We outline a derivation of a nonlinear system of equations, which finds the entries of an m×N matrix K, given
the eigenvalues of a matrix D, a diagonal N ×N matrix A and an N ×m matrix B. These matrices are related through
the matrix equation D = 2A + BK + K
tB
t
, which is sometimes called a t-Sylvester equation. The need to prescribe
the eigenvalues of the matrix D is motivated by the control of the surface roughness of certain nonlinear SPDEs (e.g.,
the stochastic Kuramoto-Sivashinsky equation) using nontrivial controls. We implement the methodology to solve
numerically the nonlinear system for various test cases, including matrices related to the control of the stochastic
Kuramoto-Sivashinsky equation and for randomly generated matrices. We study the effect of increasing the dimensions
of the system and changing the size of the matrices B and K (which correspond to using more or less controls)
and find good convergence of the solutions.