  In a recent interesting paper [8], Chen et al. initialized 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. Criteria for fundamental system-theoretic
notions such as stability, reachability and observability are established by
means of tensor decomposition. The purpose of this paper is to continue this
novel research direction. Specifically, we focus on continuous-time MLTI
control systems. We define Hamiltonian tensors and symplectic tensors and
establish the Schur-Hamiltonian tensor decomposition and symplectic tensor
singular value decomposition (SVD). Based on these we propose the algebraic
Riccati tensor equation (ARTE) and show that it has a unique positive
semidefinite solution if the system is stablizable and detectable. A
tensor-based Newton method is proposed to find numerical solutions of the ARTE.
The tensor version of the bounded real lemma is also established. A first-order
robustness analysis of the ARTE is conducted. Finally, a numerical example is
used to demonstrate the proposed theory and algorithms.
