This paper studies the discrete-time linear-quadratic optimal control problem (LQOCP) for time-delayed descriptor systems in a real Hilbert space, and obtains some novel sufficient conditions for the solvability of the discrete-time LQOCP by the generalized inverse theory and space decomposition technique. Especially, the proposed methods are simpler, easier to verify and compute, and can solve the LQOCP without imposing any range inclusion condition. In addition, our work is verified by some numerical examples.

A new model of composite networks generated by smaller simple factor networks via Lexicographic products is constructed with Laplacian dynamics. A theoretic analysis framework for the controllability of composite networks is presented and the controllability is characterized on account of the Popov-Belevitch-Hautus (PBH) test, which provides an effective technique for extending the controllability of the factors to that of the Lexicographic product network (LPN), and uncovers how the controllability of the LPN can be inferred from the controllability and spectral properties of the factor networks through analyzing the relationship between the structure of LPN and its factors’ topologies. These results obtained can then be extended to other types of graph-product networks, layered control networks, some high-order networks, and man-made network systems. This work will help one to cognize the composite networks and understand the synergistic mechanism of networks.

The average controllability can quantify the difficulty of controlling a network based on controllability Gramian metrics in terms of energy triggered by the leaders. This work addresses the average controllability of a class of composite networks generated by factor networks via N-duplication corona product, considering the simple undirected factor networks with Laplacian dynamics within a leader-follower framework, and further provides a technique to characterize the average controllability of N-duplication corona product networks (N-CPNs) by the spectral properties of factors, which greatly reduces the computational complexity and analytical difficulty. In addition, it is found that the 'size' of average controllability (i.e., trace of Gramian matrix) is dependent on., the assignment of leaders, the number of agents and the spectral properties of factors.

This paper aims to establish an analytical framework based on the Frobenius-Schur factorization for exploring the fractional powers of the coupled operator matrix ( AB)(I0) A=0AL I in X & times; X with X a Banach space. Through analyzing the representation of the resolvent of its Schur complement, we derive an explicit expression for the resolvent of A. Together with the integral representation of fractional powers of operators of positive type, this then results in an explicit formula concerning the fractional powers of A. Additionally, the obtained formula is employed to study a class of coupled evolution equations, demonstrating its effectiveness in capturing the underlying dynamics of such systems. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This article studies the energy-related controllability for a category of 'large' composite networks generated by 'small' simple factor networks with Laplacian dynamics under a leader-follower framework via corona product. Different from most existing literature on network controllability, this work characterizes the controllability of corona product networks (CPNs) from an energy point of view. This can quantify the difficulty of controlling CPNs based on controllability Gramian measures, involving average controllability and volumetric control energy, etc., where the energy is triggered by the leaders. The energy-related controllability of a CPN can be explored from the eigenvalues and eigenvectors of its factor networks. An algorithm for solving the maximum average controllability is provided, which can help one select the leaders to optimize network control and be applied in practice.

In recent years, the cooperative control of multi-agent systems (MASs) on multi-layer chain networks has become a hot research topic, since the complex connections between layers directly affect the dynamics, propagation, diffusion and consensus of the entire multi-layer chain network. In this paper, the consensus problems for first-order MASs on double-layer and three-layer chain networks are studied, respectively. The eigenvalues of the corresponding system matrices are calculated by means of matrix decomposition first and, through discussing the weight value between agents, some sufficient and necessary conditions for the consensus of MASs on undirected / directed double-layer and three-layer chain networks are established, based on Lyapunov stability theorem, which provide a simple method for judging the consensus of MASs on multi-layer chain networks. Finally, some numerical examples and simulations are presented to confirm the validity of the theoretical results, which lays a foundation for the application of MASs in complex network scenarios.

The concept of fractional powers of operators was initially developed within the framework of functional analysis and has since played a significant role in the study of evolution equations, abstract differential equations, and complex dynamical systems. Meanwhile, the operator matrix approach has emerged as a powerful and widely used tool for analyzing the structural properties of operators. Motivated by these developments, this paper focuses on the explicit computation of fractional powers of anti-triangular operator matrices. Specifically, we first derive an explicit formula for the fractional powers of off-diagonal operator matrices by employing the formal determinant of block operator matrices. Then, based on the Schur factorization, we obtain a representation for the fractional powers of the general anti-triangular case. As applications, the obtained results are further applied to certain differential equations.

This paper deals with solutions of the Yang-Baxter-like operator equation $ AXA = XAX $ AXA=XAX on the infinite dimensional Hilbert space $ \mathcal {H} $ H, where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.

This paper addresses the target controllability of first-order multi-agent systems (MASs) based on absolute protocol from viewpoints of graph theory and algebra. A sufficient and necessary condition on the target controllability of MASs is derived by computing the left Jordan chain (LJC) of the corresponding adjacency matrices. Additionally, by using the maximal almost equitable partition of nodes, some graphic target controllable conditions are obtained, so that all target nodes are reachable from the set of leaders. Also, several numerical examples are provided to show the validity of the obtained results.

Complex networks provide a crucial framework for understanding the structure and dynamic behavior of real-world systems, while consensus and controllability serve as the foundational theoretical cornerstones for achieving coordinated control and global behavior regulation in networks. In this paper, the consensus and controllability of multi-agent systems on a double-layer star topology are studied from an algebraic perspective. This study systematically investigates the consensus problem of undirected double-layer star networks under the constraint that intra-layer coupling strengths are equal based on a continuous first-order linear protocol framework, and establishes the controllability conditions for this double-layer network system by using the PBH test based on the identical intra-layer and inter-layer coupling strengths. To validate the correctness and effectiveness of the theoretical results, numerical simulations are conducted by constructing several test cases for verification. This work establishes a theoretical analysis framework for multi-agent systems on double-layer star networks, which can provide deeper insights into coordination control of complex systems.

A composite network formed by factor networks is a new type of complex network model that can describe different types of agents and multiple complicated relationships in complex systems. "Indu-Bala product" of graphs is a new graph operation with symmetry properties and good mathematical expression, which enables one to construct many pairs of respective co-spectral graphs. This work studies the controllability of a composite network formed by factor networks via Indu-Bala product based on Laplacian dynamics. The eigenvalues and eigenvectors of Indu-Bala product networks (IBPNs) can be represented by factors' eigenvalues and eigenvectors, which can reduce the computational complexity and analytical burden. Based on the PBH test, sufficient and necessary algebraic criterions for controllability of IBPNs are established, which reveal the controllability relationship between entire network and its factors. This work will help one to construct, understand and improve the composite complex network models, and gain insight into the synergistic mechanisms of the networks.

This article tackles the average controllability (a class of control energy) of neighborhood Corona product networks (NCPNs), with the aim of uncovering the association among the average controllability, the network structure and the characteristics of external control inputs, based on the controllability Gramian matrix through analyzing the eigenvalues and eigenvectors of the Laplacian matrix of the system. The obtained results indicate how the number and position of leaders to affect the average controllability of NCPNs.

This paper examines the completion of the point spectrum, residual spectrum, and continuous spectrum of an unbounded operator matrix acting in a Hilbert space. The necessary and sufficient conditions for the point spectrum of an unbounded operator matrix to be equal to the union of the point spectra of its diagonal entries are presented.