Systems of low Hankel rank: A survey (Q2784648)

Let \(T\) be a linear operator mapping a sequence of vectors \(u=[u_i]_{i=-\infty}^\infty\) to a sequence of vectors \(y=[y_k]_{k=-\infty}^\infty\): \(y=uT\). Each \(u_i\;(y_k)\) is a vector belonging to a vector space of finite dimension \(m_i\) (resp., \(n_k\)). The operator \(T\) is assumed to have a finite norm with respect to the norms \(\|u\|_2=(\sum_{i=-\infty}^\infty\|u_i\|^2_2)^{1/2}\) and \(\|y\|_2=(\sum_{k=-\infty}^\infty\|y_k\|^2_2)^{1/2}\) where \(\|u_i\|_2\) (resp., \(\|y_k\|_2\)) is the regular Euclidean norm. \(T\) is called locally finite if it possesses a time-variant system realization, i.e., \(T\) is block-upper triangular (\(T_{ik}=0\) for \(i>k\)), and there exists matrices \(\{ A_k,B_k,C_k,D_k\}\) for each integer \(k\) such that \(T_{kk}=D_k,\) and (for \(i<k\)) \(T_{ik}=B_iA_{i+1}\cdots A_{k-1}C_k\). In other words, \(T\) is represented as the transfer map of a time-variant system \(x_{k+1}=x_kA_k+u_kB_k,\;y_k=x_kC_k+u_kD_k\). If \(b_k\) are dimensions of state vectors \(x_k\) of this system, then \(A_k,B_k,C_k,D_k\) are matrices of size \(b_k\times b_{k+1},m_k\times b_{k+1},b_k\times n_k\) and \(m_k\times n_k\), respectively. The author also assumes that the realization system for \(T\) is uniformly exponentially stable, i.e., there exist uniform bounds for \(A_k,B_k,C_k,D_k\), and a real number \(\rho\) with \(0\leq\rho < 1\) such that, uniformly over \(k\), \(\limsup_{l\rightarrow\infty}\|A_{k+1}\cdots A_l\|\leq\rho^{l-k}\) (these conditions make \(T\) automatically a bounded operator on \(l_2\) sequences). The realization system is said to be of low Hankel rank if the dimensions \(b_k\) at each time \(k\) are small. In the paper under review, a survey of the main properties of low Hankel rank systems is given. Namely, the realization theory, factorization results, inversion theory, and approximation of ``large'' Hankel rank systems by systems of ``low'' Hankel rank are briefly described.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00034].