Algebraic Riccati Tensor Equations With Applications in Multilinear Control Systems

Publication Date

9-15-2025

Document Type

Article

Publication Title

SIAM Journal on Control and Optimization

Volume

63

Issue

5

DOI

10.1137/24M1640410

First Page

3378

Last Page

3406

Abstract

In a recent paper by Chen et al. [SIAM J. Control Optim., 59 (2021), pp. 749–776], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decomposition. Building on this new research direction, the purpose of our paper is to extend the study to continuous-time MLTI control systems. Specifically, we define Hamiltonian tensors and symplectic tensors, and we establish the Schur-Hamiltonian tensor decomposition and the symplectic tensor singular value decomposition (SVD). Based on these concepts, we propose the algebraic Riccati tensor equation (ARTE) and demonstrate that it has a unique positive semidefinite solution if the system is stabilizable and detectable. To find numerical solutions to the ARTE, we introduce a tensor-based Newton method. Additionally, we establish the tensor versions of the bounded real lemma and the small gain theorem. A first-order robustness analysis of the ARTE is also conducted. Finally, we provide a numerical example to illustrate the proposed theory and algorithms.

Funding Number

15213924

Funding Sponsor

Research Grants Council, University Grants Committee

Keywords

algebraic Riccati tensor equation, Einstein product, Hamiltonian tensor, multilinear time-invariant control systems, robust control

Department

Applied Data Science

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