Bo Yu

Prescribed-time consensus of multiagent systems is known for its small conservativeness in settling time. Some studies even show that the consensus happens at a prescribed instant. However, the corresponding theoretical proof is not sufficient. In this work, a consensus control based on the backstepping method is designed for multiagent systems with high-order integrator dynamics. Two Lyapunov inequalities force the consensus to happen just at the prescribed instant (prescribed-instant consensus). One of the inequalities ensures the consensus happening not after the preassigned time instant (prescribed-time consensus). The other guarantees the consensus happens not before the prescribed time instant. In the numerical example, a five-node multiagent system simulation validates the proposed prescribed-instant consensus.

A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions.

In this paper, a distributionally robust optimization model based on kernel density estimation (KDE) and mean entropic value-at-risk (EVaR) is proposed, where the ambiguity set is defined as a KDE-[ ]-divergence “ball” centered at the empirical distribution in the weighted KDE distribution function family, which is a finite-dimensional set. Instead of the joint probability distribution of the random vector, the one-dimensional probability distribution of the random loss function is approximated by the univariate weighted KDE for dimensionality reduction. Under the mild conditions of the kernel and [ ]-divergence function, the computationally tractable reformulation of the corresponding distributionally robust mean-EVaR optimization model is derived by Fenchel’s duality theory. Convergence of the optimal value and the solution set of the distributionally robust optimization problem based on KDE and mean-EVaR to those of the corresponding stochastic programming problem with the true distribution is proved. For some special cases, including portfolio selection, newsvendor problem, and linear two-stage stochastic programming problem, concrete tractable reformulations are given. Primary empirical test results for portfolio selection and project management problems show that the proposed model is promising.Funding: This work was funded by the National Natural Science Foundation of China [Grants 11971092 and 11571061] and the Fundamental Research Funds for the Central Universities [Grants DUT15RC(3)037 and DUT18RC(4)067].