:R^m ^m and gi:ℝm→ℝg_i:R^m are smooth, and in particular fi(m)=mf_i(0_m)=0_m. In what follows, every occurrence of a delayed argument y(t−d)y(t-d) is to be interpreted as y evaluated at a delayed time stamp, that is, y(t−d)y(t-d) denotes the most recent available sample of y at time t, where y may denote any signal constructed from the states and exchanged variables (e.g., xix_i, uiu_i). Then, each node i is endowed with a local cost function ϕi:ℝm→ℝ _i:R^m . Let x:=[x1,…,xn]T∈ℝnmx:=[x_1,…,x_n]^T ^nm and define the aggregate objective Φ(x):=∑i=1nϕi(xi) (x):= _i=1^n _i(x_i). The goal is to minimize the aggregate cost under consensus constraints: minx∈ℝnmΦ(x)=∑i=1nϕi(xi), _x ^nm (x)= _i=1^n _i(x_i), subject toxi=xj,∀i,j∈, to x_i=x_j, ∀ i,j∈V, (3) where the constraint xi=xjx_i=x_j, ∀i,j∈∀ i,j∈V. Denote the optimal set by: ∗:=argminx∈ℝnm,xi=xj,∀i,j∈Φ(x), ^*:= _x ^nm,x_i=x_j,∀ i,j∈V\ (x), assumed nonempty, and the optimal value by Φ∗:=minx∈ℝnm,xi=xjΦ(x)>−∞ ^*:= _x ^nm,\ x_i=x_j (x)>-∞. The control objective is to design distributed inputs uiu_i such that the NNS (2) drive the state x to the optimal consensual solution x∗∈∗x^* ^*, where x∗=[x1∗,…,xn∗]Tx^*=[x_1^*,…,x_n^*]^T and xi∗=xj∗x_i^*=x_j^*, ∀i,j∈∀ i,j∈V. Several needed assumptions are provided as follows. Assumption 1. For each node i, the scalar gain gi:ℝm→ℝg_i:R^m is C1C^1 and bounded away from zero. Namely, there exist constants 0<g¯i≤g¯i<∞0< g_i≤ g_i<∞ such that: g¯i≤|gi(x)|≤g¯i. g_i≤|g_i(x)|≤ g_i. Assumption 2. Each local cost function ϕi(x) _i(x) is twice continuously differentiable with respect to x and globally LϕiL_ _i-Lipschitz continuous gradient. For x1,x2∈ℝmx_1,x_2 ^m, the following holds: |∇ϕi(x1)−∇ϕi(x2)|≤Lϕi‖x1−x2‖.|∇ _i(x_1)-∇ _i(x_2)|≤ L_ _i||x_1-x_2||. Assumption 3. (Strong convexity condition) Each local cost function ϕi(x) _i(x) is strongly convex with parameter mϕi>0m_ _i>0. Specifically, for any x1,x2∈ℝmx_1,x_2 ^m and θ∈[0,1]θ∈[0,1], the following condition holds: ϕi(θx1+(1−θ)x2)≤ _i(θ x_1+(1-θ)x_2)≤ θϕi(x1)+(1−θ)ϕi(x2)−mϕi2θ(1−θ)‖x1−x2‖2. θ _i(x_1)+(1-θ) _i(x_2)- m_ _i2θ(1-θ)||x_1-x_2||^2. Building on the challenges outlined earlier, we now formalize the concept of input-delay tolerant semiglobal convergence to address distributed optimization problems under input delays. Definition 2. (Input-Delay Tolerant Semiglobal Convergence (IDTSC) in Distributed Optimization) Consider the distributed constrained optimization problem (2.3) implemented on a networked nonlinear system (2) with input delays. Let: ui(t)=ξi(xi(t)),u_i(t)= _i(x_i(t)), for each node i, where ξi:ℝm→ℝm _i:R^m ^m is a globally stabilizing control law. Denote by xi∗∈ℝmx_i^* ^m the optimal solution of the optimization problem (2.3). For any prescribed constant r>0r>0, define the initial condition ball: ℬr(xi∗)≜φ∈C([−d,0],ℝ):‖φ−xi∗‖≤r,B_r(x_i^*) \ ∈ C([-d,0],R):\| -x_i^*\|≤ r \, where the norm is given by ‖φ‖=sups∈[−d,0]‖φ(s)‖\| \|= _s∈[-d,0]\| (s)\|. We say that the closed-loop system exhibits input-delay tolerant semiglobal convergence (IDTSC) in distributed optimization if there exists a maximal allowable delay d¯=d¯(r)>0 d= d(r)>0 such that, for all delay values d∈(0,d¯]d∈(0, d], the following properties hold: 1. Semiglobal Attractivity: For every i∈i and any initial condition xi(s)∈ℬr(xi∗)x_i(s) _r(x_i^*), the trajectory xi(t)x_i(t) of the closed-loop system, x˙i(t)=fi(xi(t))+gi(xi(t))ξi(xi(t−d)), x_i(t)=f_i(x_i(t))+g_i(x_i(t))\, _i(x_i(t-d)), (4) satisfies limt→∞xi(t)=xi∗. _t→∞x_i(t)=x_i^*. 2. Local Stability: The closed-loop system is locally asymptotically stable at the optimal solution. To extend the scope of IDTSC beyond convex optimization, we refine the Assumption 3 on local cost functions to accommodate a broader class of nonconvex problems by employing the Polyak-Łojasiewicz (P-Ł) condition. Assumption 4. Each local cost function ϕi(xi) _i(x_i) satisfies radially unbounded and the P-Ł condition: 12‖∇ϕi(xi)‖2≥μϕi(ϕi(xi)−ϕi(xi∗)), 12\|∇ _i(x_i)\|^2≥ _ _i( _i(x_i)- _i(x_i^*)), (5) where μϕi _ _i is a positive constant, and xi∗x_i^* is an optimal solution of the optimization problem (2.3). Remark 1. The P-Ł condition in Assumption 4 provides a less restrictive alternative to strong convexity. It relaxes the requirements compared to assumptions like essential strong convexity [32], weak strong convexity [33], or the restricted secant inequality [34]. This flexibility allows the proposed framework to handle a broader class of nonconvex optimization problems [36, 37], significantly expanding its practical applicability. 2.4 Preliminary lemmas To support the theoretical analysis of the proposed algorithm, we summarize several essential lemmas that establish foundational properties and facilitate the stability and convergence proofs. Lemma 1. [35] (Cauchy-Schwarz inequality) Let x=[x1,x2,…,xn]Tx=[x_1,x_2,…,x_n]^T and y=[y1,y2,…,yn]Ty=[y_1,y_2,…,y_n]^T be vectors in ℝnR^n. Then, the following inequality holds: ∑i=1n|xi||yi|≤‖x‖‖y‖, _i=1^n|x_i||y_i|≤\|x\|\|y\|, where ‖x‖=(∑i=1n|xi|2)1/2\|x\|= ( _i=1^n|x_i|^2 )^1/2 and ‖y‖=(∑i=1n|yi|2)1/2\|y\|= ( _i=1^n|y_i|^2 )^1/2 are the Euclidean norms of x and y, respectively. Lemma 2. [21] Let f:ℝn→ℝnf:R^n ^n be a smooth mapping with f(0)=0f(0)=0. Then, there exists a smooth, non-negative scalar function Ψ:ℝn×ℝn→ℝ :R^n×R^n such that for all x,y∈ℝnx,y ^n, ‖f(x)−f(y)‖≤Ψ(x,y)‖x−y‖.\|f(x)-f(y)\|≤ (x,y)\|x-y\|. Lemma 3. [21] Let f:ℝn×ℝm→ℝf:R^n×R^m be a smooth function. Then, there exist smooth scalar functions a(x)≥0a(x)≥ 0, b(y)≥0b(y)≥ 0, c(x)≥1c(x)≥ 1, and d(y)≥1d(y)≥ 1 for all x∈ℝnx ^n and y∈ℝmy ^m, such that |f(x,y)|≤a(x)+b(y)and|f(x,y)|≤c(x)d(y).|f(x,y)|≤ a(x)+b(y) and |f(x,y)|≤ c(x)d(y). Lemma 4. [36, 7, 37] Given Assumption 4 (P-Ł condition), the distributed optimization problem (2.3) associated with the NNS (2) has at least one optimal solution x∗x^*. At this optimal point, the gradient vector field satisfies ∇Φ(x∗)=0∇ (x^*)=0. Lemma 5. [38] (Razumikhin theorem) Consider a nonlinear system with time-delay described by: x˙(t)=f(xt), x(t)=f(x_t), where f:C([−d,0],ℝn)→ℝnf:C([-d,0],R^n) ^n is a locally Lipschitz continuous function with f(0)=0f(0)=0, and the initial condition x(θ)=φ(θ)x(θ)= (θ) is continuous for θ∈[−d,0]θ∈[-d,0]. The system is globally asymptotically stable if there exist: 1. A first-order continuously derivable (i.e., C1C^1) function V:ℝn→ℝV:R^n . 2. A continuous (i.e., C0C^0) function W:ℝn→ℝW:R^n , both positive definite and proper. 3. A C0C^0 non-decreasing function p(s)p(s) such that p(s)>sp(s)>s for s>0s>0. such that for all θ∈[−d,0]θ∈[-d,0], the following condition holds: V˙(x(t))≤−W(x(t))wheneverV(x(t+θ))≤p(V(x(t))). V(x(t))≤-W(x(t)) V(x(t+θ))≤ p(V(x(t))). Lemma 6. [39] (Input-to-State Stability) Consider the nonlinear dynamical system x˙(t)=f(x(t),u(t)), x(t)=f(x(t),u(t)), where x(t)∈ℝnx(t) ^n denotes the state vector, u(t)∈ℝmu(t) ^m represents the input vector, and the function f:ℝn×ℝm→ℝnf:R^n×R^m ^n is locally Lipschitz continuous in its arguments. Assume the existence of a continuously differentiable Lyapunov function V:ℝn→ℝV:R^n , a class K function γ, and a positive constant α>0α>0 such that for all x∈ℝnx ^n and u∈ℝmu ^m, the following conditions are satisfied: a) α1(‖x‖)≤V(x)≤α2(‖x‖) _1(\|x\|)≤ V(x)≤ _2(\|x\|), where α1,α2∈∞ _1, _2 _∞. b) ∇V(x)⋅f(x,u)≤−αV(x)+γ(‖u‖)∇ V(x)· f(x,u)≤-α V(x)+γ(\|u\|). Under these conditions, the system is input-to-state stable (ISS) with respect to the input u(t)u(t). Specifically, there exist functions β∈ℒβ and γ~∈ γ such that for any initial state x(0)x(0) and input u(t)u(t), the following bound holds for all t≥0t≥ 0: ‖x(t)‖≤β(‖x(0)‖,t)+γ~(sup0≤τ≤t‖u(τ)‖),∀t≥0. \|x(t)\|≤β(\|x(0)\|,t)+ γ ( _0≤τ≤ t\|u(τ)\| ), ∀ t≥ 0. (6) 3 Main Results This section presents the main contributions of the paper, focusing on the development of a novel SIDT algorithm. The algorithm addresses both distributed convex and nonconvex optimization problems (2.3) in NNS (2) with input delays. 3.1 SIDT algorithm design To achieve IDTSC, we propose a novel SIDT algorithm designed to solve distributed optimization problems (2.3) in delay-affected NNSs (2). The control input for each node is formulated as: ui(t−d)= u_i(t-d)= gi(xi(t−d))−1(u¯i(t−d)−fi(xi(t−d))), g_i(x_i(t-d))^-1( u_i(t-d)-f_i(x_i(t-d))), (7) where the system terms fif_i and gig_i are known, and u¯i(t−d) u_i(t-d) is an auxiliary control term given by: u¯i(t−d)= u_i(t-d)= ϑ∑j=1naij(xj(t−d)−xi(t−d)) _j=1^na_ij(x_j(t-d)-x_i(t-d)) +ε∑j∈i[℘ij(t−d)⋅sign(xj(t−d)−xi(t−d))]+u¯η,i(t−d), + _j∈N_i [ _ij(t-d)· sign(x_j(t-d)-x_i(t-d)) ]+ u_η,i(t-d), and u¯η,i(t−d)=−k0ηi(xi(t−d)). u_η,i(t-d)=-k_0 _i(x_i(t-d)). The terms are specified as follows: ϑ>0 >0 is a gain parameter, ℘ij(t−d)=‖u¯η,i(t−d)‖+‖u¯η,j(t−d)‖ _ij(t-d)=\| u_η,i(t-d)\|+\| u_η,j(t-d)\|, ηi(xi(t−d))=∇ϕi(xi(t−d)) _i(x_i(t-d))=∇ _i(x_i(t-d)), k0>0k_0>0 is a scaling parameter for the gradient-based term, and ε is a control gain defined later. In scenarios where there are no input delays d=0d=0, the SIDT algorithm (7) simplifies to: ui(t)=gi(xi(t))−1(−fi(xi(t))+u¯i(t)), u_i(t)=g_i(x_i(t))^-1(-f_i(x_i(t))+ u_i(t)), (8) where u¯i(t)= u_i(t)= ϑ∑j=1naij(xj(t)−xi(t))+ε∑j∈i[℘ij(t)⋅sign(xj(t)−xi(t))]+u¯η,i, _j=1^na_ij(x_j(t)-x_i(t))+ _j∈N_i [ _ij(t)· sign(x_j(t)-x_i(t)) ]+ u_η,i, u¯η,i= u_η,i= −k0ηi(xi(t)), -k_0 _i(x_i(t)), and ηi(xi(t))= _i(x_i(t))= ∇ϕi(xi(t)). ∇ _i(x_i(t)). 3.2 SIDT algorithm for distributed convex optimization problem In this subsection, we analyze the application of the SIDT algorithm (7) to solve the distributed convex optimization problem (2.3) in NNSs (2) with input delays. The analysis leads to the following result: Theorem 1. Assuming Assumptions 1-3 are satisfied, if ε≥2n ≥ 2n holds, then the application of the SIDT algorithm (7) to the distributed optimization problem (2.3) of the NNS (2) with input delays guarantees that the system exhibits practical IDTSC. Proof. The proof proceeds in two main steps. Step 1: Establish that the NNS (2) without the input delay d, utilizing the SIDT algorithm (8), achieves global asymptotic and local exponential stability at the optimal solution of the distributed optimization problem (2.3). Step 2: Demonstrate that the SIDT algorithm (7) ensures practical IDTSC in the presence of input delay d. Step 1 (Stability without input delay): Consider the NNS (2) without input delay, formulated as: x˙i(t)=fi(xi(t))+gi(xi(t))ui(t),i=1,…,n. x_i(t)=f_i(x_i(t))+g_i(x_i(t))u_i(t), i=1,…,n. (9) Substituting the control input ui(t)u_i(t) from (8) into (9): x˙i(t)=fi(xi(t))+gi(xi(t))ξi(xi(t)). x_i(t)=f_i(x_i(t))+g_i(x_i(t)) _i(x_i(t)). (10) Construct the following Lyapunov function candidate as follows: Vpre=12∑i=1nexiTexi, V_pre= 12 _i=1^ne_x_i^Te_x_i, (11) where exi=xi−1n∑j=1nxje_x_i=x_i- 1n _j=1^nx_j. The derivative of (11) along the trajectories of (10) yields: V˙pre= V_pre= ∑i=1nexiT⋅e˙xi _i=1^ne_x_i^T· e_x_i = = ∑i=1nexiT(x˙i−1n∑j=1nx˙j). _i=1^ne_x_i^T( x_i- 1n _j=1^n x_j). (12) Since ∑i=1nexi=0 _i=1^ne_x_i=0, it follows that: −1n∑i=1nexi⋅(∑j=1nx˙j)=0,- 1n _i=1^ne_x_i·( _j=1^n x_j)=0, implying that: V˙pre= V_pre= ∑i=1nexiTx˙i _i=1^ne_x_i^T x_i = = −ϑ∑i=1n∑j=1nexiTaij(xi(t)−xj(t))−ε∑i=1n∑j∈iexiT℘ij(t)sign(xi(t)−xj(t)) - _i=1^n _j=1^ne_x_i^Ta_ij(x_i(t)-x_j(t))- _i=1^n _j∈N_ie_x_i^T _ij(t) sign(x_i(t)-x_j(t)) +∑i=1nexiTu¯η,i. + _i=1^ne_x_i^T u_η,i. (13) To analyze the term I1=−ϑ∑i=1nexiT∑j=1naij(xi(t)−xj(t))I_1=- _i=1^ne_x_i^T _j=1^na_ij(x_i(t)-x_j(t)) in (3.2), one has: I1= I_1= −ϑ2∑i=1n∑j=1naijexiT(xi−xj)−ϑ2∑i=1n∑j=1naijexjT(xj−xi) - 2 _i=1^n _j=1^na_ije_x_i^T(x_i-x_j)- 2 _i=1^n _j=1^na_ije_x_j^T(x_j-x_i) = = −ϑ2∑i=1n∑j=1naij(xi−xj)T(xi−xj) - 2 _i=1^n _j=1^na_ij(x_i-x_j)^T(x_i-x_j) = = −ϑexTℒ¯ex - e_x^T Le_x ≤ ≤ −2ϑλ2(ℒ¯)Vpre, -2 _2( L)V_pre, (14) where ex=[ex1,ex2,…,exn]Te_x=[e_x_1,e_x_2,…,e_x_n]^T, ℒ¯=ℒ⊗Im L=L _m, and λ2(ℒ¯) _2( L) denotes the smallest non-zero eigenvalue of ℒ¯ L. For I2=−ε∑i=1n∑j∈iexiT℘ij(t)(xi(t)−xj(t))I_2=- _i=1^n _j∈N_ie_x_i^T _ij(t)(x_i(t)-x_j(t)), it yields that: I2= I_2= −ε2∑i=1n∑j∈iexiT℘ijsign(xi−xj)−ε2∑i=1n∑j∈iexjT℘jisign(xj−xi) - 2 _i=1^n _j∈N_ie_x_i^T _ij sign(x_i-x_j)- 2 _i=1^n _j∈N_ie_x_j^T _ji sign(x_j-x_i) = = −ε2∑i=1n∑j∈i℘ij(xi−xj)Tsign(xi−xj) - 2 _i=1^n _j∈N_i _ij(x_i-x_j)^T sign(x_i-x_j) ≤ ≤ −ε2∑i=1n∑j∈i‖xi−xj‖‖u¯η,i‖. - 2 _i=1^n _j∈N_i\|x_i-x_j\|\| u_η,i\|. (15) To analyze the term I3=−∑i=1nexiTu¯η,iI_3=- _i=1^ne_x_i^T u_η,i, note that: maxi,j∈‖xi(t)−xj(t)‖≤∑i=1n∑j∈i‖xi−xj‖, _i,j∈V\|x_i(t)-x_j(t)\|≤ _i=1^n _j∈N_i\|x_i-x_j\|, which implies that: I3≤ I_3≤ ∑i=1n‖exi‖‖u¯η,i‖ _i=1^n\|e_x_i\|\| u_η,i\| ≤ ≤ ∑i=1n(∑i=1n∑j∈i‖xi−xj‖)‖u¯η,i‖ _i=1^n ( _i=1^n _j∈N_i\|x_i-x_j\| )\| u_η,i\| ≤ ≤ n∑i=1n∑j∈i‖xi−xj‖‖u¯η,i‖. n _i=1^n _j∈N_i\|x_i-x_j\|\| u_η,i\|. (16) According to the condition ε≥2n ≥ 2n, summing up equations (3.2)-(3.2) and substituting them into (3.2) yields: V˙pre≤ V_pre≤ −2ϑλ2(ℒ¯)Vpre. -2 _2( L)V_pre. (17) From (17), together with the quadratic bounds c1‖ex(t)‖2≤Vpre(t)≤c2‖ex(t)‖2c_1\|e_x(t)\|^2≤ V_pre(t)≤ c_2\|e_x(t)\|^2 (for some c1c_1, c2>0c_2>0), we obtain that exie_x_i globally converges exponentially to origin, namely, limt→∞‖xi(t)−1n∑j=1nxj‖=0 _t→∞\|x_i(t)- 1n _j=1^nx_j\|=0, ∀i∈∀ i∈V. Since limt→∞‖xi(t)−1n∑j=1nxj‖=0 _t→∞\|x_i(t)- 1n _j=1^nx_j\|=0, ∀i∈∀ i∈V, it follows that: xi(t)=xj(t),∀i,j∈,as t→∞.x_i(t)=x_j(t), ∀ i,j∈V, ~t→∞. (18) Equivalently, the state converges to the consensus subspace: :=x∈ℝnm:xi=xj,∀i,j∈.C:=\x ^nm:x_i=x_j,∀ i,j \. Let ex(t)=Px(t)e_x(t)=Px(t), where P=Pn⊗ImP=P_n _m and Pn=In−1nnnTP_n=I_n- 1n1_n1_n^T. Note that the disagreement dynamics evolve in the orthogonal complement of spann⊗ℝmspan\1_n\ ^m. In particular, P(n⊗y)=0,∀y∈ℝm,(nT⊗Im)(ℒ⊗Im)=0,P(1_n y)=0,~∀ y ^m, (1_n^T _m)(L _m)=0, so any auxiliary input that acts through the consensus channel (i.e., takes values in spann⊗ℝmspan\1_n\ ^m, including the term u¯η,i(t−d) u_η,i(t-d) as designed in (7)-(8)) is annihilated by P and does not enter the exe_x-subsystem. Therefore, (17) guarantees exponential decay of the disagreement independently of the specific variation of u¯η,i(t−d) u_η,i(t-d), and the closed-loop trajectories necessarily enter (and remain in) C. Then, select a new Lyapunov function candidate as follows: V1=Φ(x(t))−Φ(x∗(t)), V_1= (x(t))- (x^*(t)), (19) where x∗(t)∈ℝnmx^*(t) ^nm is the optimal solution of the distributed optimization problem (2.3). According to Assumption 2, the aggregate cost function Φ(x(t)) (x(t)) is LΦL_ -Lipschitz continuous in its gradient: ‖∇Φ(x)−∇Φ(y)‖≤LΦ‖x−y‖,∀x,y∈ℝnm, \|∇ (x)-∇ (y)\|≤ L_ \|x-y\|, ∀ x,y ^nm, (20) where LΦ=maxi=1,…,nLϕiL_ = _i=1,…,n\L_ _i\, ∀i∈∀ i∈V. Using the first-order Taylor expansion for Φ : Φ(y)≤ (y)≤ Φ(x)+∇Φ(x)T(y−x)+∫01(∇Φ(x+t(y−x))−∇Φ(x))T(y−x)t. (x)+∇ (x)^T(y-x)+ ^1_0(∇ (x+t(y-x))-∇ (x))^T(y-x)dt. (21) The LΦL_ -Lipschitz continuity of ∇Φ(x)∇ (x ) (20) allows us to bound the integral term: ∫01‖∇Φ(x+t(y−x))−∇Φ(x)‖t≤ ^1_0\|∇ (x+t(y-x))-∇ (x)\|dt≤ ∫01LΦt‖y−x‖t ^1_0L_ t\|y-x\|dt = = LΦ2‖y−x‖2. L_ 2\|y-x\|^2. (22) Substituting back, we get: ∫01(∇Φ(x+t(y−x))−∇Φ(x))T(y−x)t≤ ^1_0(∇ (x+t(y-x))-∇ (x))^T(y-x)dt≤ LΦ2‖y−x‖2. L_ 2\|y-x\|^2. (23) Substituting (23) into (21), it obtains that: Φ(y)≤ (y)≤ Φ(x)+∇Φ(x)T(y−x)+LΦ2‖y−x‖2. (x)+∇ (x)^T(y-x)+ L_ 2\|y-x\|^2. (24) Setting x=x∗x=x^* and noting ∇Φ(x∗)=0∇ (x^*)=0, we get: Φ(x)≤ (x)≤ Φ(x∗)+LΦ2‖x−x∗‖2. (x^*)+ L_ 2\|x-x^*\|^2. (25) This provides an upper bound: Vnd(t)=Φ(x(t))−Φ(x∗(t))≤ V_nd(t)= (x(t))- (x^*(t))≤ LΦ2‖x−x∗‖2. L_ 2\|x-x^*\|^2. (26) Define an auxiliary function as: λ(θ)=Φ((1−θ)x+θy),∀x,y∈ℝnm, λ(θ)= ((1-θ)x+θ y), ∀ x,y ^nm, (27) where θ is defined in Assumption 3, and λ(θ)λ(θ) is differentiable on the interval θ∈[0,1]θ∈[0,1]. According to Assumption 3: λ(θ)≤(1−θ)Φ(x)+θΦ(y)−mΦ2θ(1−θ)‖y−x‖2. λ(θ)≤(1-θ) (x)+θ (y)- m_ 2θ(1-θ)\|y-x\|^2. (28) Take the derivative of left-hand sides of (28) with respect to θ: λ′(θ)=∇Φ((1−θ)x+θy)T(y−x). λ (θ)=∇ ((1-θ)x+θ y)^T(y-x). (29) At θ=0θ=0, we have: λ′(0)=∇Φ(x)T(y−x). λ (0)=∇ (x)^T(y-x). (30) For the right-hand side of (28), taking the derivative gives: λr′(θ)= λ _r(θ)= −Φ(x)+Φ(y)−mΦ2[(1−2θ)‖y−x‖2−2θ(1−θ)‖y−x‖2], - (x)+ (y)- m_ 2[(1-2θ)\|y-x\|^2-2θ(1-θ)\|y-x\|^2], where λr(θ)=(1−θ)Φ(x)+θΦ(y)−[mΦθ(1−θ)‖y−x‖2]/2 _r(θ)=(1-θ) (x)+θ (y)-[m_ θ(1-θ)\|y-x\|^2]/2. At θ=0θ=0, this becomes: λr′(0)= λ _r(0)= −Φ(x)+Φ(y)−mΦ2‖y−x‖2. - (x)+ (y)- m_ 2\|y-x\|^2. (31) Since λ(θ)≤λr(θ)λ(θ)≤ _r(θ), at θ=0θ=0, it follows: λ′(0)≤λr′(0)λ (0)≤λ _r(0). By substituting (30) and (31), we have: ∇Φ(x)T(y−x)≤−Φ(x)+Φ(y)−mΦ2‖y−x‖2, ∇ (x)^T(y-x)≤- (x)+ (y)- m_ 2\|y-x\|^2, which can further derives that: Φ(y)≥Φ(x)+∇Φ(x)T(y−x)+mΦ2‖y−x‖2. (y)≥ (x)+∇ (x)^T(y-x)+ m_ 2\|y-x\|^2. (32) Similarly to (24)-(25), setting x=x∗x=x^* in equation (32), the inequality transforms into the following form: Φ(x)≥ (x)≥ Φ(x∗)+mΦ2‖x−x∗‖2. (x^*)+ m_ 2\|x-x^*\|^2. (33) By combining this result (33) with equation (26): mΦ2‖x−x∗‖2≤V1(t)≤LΦ2‖x−x∗‖2. m_ 2\|x-x^*\|^2≤ V_1(t)≤ L_ 2\|x-x^*\|^2. (34) Let: si=ϑ∑j=1naij(xj(t)−xi(t))+ε∑j∈i[℘ij(t)sign(xj(t)−xi(t))].s_i= _j=1^na_ij(x_j(t)-x_i(t))+ _j∈N_i[ _ij(t) sign(x_j(t)-x_i(t))]. The derivative of (19) along the trajectories of (10) is given by: V˙1 V_1 =dt(Φ(x(t))−Φ(x∗(t))) = ddt( (x(t))- (x^*(t))) =−k0∑i=1nηiT(xi(t))ηi(xi(t))+∑i=1nηiT(xi(t))si(t) =-k_0 _i=1^n _i^T(x_i(t)) _i(x_i(t))+ _i=1^n _i^T(x_i(t))s_i(t) ≤−k0(∇TΦ(x)∇Φ(x))+∑i=1nηiT(xi(t))si(t). ≤-k_0(∇^T (x)∇ (x))+ _i=1^n _i^T(x_i(t))s_i(t). (35) Based on Assumption 3 and (32), setting y=x∗y=x^*, we have: Φ(x)−Φ(x∗)≥∇Φ(x)T(x−x∗)+mΦ2‖x−x∗‖2. (x)- (x^*)≥∇ (x)^T(x-x^*)+ m_ 2\|x-x^*\|^2. (36) By discarding the non-negative term mΦ‖x−x∗‖2/2m_ \|x-x^*\|^2/2, this reduces to: Φ(x)−Φ(x∗)≥∇Φ(x)T(x−x∗). (x)- (x^*)≥∇ (x)^T(x-x^*). (37) Using the Cauchy-Schwarz inequality, (37) can be further derived as: Φ(x)−Φ(x∗)≥−‖∇Φ(x)‖‖x−x∗‖. (x)- (x^*)≥-\|∇ (x)\|\|x-x^*\|. Taking the derivative of both sides of equation (33), we get: ‖∇Φ(x)‖≥mΦ‖x−x∗‖. \|∇ (x)\|≥ m_ \|x-x^*\|. (38) Squaring both sides, it yields: ‖∇Φ(x)‖2≥mΦ2‖x−x∗‖2.\|∇ (x)\|^2≥ m_ ^2\|x-x^*\|^2. Substituting ‖x−x∗‖2≥2(Φ(x)−Φ(x∗))/LΦ\|x-x^*\|^2≥2( (x)- (x^*))/L_ from (25), one has: ∇TΦ(x)∇Φ(x)≥κΦ(Φ(x)−Φ(x∗)), ∇^T (x)∇ (x)≥ _ ( (x)- (x^*)), where κΦ=2mΦ2/LΦ _ =2m_ ^2/L_ . Substituting into equation (37) obtains: V˙1 V_1 ≤−k0κΦ(Φ(x)−Φ(x∗))+∑i=1nηi(xi(t))si(t) ≤-k_0 _ ( (x)- (x^*))+ _i=1^n _i(x_i(t))s_i(t) ≤−βΦV1+∑i=1nηiT(xi(t))si(t), ≤- _ V_1+ _i=1^n _i^T(x_i(t))s_i(t), (39) where βΦ=k0κΦ>0 _ =k_0 _ >0. Since exie_x_i is globally exponential convergence, si(t)s_i(t) also converges exponentially, namely, ‖s(t)‖≤‖s(0)‖e−αst\|s(t)\|≤\|s(0)\|e^- _st, with s(t)=[s1(t),s2(t),…,sn(t)]Ts(t)=[s_1(t),s_2(t),…,s_n(t)]^T and αs=2ϑ1λ2(ℒ¯) _s=2 _1 _2( L). Based on Assumption 2, there exist a positive constant η¯ η satisfying that η¯≥‖∇Φ(x)‖ η≥\|∇ (x)\|. Using the Cauchy-Schwarz inequality, ∑i=1nηiT(xi(t))si(t)≤ _i=1^n _i^T(x_i(t))s_i(t)≤ ‖∇Φ(x)‖‖s(t)‖≤η¯‖s(t)‖. \|∇ (x)\|\|s(t)\|≤ η\|s(t)\|. (40) Substituting (40) into (3.2) obtains: V˙1≤ V_1≤ −βΦV1+η¯‖s(t)‖ - _ V_1+ η\|s(t)\| = = −βΦV1+γs(‖s(t)‖), - _ V_1+ _s(\|s(t)\|), (41) where γs(‖s(t)‖)=η¯‖s(t)‖ _s(\|s(t)\|)= η\|s(t)\| is obviously a class K function. From Lemma 6, and the form of (34) and (3.2), it can be concluded that the system (10) is ISS with respect to input s(t)s(t). Using the ISS representation: ‖x(t)−x∗‖≤βISS(‖x(0)−x∗‖,t)+γs(sup0≤τ≤t‖s(τ)‖), \|x(t)-x^*\|≤ _ISS(\|x(0)-x^*\|,t)+ _s( _0≤τ≤ t\|s(τ)\|), and substituting ‖s(t)‖≤‖s(0)‖e−αst\|s(t)\|≤\|s(0)\|e^- _st, ∀t≥0∀ t≥ 0, we have: ‖x(t)−x∗‖≤βISS(‖x(0)−x∗‖,t)+γs(‖s(0)‖e−αst). \|x(t)-x^*\|≤ _ISS(\|x(0)-x^*\|,t)+ _s(\|s(0)\|e^- _st). Due to βISS∈ℒ _ISS , it satisfies that βISS(σ,t)→0 _ISS(σ,t)→ 0 when t→∞t→∞ for all σ>0σ>0. Additionally, γs(‖s(0)‖e−αst)→0 _s(\|s(0)\|e^- _st)→ 0 with t→∞t→∞. Thus, it demonstrated that limt→∞‖x(t)−x∗‖=0 _t→∞\|x(t)-x^*\|=0 as well as the positive definiteness and boundedness of V1V_1, indicating that the system is globally asymptotically stable on x∗x^*. In the neighborhood of x∗x^*, the influence of s(t)s(t) for x(t)x(t) tends to 0, implying that the input item s(t)s(t) can be ignored. Therefore, it yields: V˙1≤ V_1≤ −βΦV1. - _ V_1. (42) Using (42) and (34), it follows that x∗x^*, x(t)x(t) exponentially converge to x∗x^* locally. Thus, combining the analysis of (3.2)-(3.2) and (42), the closed-loop system (10) is GALE stability at the optimal solution point x∗x^*. Step 2 (Stability with input delay): Before proceeding, we clarify how the discontinuous sign-based channel is treated in the delay analysis. The SIDT algorithm (7) contains the ideal sign function, and hence the resulting closed-loop vector field is generally not locally Lipschitz on the switching surfaces. Therefore, the following proof does not invoke the classical Razumikhin theorem by requiring the whole closed-loop functional to be locally Lipschitz. Instead, we use a Razumikhin-type comparison argument together with a decomposition of the feedback into a locally Lipschitz part and a uniformly bounded discontinuous part. More precisely, the delayed feedback mismatch is estimated by a Lipschitz term associated with the smooth channel and a bounded residual term induced by the sign-based channel. Building on the results of Lemma 8.1 in [21], we establish that there exists a threshold d¯c1 d_c1 which depends on the initial domain radius r. Specifically, for any delay d∈(0,d¯c1]d∈(0, d_c1], all solution trajectories of the time-delay nonlinear system (2), initialized with xi(s)x_i(s) satisfying sups∈[−d,0]‖xi(s)‖≤r _s∈[-d,0]\|x_i(s)\|≤ r, remain well-defined and do not experience finite escape time over the interval [0,d][0,d]. Furthermore, these trajectories are bounded by 2r2r, where r is an arbitrarily large positive real number. Using the Cauchy-Schwarz inequality, consider a Lyapunov function candidate V1V_1 as (19). Define the level set Ω0,V1=x∈ℝn|V1(x)≤r0,V1 _0,V_1=\x ^n|V_1(x)≤ r_0,V_1\, where r0,V1r_0,V_1 is a positive constant satisfying: r0,V1=LΦ2max(∥2nr−x∗∥,∥−2nr−x∗∥)2.r_0,V_1= L_ 2 (\|2 nr-x^*\|,\|-2 nr-x^*\|)^2. This result suggests that, for sufficiently small delays d>0d>0, the system’s state xi(t)x_i(t), ∀t∈[0,d]∀ t∈[0,d] remains bounded and the solutions do not diverge within this interval. Next, we extend the analysis to the time domain t∈[d,+∞)t∈[d,+∞). By leveraging the dynamics of the closed-loop system: x˙i(t)=fi(xi(t))+gi(xi(t))ξi(xi(t−d)), x_i(t)=f_i(x_i(t))+g_i(x_i(t)) _i(x_i(t-d)), (43) it can be deduced from (3.2), the mean value theorem and Lemma 1 that for t>dt>d, we derive that: V˙1≤ V_1≤ −a1‖x−x∗‖2+‖∂V1∂x‖∑i=1n‖gi(xi)‖‖ξi(xi(t−d))−ξi(xi)‖ -a_1\|x-x^*\|^2+ \| ∂ V_1∂ x \| _i=1^n\|g_i(x_i)\|\| _i(x_i(t-d))- _i(x_i)\| ≤ ≤ −a1‖x−x∗‖2+‖∂V1∂x‖‖g(x)‖‖ξ(x(t−d))−ξ(x)‖, -a_1\|x-x^*\|^2+ \| ∂ V_1∂ x \|\|g(x)\|\|ξ(x(t-d))-ξ(x)\|, (44) where a1=βΦmΦ/2a_1= _ m_ /2, g(x)=[g1(x1),g2(x2),…,gn(xn)]Tg(x)=[g_1(x_1),g_2(x_2),…,g_n(x_n)]^T, and ξ=[ξ1,ξ2,…,ξn]Tξ=[ _1, _2,…, _n]^T. Define an auxiliary error as ei=xi−xi∗e_i=x_i-x_i^* and an auxiliary vector e=[e1,e2,…,en]Te=[e_1,e_2,…,e_n]^T. Split the control law as ξ(x)=ξsm(x)+ξsgn(x)ξ(x)=ξ^sm(x)+ξ^sgn(x), where ξsm(x)ξ^sm(x) collects the smooth feedback terms and ξsgn(x)ξ^sgn(x) denotes the sign-based consensus-enhancing term. On the compact set Ω0,V1 _0,V_1, the smooth component ξsmξ^sm is locally Lipschitz, while the sign-based component ξsgnξ^sgn is uniformly bounded but may be discontinuous on the switching surfaces. Therefore, we do not impose a Lipschitz estimate on ξsgnξ^sgn. Instead, its possible jump is absorbed into a bounded residual. Since the solution trajectories remain in the compact set Ω0,V1 _0,V_1, there exist positive constants Lξ,rL_ξ,r, L℘,rL_ ,r, g¯r g_r, and ℘¯r _r such that ‖gi(xi)−1‖≤g¯r\|g_i(x_i)^-1\|≤ g_r, |℘ij(x)−℘ij(y)|≤L℘,r‖x−y‖| _ij(x)- _ij(y)|≤ L_ ,r\|x-y\| and ℘ij(x)≤℘¯r _ij(x)≤ _r, for all x,y∈Ω0,V1x,y∈ _0,V_1. From Lemmas 2 and 3, the local Lipschitz property of ξsmξ^sm, and the uniform boundedness of ξsgnξ^sgn on Ω0,V1 _0,V_1, there exist a smooth function β1(x(t−d))≥1 _1(x(t-d))≥ 1 and constants b1,b2>0b_1,b_2>0, independent of d, such that: ‖ξ(x(t−d))−ξ(x)‖≤ \|ξ(x(t-d))-ξ(x)\|≤ b1β1(x(t−d))‖e(t−d)−e(t)‖+b2. b_1 _1(x(t-d))\|e(t-d)-e(t)\|+b_2. (45) where b1=Lξ,r+εg¯rΔGL℘,rb_1=L_ξ,r+ g_r _GL_ ,r and b2=2εg¯rΔG℘¯rb_2=2 g_r _G _r with ΔG=maxi|i| _G= _i|N_i|. From (34) and (3.2), it follows that, for some a0>0a_0>0, ‖∂V1∂x‖|Ω0,V1≤a0‖e‖. . \| ∂ V_1∂ x \| |_ _0,V_1≤ a_0\|e\|. (46) Assume there exists a positive function β2(x)≥1 _2(x)≥ 1 satisfying g(x)≤β2(x)g(x)≤ _2(x). Thus, based on (3.2)-(46), for t≥dt≥ d, it obtains: V˙1(x)|Ω0,V1≤ . V_1(x) |_ _0,V_1≤ −a1‖e‖2+a0β2(x)‖e‖(b1β1(x(t−d))‖e(t−d)−e(t)‖+b2) -a_1\|e\|^2+a_0 _2(x)\|e\|(b_1 _1(x(t-d))\|e(t-d)-e(t)\|+b_2) ≤ ≤ −a12‖e‖2+a¯2β12(x(t−d))‖e(t−d)−e(t)‖2+a¯3, - a_12\|e\|^2+ a_2 _1^2(x(t-d))\|e(t-d)-e(t)\|^2+ a_3, (47) for some a¯2,a¯3>0 a_2, a_3>0 independent of the time delay d. Here, a¯3 a_3 is the residual term induced by the discontinuous sign-based channel and the boundedness of β2(x) _2(x) on Ω0,V1 _0,V_1. Let ℓi(xi(t),ui(t−d))=fi(xi(t))+gi(xi(t))ui(t−d) _i(x_i(t),u_i(t-d))=f_i(x_i(t))+g_i(x_i(t))u_i(t-d). Due to the smoothness of fi(xi(t))f_i(x_i(t)) and gi(xi(t))g_i(x_i(t)) with fi(m)=mf_i(0_m)=0_m, one has ℓi(m,m)=m _i(0_m,0_m)=0_m. Based on Lemmas 2 and 3, it implies that there exist ϖ1(x(τ))≥1 _1(x(τ))≥ 1, ϖ2(x(τ−d))≥1 _2(x(τ-d))≥ 1, and a constant ϖ0>0 _0>0, such that: ‖e(t−d)−e(t)‖2|Ω0,V1≤ .\|e(t-d)-e(t)\|^2 |_ _0,V_1≤ d∫t−d‖e˙(τ)‖2τ d _t^t-d\| e(τ)\|^2dτ ≤ ≤ d∫t−dt(ϖ1(x(τ))∥e(τ)∥+ϖ2(x(τ−d))∥e(τ−d)∥ d _t-d^t( _1(x(τ))\|e(τ)\|+ _2(x(τ-d))\|e(τ-d)\| +ϖ0)2dτ + _0)^2dτ ≤ ≤ 3d∫t−dt(ϖ12(x(τ))∥e(τ)∥2+ϖ22(x(τ−d))∥e(τ−d)∥2 3d _t-d^t( _1^2(x(τ))\|e(τ)\|^2+ _2^2(x(τ-d))\|e(τ-d)\|^2 +ϖ02)dτ + _0^2)dτ ≤ ≤ 3d2supℏ∈[−2d,0]ϖ3(x(t+ℏ))e(t+ℏ)∥2+3ϖ02d2, 3d^2 _ ∈[-2d,0] _3(x(t+ ))e(t+ )\|^2+3 _0^2d^2, (48) where ℓ=[ℓ1,ℓ2,…,ℓn]T =[ _1, _2,…, _n]^T, and ϖ3(∙)=ϖ12(∙)+ϖ22(∙) _3( )= _1^2( )+ _2^2( ) is a smooth scalar function. Substituting (3.2) into (3.2) obtains that, for all t∈[d,+∞)t∈[d,+∞), V˙1(x)|Ω0,V1≤ . V_1(x) |_ _0,V_1≤ −a12‖e‖2+3d2a¯2β12(x(t−d))supℏ∈[−2d,0]ϖ3(x(t+ℏ))‖e(t+ℏ)‖2 - a_12\|e\|^2+3d^2 a_2 _1^2(x(t-d)) _ ∈[-2d,0] _3(x(t+ ))\|e(t+ )\|^2 +a¯3+3a¯2β12(x(t−d))ϖ02d2. + a_3+3 a_2 _1^2(x(t-d)) _0^2d^2. (49) Next, choose p(s)=3sp(s)=3s, which satisfies p(s)>sp(s)>s for all s>0s>0. In the following Razumikhin-type comparison, suppose that, for all ℏ∈[−2d,0] ∈[-2d,0], V1(x(t+ℏ))<p[V1(x(t))]=3V1(x(t)). V_1(x(t+ ))<p[V_1(x(t))]=3V_1(x(t)). (50) Then, using the quadratic bounds in (34), one obtains: mΦ2‖e(t+ℏ)‖2≤ m_ 2\|e(t+ )\|^2≤ V1(x(t+ℏ)) V_1(x(t+ )) < < 3V1(x(t)) 3V_1(x(t)) ≤ ≤ 3LΦ2‖e(t)‖2, 3L_ 2\|e(t)\|^2, (51) for all t∈[d,+∞)t∈[d,+∞) and ℏ∈[−2d,0] ∈[-2d,0]. Thus, one has: supℏ∈[−2d,0]‖e(t+ℏ)‖2|Ω0,V1≤3LΦmΦ‖e(t)‖2, . _ ∈[-2d,0]\|e(t+ )\|^2 |_ _0,V_1≤ 3L_ m_ \|e(t)\|^2, (52) and supℏ∈[−2d,0]‖e(t+ℏ)‖|Ω0,V1≤ . _ ∈[-2d,0]\|e(t+ )\| |_ _0,V_1≤ 6mΦV1(x(t))|Ω0,V1 6m_ .V_1(x(t)) |_ _0,V_1 ≤ ≤ 6r0,V1mΦ. 6r_0,V_1m_ . (53) Substituting (52)-(3.2) into (3.2), it yields that for all t∈[d,+∞)t∈[d,+∞), V˙1(x)|Ω0,V1≤ . V_1(x) |_ _0,V_1≤ −a12‖e(t)‖2+3d2a¯2a4a5‖e(t)‖2+a¯3+3d2a¯2a6ϖ02, - a_12\|e(t)\|^2+3d^2 a_2a_4a_5\|e(t)\|^2+ a_3+3d^2 a_2a_6 _0^2, (54) where a4=sup‖ϵ‖≤6r0,V1mΦβ12(ϵ)⋅sup‖ϵ‖≤6r0,V1mΦϖ3(ϵ)a_4= _\|ε\|≤ 6r_0,V_1m_ _1^2(ε)· _\|ε\|≤ 6r_0,V_1m_ _3(ε), a5=3LΦmΦa_5= 3L_ m_ and a6=sup‖ϵ‖≤6r0,V1mΦβ12(ϵ)a_6= _\|ε\|≤ 6r_0,V_1m_ _1^2(ε). Noting that a4a_4, a5a_5, and a6a_6 are independent of d, we provide a parameter a¯=3a¯2a4a5 a=3 a_2a_4a_5 and the maximal delay d¯c2=a14a¯ d_c2= a_14 a, depending on r0,V1r_0,V_1 thereby on r>0r>0. As a consequence, for d≤d¯c2d≤ d_c2, (54) is reduced to: V˙1(x)|Ω0,V1≤ . V_1(x) |_ _0,V_1≤ −a14‖e‖2+Cs(d),∀t∈[d,+∞), - a_14\|e\|^2+C_s(d), ∀ t∈[d,+∞), (55) where Cs(d)=a¯3+3d2a¯2a6ϖ02C_s(d)= a_3+3d^2 a_2a_6 _0^2. By (34), one has ‖e‖2≥2LΦV1(x)\|e\|^2≥ 2L_ V_1(x). Substituting this estimate into (55) gives, ∀t∈[d,+∞)∀ t∈[d,+∞), V˙1(x)|Ω0,V1≤−a12LΦV1(x)+Cs(d), . V_1(x) |_ _0,V_1≤- a_12L_ V_1(x)+C_s(d), (56) Let d¯=mind¯c1,d¯c2>0 d= \ d_c1, d_c2\>0. Then, for any d∈(0,d¯]d∈(0, d], the solution trajectories of the closed-loop system (43), starting from any continuous initial history x(s)∈C([−d,0],ℝn)x(s)∈ C([-d,0],R^n) with ‖x(s)‖C≤nr\|x(s)\|_C≤ nr, are well defined and remain in the compact set Ω0,V1 _0,V_1 for all t≥0t≥ 0. Moreover, the Razumikhin-type estimate (56) holds on Ω0,V1 _0,V_1, where Cs(d)C_s(d) collects the bounded residual induced by the discontinuous sign-based channel and the delay-dependent mismatch. Applying the comparison principle to (56), one obtains: lim supt→∞V1(x(t))≤2LΦa1Cs(d). _t→∞V_1(x(t))≤ 2L_ a_1C_s(d). Combining this estimate with (34) further yields: lim supt→∞‖x(t)−x∗‖≤4LΦa1mΦCs(d). _t→∞\|x(t)-x^*\|≤ 4L_ a_1m_ \,C_s(d). (57) Therefore, the closed-loop system (43) is uniformly ultimately bounded with respect to the optimal solution x∗x^*. Hence, the NNS (2) exhibits practical IDTSC for distributed convex optimization as long as d≤d¯d≤ d. In particular, the state xi(t)x_i(t) semiglobally converges to a neighborhood of xi∗x_i^*, whose size is determined by the residual term Cs(d)C_s(d). This residual originates from the bounded treatment of the discontinuous sign-based channel and the delay-dependent mismatch estimate, and thus the obtained ultimate bound is conservative. ∎ Remark 2. The present analysis is developed for a constant input delay. As a potential extension, the time-varying Razumikhin stability theorem in [40] offers a possible route to treat bounded time-varying input delays within a similar proof architecture. Specifically, one may attempt to adapt Step 2 of Theorem 1 by (i) replacing fixed-delay shift estimates with bounds based on a delay envelope, and (i) performing a Razumikhin comparison on a fixed window determined by the maximal delay. A sketch of the required modifications is given below. Consider a time-varying delay signal d(⋅)d(·) that is continuous (or piecewise continuous) and satisfies: 0≤d(t)≤d¯max,∀t≥0,0≤ d(t)≤ d_ , ∀ t≥ 0, where d¯max d_ is a known delay envelope. In our semiglobal framework, one would further require d¯max≤d¯(r) d_ ≤ d(r) so that the constant-delay admissibility margin remains respected. In the constant-delay analysis, the delay enters through the standard estimate ‖x(t)−x(t−d)‖≤dsuph∈[−d,0]‖x˙(t+h)‖\|x(t)-x(t-d)\|≤ d _h∈[-d,0]\| x(t+h)\|. For d(t)d(t), the corresponding bound is: ‖x(t)−x(t−d(t))‖≤ \|x(t)-x(t-d(t))\|≤ ∫t−d(t)t‖x˙(σ)‖σ _t-d(t)^t\| x(σ)\|\,dσ ≤ ≤ d¯maxsuph∈[−d¯max,0]‖x˙(t+h)‖. d_ _h∈[- d_ ,0]\| x(t+h)\|. (58) Thus, occurrences of d and suph∈[−d,0](⋅) _h∈[-d,0](·) in Step 2 can be conservatively replaced by d¯max d_ and suph∈[−d¯max,0](⋅) _h∈[- d_ ,0](·), respectively. The Razumikhin comparison used in our proof (the p(⋅)p(·)-argument) would be imposed on the fixed window [−2d¯max,0][-2 d_ ,0]: V(t+θ)≤p(V(t)),∀θ∈[−2d¯max,0], V(t+θ)≤ p (V(t) ), ∀θ∈[-2 d_ ,0], (59) which yields a bound of the form supθ∈[−2d¯max,0]V(t+θ)≤ccmpV(t) _θ∈[-2 d_ ,0]V(t+θ)≤ c_ cmpV(t) for some ccmp>1c_ cmp>1. With (2)-(59), the delayed closed loop can be rewritten in the standard functional form x˙(t)=f(t,xt) x(t)=f(t,x_t) on C([−d¯max,0])C([- d_ ,0]). One may then apply the time-varying Razumikhin stability theorem (e.g., Theorem 1 in [40]) to close the comparison argument and establish convergence under d(t)d(t). A complete extension would require re-deriving the admissibility condition and the comparison constants for the time-varying delay case, which is challenging and is left for future work. Remark 3. This paper characterizes an explicit admissible input-delay margin d¯=d¯(r) d= d(r) for the proposed SIDT algorithm (7), in the semiglobal sense that the tolerable delay depends on the prescribed initial radius r of (2). In particular, d¯(r) d(r) is inversely proportional to r that a larger admissible initial radius leads to a smaller delay margin. Importantly, the controller implementation is delay-independent in that it does not require the knowledge of the realized delay value and does not involve delay-dependent gain tuning or structural modification, and the same controller is applied for any admissible delay satisfying d≤d¯(r)d≤ d(r). 3.3 SIDT algorithm for distributed nonconvex optimization problem In this subsection, the application of the SIDT algorithm (7) for solving the distributed nonconvex optimization in delay-affected NNSs (2) is investigated. Theorem 2. Suppose that Assumptions 1, 2, and 4 hold. For the distributed nonconvex optimization problem (2.3) implemented on the NNS (2), if ε≥2n ≥ 2n, then the SIDT algorithm (7) renders the closed-loop system input-delay tolerant in the practical semiglobal sense. Proof. The proof is similar to the proof of Theorem 1, which can be divided into two steps. Step 1: Demonstrate that the NNSs (2) without input delay is global asymptotic and local exponential stability under the regulation of the SIDT algorithm (8). Considering the non-delayed NNSs (9) and closed-loop system (10), similarly to the certification process of (11)-(17), it obtains that exie_x_i globally converges exponentially to origin. Before moving on, according to [36], the P-Ł inequality implies that every stationary point is globally optimal. Specifically, if ∇Φ(y)=0∇ (y)=0, then the P-Ł inequality forces Φ(y)=Φ∗ (y)= ^*. Consequently, non-optimal critical points, saddle points included, are excluded on any set where P-Ł holds. In this theorem, the optimizer is assumed to be unique, i.e., X∗=x∗X^*=\x^*\. Next, we propose the Lyapunov function candidate: V2=Φ(x(t))−Φ(x∗). V_2= (x(t))- (x^*). (60) According to (3.2), the derivative of (60) along the trajectories of (10) yields that: V˙2 V_2 ≤−k0(∇TΦ(x)∇Φ(x))+∑i=1nηiT(xi(t))si(t), ≤-k_0(∇^T (x)∇ (x))+ _i=1^n _i^T(x_i(t))s_i(t), (61) where si(t)= s_i(t)= ε∑j∈i[℘ij(t)sign(xj(t)−xi(t))]+ϑ∑j=1naij(xj(t)−xi(t)). _j∈N_i[ _ij(t) sign(x_j(t)-x_i(t))]+ _j=1^na_ij(x_j(t)-x_i(t)). Based on (5) in Assumption 4, we have: ‖∇Φ(x)‖2≥2μΦ(Φ(x)−Φ(x∗)), \|∇ (x)\|^2≥ 2 _ ( (x)- (x^*)), (62) where μΦ=min1≤i≤nμϕi _ = _1≤ i≤ n _ _i. Substituting (62) into (61) obtains: V˙2≤ V_2≤ −k0(∇TΦ(x)∇Φ(x))+∑i=1nηiT(xi(t))si(t) -k_0(∇^T (x)∇ (x))+ _i=1^n _i^T(x_i(t))s_i(t) ≤ ≤ −2k0μΦ(Φ(x)−Φ(x∗))+∑i=1nηiT(xi(t))si(t) -2k_0 _ ( (x)- (x^*))+ _i=1^n _i^T(x_i(t))s_i(t) ≤ ≤ −βΦ,2V2+∑i=1nηiT(xi(t))si(t), - _ ,2V_2+ _i=1^n _i^T(x_i(t))s_i(t), (63) where βΦ,2=2k0μΦ>0 _ ,2=2k_0 _ >0. Due to the globally exponential convergence of exie_x_i, it deduces that: V˙2≤ V_2≤ −βΦ,2V2+‖∇Φ(x)‖‖s(t)‖ - _ ,2V_2+\|∇ (x)\|\|s(t)\| ≤ ≤ −βΦ,2V2+η¯‖s(t)‖ - _ ,2V_2+ η\|s(t)\| = = −βΦ,2V2+γs(‖s(t)‖), - _ ,2V_2+ _s(\|s(t)\|), (64) where γs(‖s(t)‖)=η¯‖s(t)‖ _s(\|s(t)\|)= η\|s(t)\| is obviously a class K function. Similarly to the analysis of (3.2)-(42), it thus follows from (3.3) that, in the vicinity of x∗x^*, the Lyapunov function V2V_2 satisfies the inequality: V˙2≤−βΦ,2V2. V_2≤- _ ,2V_2. (65) Drawing upon the results from [36] and Lemma 4, it can be deduced that any stationary point, where ∇Φ(x)=0∇ (x)=0, is necessarily a global optimal solution. Consequently, within the global domain of x, it follows from (3.3)-(65) that: limt→∞‖x(t)−x∗‖=0 _t→∞\|x(t)-x^*\|=0. Additionally, based on Assumption 4, Φ(x) (x) is radial unbounded, implying the Lyapunov function V2V_2 (60) is also exhibits radial unboundedness. Specifically, one has: lim‖x‖→∞V2=∞. _\|x\|→∞V_2=∞. Therefore, it can be concluded that: a) V2V_2 is both positive definite and radially unbounded. b) limt→∞‖x(t)−x∗‖=0 _t→∞\|x(t)-x^*\|=0. c) in the neighborhood of x∗x^*, V2V_2 satisfies V˙2≤−βΦ,2V2 V_2≤- _ ,2V_2. Thus, the closed-loop system (10) is GALE stable at x∗x^*. Step 2: Because the SIDT algorithm (7) contains an ideal sign-based channel, the delayed closed-loop vector field is generally discontinuous on the switching surfaces. Therefore, the following argument is not based on the local Lipschitz continuity of the whole closed-loop functional. Instead, as in Theorem 1, we use a Razumikhin-type comparison estimate, where the smooth part of the feedback is treated through a local Lipschitz bound and the sign-based part is absorbed into a bounded residual. From Step 1, the nominal closed loop admits, on the compact set Ω0,V2 _0,V_2, the estimate: ∇Φ(x)T[f(x)+g(x)ξ(x)]≤−βΦ,2V2(x),∇ (x)^T[f(x)+g(x)ξ(x)]≤- _ ,2V_2(x), possibly after reducing βΦ,2 _ ,2. In alignment with the analytical proof in Theorem 1, consider the the delayed closed loop (43), differentiate V2V_2 along trajectories and add-subtract ξ(x(t))ξ(x(t)): V˙2= V_2= ∇Φ(x(t))T[f(x(t))+g(x(t))ξ(x(t−d))] ∇ (x(t))^T[f(x(t))+g(x(t))ξ(x(t-d))] = = ∇Φ(x(t))T[f(x(t))+g(x(t))ξ(x(t))]⏟≤−βΦ,2V2(t)by (65) ∇ (x(t))^T[f(x(t))+g(x(t))ξ(x(t))]_≤\,- _ ,2V_2(t)\ by T2-V25 +∇Φ(x(t))Tg(x(t))(ξ(x(t−d))−ξ(x(t))). +∇ (x(t))^Tg(x(t))(ξ(x(t-d))-ξ(x(t))). (66) Let Ω0,V2:=x∈ℝn:V2(x)≤r0,V2 _0,V_2:=\x ^n:V_2(x)≤ r_0,V_2\ be a compact sublevel set. Since the closed-loop trajectories remain in Ω0,V2 _0,V_2, the smooth part of ξ(⋅)ξ(·) is Lipschitz on Ω0,V2 _0,V_2, while the sign-based channel is uniformly bounded there. Hence, there exist positive constants Lξ,r>0L_ξ,r>0 and Mξ,r>0M_ξ,r>0 such that ‖ξ(x(t−d))−ξ(x(t))‖≤Lξ,r‖x(t)−x(t−d)‖+Mξ,r\|ξ(x(t-d))-ξ(x(t))\|≤ L_ξ,r\|x(t)-x(t-d)\|+M_ξ,r, ∀x(t),x(t−d)∈Ω0,V2∀ x(t),x(t-d)∈ _0,V_2. Moreover, let Cg=supx∈Ω0,V2‖g(x)‖C_g= _x∈ _0,V_2\|g(x)\|, C0=CgLξ,rC_0=C_gL_ξ,r and C1=CgMξ,rC_1=C_gM_ξ,r. Hence, from (3.3), V˙2(x)|Ω0,V2≤ . V_2(x) |_ _0,V_2≤ −βΦ,2V2+C1‖∇Φ(x(t))‖+C0‖∇Φ(x(t))‖‖x(t)−x(t−d)‖. - _ ,2V_2+C_1\|∇ (x(t))\|+C_0\|∇ (x(t))\|\|x(t)-x(t-d)\|. (67) Using the standard trajectory estimate, ‖x(t)−x(t−d)‖≤dsuph∈[−d,0]‖x˙(t+h)‖, \|x(t)-x(t-d)\|≤ d _h∈[-d,0]\| x(t+h)\|, (68) and noting that the delayed closed-loop vector field consists of a smooth part plus a bounded sign-based channel, there exist constants K∇,1>0K_∇,1>0 and K∇,2>0K_∇,2>0, independent of d, such that: ‖x˙(τ)‖2≤K∇,1‖∇Φ(x(τ))‖2+K∇,2, \| x(τ)\|^2≤ K_∇,1\|∇ (x(τ))\|^2+K_∇,2, (69) whenever x(τ)∈Ω0,V2x(τ)∈ _0,V_2. Combining (68)-(69) with (67) and applying Young’s inequality with a tuned parameter yields: V˙2(x)|Ω0,V2≤ . V_2(x) |_ _0,V_2≤ −βΦ,22V2−βΦ,24μΦ‖∇Φ(x(t))‖2+C1‖∇Φ(x(t))‖ - _ ,22V_2- _ ,24 _ \|∇ (x(t))\|^2+C_1\|∇ (x(t))\| +C0‖∇Φ(x(t))‖(dsuph∈[−d,0]‖x˙(t+h)‖) +C_0\|∇ (x(t))\|(d _h∈[-d,0]\| x(t+h)\|) ≤ ≤ −βΦ,22V2+2μΦC02βΦ,2d2suph∈[−d,0]‖x˙(t+h)‖2+2μΦC12βΦ,2, - _ ,22V_2+ 2 _ C_0^2 _ ,2d^2 _h∈[-d,0]\| x(t+h)\|^2+ 2 _ C_1^2 _ ,2, (70) where the P-Ł inequality ‖∇Φ(x)‖2≥2μΦV2(x)\|∇ (x)\|^2≥ 2 _ V_2(x) is utilized to cancel the mixed term. Since Ω0,V2 _0,V_2 is compact and ∇Φ(⋅)∇ (·) is continuous, there exists a constant Gr>0G_r>0 such that: ‖∇Φ(x)‖≤Gr,∀x∈Ω0,V2.\|∇ (x)\|≤ G_r, ∀ x∈ _0,V_2. Invoking (69), one has: suph∈[−d,0]‖x˙(t+h)‖2|Ω0,V2≤K∇,1Gr2+K∇,2. . _h∈[-d,0]\| x(t+h)\|^2 |_ _0,V_2≤ K_∇,1G_r^2+K_∇,2. (71) Substituting (71) into (3.3) yields: V˙2(x(t))|Ω0,V2≤−βΦ,22V2(x(t))+CPL(d), . V_2(x(t)) |_ _0,V_2≤- _ ,22V_2(x(t))+C_PL(d), (72) where CPL(d)=2μΦC12βΦ,2+2μΦC02d2βΦ,2(K∇,1Gr2+K∇,2).C_PL(d)= 2 _ C_1^2 _ ,2+ 2 _ C_0^2d^2 _ ,2 (K_∇,1G_r^2+K_∇,2 ). Choose the compact sublevel set Ω0,V2 _0,V_2 such that it contains the prescribed initial history and satisfies: supθ∈[−d,0]V2(x(θ))<r0,V2. _θ∈[-d,0]V_2(x(θ))<r_0,V_2. Moreover, by increasing r0,V2r_0,V_2 if necessary, assume that: r0,V2>2βΦ,2CPL(d).r_0,V_2> 2 _ ,2C_PL(d). Then, on the boundary V2(x)=r0,V2V_2(x)=r_0,V_2, it follows from (72) that: V˙2(x(t))|V2=r0,V2≤ . V_2(x(t)) |_V_2=r_0,V_2≤ −βΦ,22r0,V2+CPL(d) - _ ,22r_0,V_2+C_PL(d) < < 0. 0. (73) Therefore, the sublevel set Ω0,V2 _0,V_2 is positively invariant for the delayed closed-loop system as long as the solution is well defined. Together with the short-time well-posedness bound established in Theorem 1, there exists a delay bound: d¯=mind¯c1,d¯c2>0, d= \ d_c1, d_c2\>0, (74) depending on the prescribed initial radius r, such that for every d∈(0,d¯]d∈(0, d], the solution of the delayed closed-loop system (43), starting from any continuous initial history x(θ)∈C([−d,0],ℝn)x(θ)∈ C([-d,0],R^n), is well defined and remains in Ω0,V2 _0,V_2 for all t≥0t≥ 0. Here, d¯c1 d_c1 guarantees the short-time well-posedness on [0,d][0,d], while d¯c2 d_c2 is chosen such that the positive-invariance condition (3.3) holds on Ω0,V2 _0,V_2. Applying the comparison principle to (72), for all t≥dt≥ d, one obtains: V2(x(t))≤ V_2(x(t))≤ e−βΦ,22(t−d)V2(x(d)) e^- _ ,22(t-d)V_2(x(d)) +2βΦ,2(1−e−βΦ,22(t−d))CPL(d). + 2 _ ,2 (1-e^- _ ,22(t-d) )C_PL(d). (75) Consequently, lim supt→∞V2(x(t))≤2βΦ,2CPL(d). _t→∞V_2(x(t))≤ 2 _ ,2C_PL(d). (76) Since the optimizer is unique, i.e., X∗=x∗X^*=\x^*\, and V2(x)=Φ(x)−Φ(x∗)V_2(x)= (x)- (x^*) is positive definite with respect to x∗x^* on Ω0,V2 _0,V_2, there exists a class-K function αV2 _V_2 such that: αV2(‖x−x∗‖)≤V2(x),∀x∈Ω0,V2. _V_2(\|x-x^*\|)≤ V_2(x), ∀ x∈ _0,V_2. (77) Combining (76) with (77) yields: lim supt→∞‖x(t)−x∗‖≤αV2−1(2βΦ,2CPL(d)). _t→∞\|x(t)-x^*\|≤ _V_2^-1 ( 2 _ ,2C_PL(d) ). (78) Therefore, under Assumption 4, the delayed closed-loop NNS (43) is input-delay tolerant in the practical semiglobal sense for distributed nonconvex optimization. Specifically, for every prescribed compact initial set, there exists an admissible delay bound d¯>0 d>0 such that, for all d∈(0,d¯]d∈(0, d], the state trajectory remains well defined, stays bounded, and converges to a neighborhood of the unique optimizer x∗x^*. The size of this ultimate neighborhood is characterized by CPL(d)C_PL(d), which collects the bounded residual induced by the discontinuous sign-based channel and the delay-dependent mismatch estimate. Thus, the obtained convergence result is practical rather than exact, and the ultimate bound is conservative. ∎ Remark 4. While the SIDT algorithm (7) demonstrates versatility in handling both types of optimization problems, its tolerance to input delays varies between the convex and non-convex conditions. Specifically, the algorithm exhibits different levels of robustness to input delays (namely, the different upper bound delay d¯ d in (54)-(55) and (74), respectively) depending on the convexity properties of the optimization problem. Remark 5. The discontinuity of sign(⋅)sign(·) in (7)-(8) may induce chattering. A standard remedy is to replace sign(s)sign(s), for all s∈ℝms ^m, by a continuous boundary-layer approximation: satσ(s)=smaxσ,‖s‖,σ>0. _σ(s)= s \σ,\|s\|\, σ>0. These proxies satisfy that ‖satσ(s)‖≤1\|sat_σ(s)\|≤ 1, sTsatσ(s)≥‖s‖−σs^Tsat_σ(s)≥\|s\|-σ and satσ(s)→s/‖s‖sat_σ(s)→ s/\|s\| pointwise for s≠0s≠ 0 as σ↘0σ 0. It is noted that our proof in Theorems 1 and 2 use only boundedness and this sector inequality. Therefore, the analysis carries over verbatim with signsign replaced by satσsat_σ. In particular, for Theorems 1 and 2, there exist constants c1,c2>0c_1,c_2>0 (independent of σ) and a delay bound d¯>0 d>0 such that, for all d≤d¯d≤ d, V˙l≤−c1Vl+c2σ, V_l≤-c_1V_l+c_2σ, whence Vl≤e−c1(t−t0)Vl(t0)+c2c1σ, V_l≤ e^-c_1(t-t_0)V_l(t_0)+ c_2c_1σ, where l∈1,2l∈\1,2\. Namely, practical consensus in the Lyapunov metric within an (σ)O(σ) neighborhood. Remark 6. The SIDT algorithm gains k0k_0, ϑ , ε in (7) is employed to obtain explicit admissible delay margins d¯ d in Theorems 1 and 2. As usual, there is a speed-delay trade-off such that increasing the gains accelerates convergence but also enlarges the constants a¯ a in (54) and βΦ,2 _ ,2 in (72)-(3.3) that ultimately shrink d¯ d. Moreover, the condition ε≥2n ≥ 2n in Theorems 1 and 2 is a uniform, topology-agnostic bound ensuring the consensus term dominates aggregated disagreement, which is conservative and scales with network size. In large networks this pushes the minimal admissible ε upward and thereby reduces the delay tolerance. Hence, in practice one should balance network size and sparsity against the required delay robustness when selecting gains. Furthermore, a systematic treatment of adaptive/scheduled gains and topology-dependent bounds that avoid explicit n-scaling is a promising direction for future work. 4 Numerical Examples Figure 1: Illustration of communication network in the NNS (2). In this section, we present two numerical examples to demonstrate the proposed SIDT algorithm (7) within the frameworks of convex and nonconvex optimization, respectively. Both examples are grounded in the identical NNS (2) and utilize the same control parameters in (7). We begin by outlining the shared NNS’s configuration and control parameter settings that are fundamental to both optimization scenarios. As shown in Fig. 1, consider a NNS (2) consisting of 55 nodes communicating under the undirected graph G such that: ℒ=[3−1−1−10−12−100−1−1200−1002−1000−11], L= bmatrix3&-1&-1&-1&0\\ -1&2&-1&0&0\\ -1&-1&2&0&0\\ -1&0&0&2&-1\\ 0&0&0&-1&1\\ bmatrix, (79) where ℒL is the Laplacian matrix of the graph G. The dynamical model of the NNS (2) with m=1m=1 is presented as: x˙i(t)= x_i(t)= xi2(t)+ui(t−d),i=1,…,n, x_i^2(t)+u_i(t-d), i=1,…,n, (80) where d is the input delay. For each example, we determine the admissible delay threshold d¯ d separately, since the objective’s curvature properties (e.g., strong convexity and P-Ł conditions) affect the comparison constants that define d¯ d. It is important to emphasize that although we set d, the controller does not require explicit knowledge of d during simulations. Instead, it operates solely based on the system state affected by the delay d. Then, the setting of control parameters in (7) is proposed as: ϑ=0.1 =0.1, ε=11 =11, and k0=0.1k_0=0.1. To proceed, two numerical examples are presented as follows. Table 1: The local convex cost function of NNS (80) Node index Local cost function 1 ϕ1=x12+x1+1 _1=x_1^2+x_1+1 2 ϕ2=x22+2x2+2 _2=x_2^2+2x_2+2 3 ϕ3=x32+3x3+3 _3=x_3^2+3x_3+3 4 ϕ4=x42+4x4+4 _4=x_4^2+4x_4+4 5 ϕ5=x52+5x5+5 _5=x_5^2+5x_5+5 Figure 2: Evolution of system states xi(t)x_i(t) and the state errors exi(t)e_x_i(t). (a): Evolution of xi(t)x_i(t), ∀i∈∀ i∈V. (b): Evolution of exi(t)e_x_i(t), ∀i∈∀ i∈V. Figure 3: Evolution of local and global cost functions. (a): Evolution of ϕi(xi) _i(x_i), ∀i∈∀ i∈V. (b): Evolution of Φ(x) (x). Example 1. (Example for distributed convex optimization) In this numerical example, the SIDT algorithm (7) is utilized for the distributed convex optimization in the delay-affected NNS (80). The local cost functions are respectively provided in Table 1. We set the initial radius r=5r=5, namely, initial at (−5,5)(-5,5), and select the delay parameter d randomly in the range 0.01,0.03,0.05\0.01,0.03,0.05\. The simulation results of the numerical examples are presented in Figs. 2 and 3. By Fig. 2, it illustrates the temporal evolution of the system states xi(t)x_i(t) and the state errors exi(t)e_x_i(t). It is evident that the states converge to consensus, thereby satisfying the constraints of the distributed optimization problem (2.3), and practically achieving the optimal solution x∗=−1.47x^*=-1.47. As shown in Fig. 3, it depicts the dynamics of both the local cost functions ϕi(xi) _i(x_i), ∀i∈∀ i∈V, and global cost functions Φ(x) (x) over time. Based on these results, it can be concluded that the SIDT algorithm (7) effectively regulates delay-affected NNS (80) to attain IDTSC in the practical sense for distributed optimization. Table 2: The local nonconvex cost function of NNS (80) Node index Local cost function 1 ϕ1=(x1−1)2+2sin2(x1−1) _1=(x_1-1)^2+2 ^2(x_1-1) 2 ϕ2=(x2−2)2+2sin2(x2−2) _2=(x_2-2)^2+2 ^2(x_2-2) 3 ϕ3=(x3−1)2+2sin2(x3−1) _3=(x_3-1)^2+2 ^2(x_3-1) 4 ϕ4=(x4−2)2+2sin2(x4−2) _4=(x_4-2)^2+2 ^2(x_4-2) 5 ϕ5=(x5−1)2+2sin2(x5−1) _5=(x_5-1)^2+2 ^2(x_5-1) Figure 4: Evolution of system states xi(t)x_i(t) and the state errors exi(t)e_x_i(t). (a): Evolution of xi(t)x_i(t), ∀i∈∀ i∈V. (b): Evolution of exi(t)e_x_i(t), ∀i∈∀ i∈V. Figure 5: Evolution of local and global cost functions. (a): Evolution of ϕi(xi) _i(x_i), ∀i∈∀ i∈V. (b): Evolution of Φ(x) (x). Example 2. (Example for distributed nonconvex optimization) In this numerical example, the SIDT algorithm (7) is applied to a distributed nonconvex optimization problem within a delay-affected NNS (80). The local and global cost functions for this scenario are detailed in Table 2. We initialize the system with a radius r=5r=5, namely, initial at (−5,5)(-5,5), and select the delay parameter d randomly in the range d∈0.01,0.02,0.03d∈\0.01,0.02,0.03\. The simulation results are shown in Fig.s 4 and 5. Fig. 4 shows the temporal evolution of the system states xi(t)x_i(t) and state errors exi(t)e_x_i(t). It clearly demonstrates that the states xi(t)x_i(t) converge to consensus, thereby satisfying the consensus constraints and practically converging the optimal solution x∗=1.4x^*=1.4. In Fig. 5, it presents the evolution of the local cost functions ϕi(xi) _i(x_i), for all i∈i , and the global cost function Φ(x) (x) over time. These phenomena substantiate that the SIDT algorithm (7) effectively controls the delay-affected NNS (80), achieving IDTSC in the practical sense for distributed nonconvex optimization. 5 Conclusion In this paper, we have introduced the IDTSC concept, providing a robust foundation for analyzing distributed optimization problems in NNSs with input delays and consensus constraints. To achieve IDTSC practically in constrained optimization problem of delay-affected NNSs, a novel SIDT algorithm has been proposed. Furthermore, it has been demonstrated that the proposed algorithm can be extended to nonconvex optimization problems under P-Ł condition, showcasing the algorithm’s robustness and adaptability in less restrictive optimization environments. Comprehensive simulations have validated the efficacy of the proposed algorithm, affirming its capability to achieve IDTSC in both convex and nonconvex distributed optimization scenarios within delay-affected NNSs. 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