On the numerical solution of a T-Sylvester type matrix equation arising in the control of stochastic partial differential equations

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

Mean field limits for interacting diffusions in a two-scale potential

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in [13]. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions