The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation $\boldsymbol{K} \boldsymbol{M}+\boldsymbol{M} \boldsymbol{K}=\boldsymbol{r}\, \boldsymbol{s}^{T}$ we introduce a scalar function $S^{(i,j)}=\boldsymbol{s}^{T}\, \boldsymbol{K}^j(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^i\boldsymbol{r}$ which is defined as same as in discrete case. $S^{(i,j)}$ satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on $\boldsymbol{r}$ and $\boldsymbol{s}$, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of $S^{(i,j)}$ defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property $S^{(i,j)}=S^{(i,j)}$. The solution $\boldsymbol{M}$ provides $\tau$ function by $\tau=|\boldsymbol{I}+\boldsymbol{M}|$. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.Comment: 23 page

A priori Estimate for the solution of Sylvester equation

For A, B and Y operators in B(H) it's well known the importance of sylvester equation AX - XB= Y in control theory and its applications. In this paper -using integral calculus- we were able to give a priori estimate of the solution of famous sylvester equation when A and B are selfadjoint operators, some other results are also given

New Noise-Tolerant ZNN Models With Predefined-Time Convergence for Time-Variant Sylvester Equation Solving

Sylvester equation is often applied to various fields, such as mathematics and control systems due to its importance. Zeroing neural network (ZNN), as a systematic design method for time-variant problems, has been proved to be effective on solving Sylvester equation in the ideal conditions. In this paper, in order to realize the predefined-time convergence of the ZNN model and modify its robustness, two new noise-tolerant ZNNs (NNTZNNs) are established by devising two novelly constructed nonlinear activation functions (AFs) to find the accurate solution of the time-variant Sylvester equation in the presence of various noises. Unlike the original ZNN models activated by known AFs, the proposed two NNTZNN models are activated by two novel AFs, therefore, possessing the excellent predefined-time convergence and strong robustness even in the presence of various noises. Besides, the detailed theoretical analyses of the predefined-time convergence and robustness ability for the NNTZNN models are given by considering different kinds of noises. Simulation comparative results further verify the excellent performance of the proposed NNTZNN models, when applied to online solution of the time-variant Sylvester equation

Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case $n=4$, the proofs for the case $n=5$ and the case of arbitrary dimension $n$. The results can be used in symbolic computation.Comment: 19 page