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This study addresses a distributed state feedback controller design problem for continuous-time linear time-invariant systems by means of linear matrix inequalities (LMI). As the exact convexification is still open, the block-diagonal relaxation of Lyapunov functions has been prevalent despite its conservatism. In this work, we target a class of non-block-diagonal Lyapunov functions that has the same sparsity as distributed controllers. By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, we first present a (nonlinear) matrix inequality for stabilizing distributed controllers with such Lyapunov functions, which boils down to a necessary and sufficient condition for such controllers if the sparsity pattern is chordal. As a relaxation of the inequality, we derive an LMI that completely covers the conventional relaxation and then provide analogous results for $H_\infty$ control. Lastly, numerical examples underscore the efficacy of our results., Comment: 8 pages

In this study, we explore distributed optimization problems with clique-wise coupling through the lens of operator splitting. This framework of clique-wise coupling extends beyond conventional pairwise coupled problems, encompassing consensus optimization and formation control, and is applicable to a wide array of examples. We first introduce a matrix, called the clique-wise duplication (CD) matrix, which enables decoupled reformulations for operator splitting methods and distributed computation. Leveraging this matrix, we propose a new distributed optimization algorithm via Davis-Yin splitting (DYS), a versatile three-operator splitting method. We then delve into the properties of this method and demonstrate how existing consensus optimization methods (NIDS, Exact Diffusion, and Diffusion) can be derived from our proposed method. Furthermore, being inspired by this observation, we derive a Diffusion-like method, the clique-based projected gradient descent (CPGD), and present Nesterov's acceleration and in-depth convergence analysis for various step sizes. The paper concludes with numerical examples that underscore the efficacy of our proposed method., Comment: 16pages

In this study, we propose a design methodology of distributed controllers for multi-agent systems on a class of directed interaction networks by extending the gradient-flow method. Although the gradient-flow method is a common design tool for distributed controllers, it is inapplicable to directed networks. First, we demonstrate how to construct a distributed controller for systems over a class of time-invariant directed graphs. Subsequently, we establish better convergence properties and performance enhancement than the conventional gradient-flow method. To illustrate its application in time-varying networks, we address the dynamic matching problem of two distinct groups of agents with different sensing ranges. This problem is a novel coordination task that involves pairing agents from two distinct groups to achieve a convergence of the paired agents' states to the same value. Accordingly, we apply the proposed method to this problem and provide sufficient conditions for successful matching. Lastly, numerical examples for systems on both time-invariant and time-varying networks demonstrate the effectiveness of the proposed method., Comment: 13 pages. https://ieeexplore.ieee.org/document/10453999

This study addresses a distributed optimization with a novel class of coupling of variables, called clique-wise coupling. A clique is a node set of a complete subgraph of an undirected graph. This setup is an extension of pairwise coupled optimization problems (e.g., consensus optimization) and allows us to handle coupling of variables consisting of more than two agents systematically. To solve this problem, we propose a clique-based linearized ADMM algorithm, which is proved to be distributed. Additionally, we consider objective functions given as a sum of nonsmooth and smooth convex functions and present a more flexible algorithm based on the FLiP-ADMM algorithm. Moreover, we provide convergence theorems of these algorithms. Notably, all the algorithmic parameters and the derived condition in the theorems depend only on local information, which means that each agent can choose the parameters in a distributed manner. Finally, we apply the proposed methods to a consensus optimization problem and demonstrate their effectiveness via numerical experiments.

This paper addresses a distributed convex optimization problem with a class of coupled constraints, which arise in a multi-agent system composed of multiple communities modeled by cliques. First, we propose a fully distributed gradient-based algorithm with a novel operator inspired by the convex projection, called the clique-based projection. Next, we scrutinize the convergence properties for both diminishing and fixed step sizes. For diminishing ones, we show the convergence to an optimal solution under the assumptions of the smoothness of an objective function and the compactness of the constraint set. Additionally, when the objective function is strongly monotone, the strict convergence to the unique solution is proved without the assumption of compactness. For fixed step sizes, we prove the non-ergodic convergence rate of O(1/k) concerning the objective residual under the assumption of the smoothness of the objective function. Furthermore, we apply Nesterov's acceleration method to the proposed algorithm and establish the convergence rate of O(1/k^2). Numerical experiments illustrate the effectiveness of the proposed method., Comment: 8 pages, 5 figures

This letter deals with the optimization problems of stochastic switching buffer networks, where the switching law is governed by Markov process. The dynamical buffer network is introduced, and its application in modeling the car-sharing network is also presented. To address the nonconvexity for getting a solution as close-to-the-global-optimal as possible of the optimization problem, we adopt a succinct but effective nonconvex optimization method called \emph{ DC (difference of convex functions) programming}. By resorting to the log-log convexity of a class of nonlinear functions called posynomials, the optimization problems can be reduced to DC programming problems. Finally, we verify the effectiveness of our results by simulation experiments.

This paper treats an optimal scheduling problem of control nodes in networked systems. We newly introduce both the L0 and l0 constraints on control inputs to extract a time-varying small number of effective control nodes. As the cost function, we adopt the trace of the controllability Gramian to reduce the required control energy. Since the formulated optimization problem is combinatorial, we introduce a convex relaxation problem for its computational tractability. After a reformulation of the problem into an optimal control problem to which Pontryagin's maximum principle is applicable, we give a sufficient condition under which the relaxed problem gives a solution of the main problem. Finally, the proposed method is applied to a rebalancing problem of a mobility network., Comment: 8 pages, 6 figures. Supplemental material 3 pages

Motivated by the fact that intelligent traffic control systems have become inevitable demand to cope with the risk of traffic congestion in urban areas, this paper develops a distributed control strategy for urban traffic networks. Since these networks contain a large number of roads having different directions, each of them can be described as a multi-agent system. Thus, a coordination among traffic flows is required to optimize the operation of the overall network. In order to determine control decisions, we describe the objective of improving traffic conditions as a constrained optimization problem with respect to downstream traffic flows. By applying the gradient projection method and the minimal polynomial of a matrix pair, we propose algorithms that allow each road cell to determine its control decision corresponding to the optimal solution while using only its local information. The effectiveness of our proposed algorithms is validated by numerical simulations.