Optimization of the solution of the parameter-dependent Sylvester equation and applications

This paper deals with an efficient algorithm for optimization of the solution of the parameter-dependent Sylvester equation (A(0) - upsilon C1C2T)X(upsilon) + X(upsilon)(B-0 - upsilon D1D2T) = E, where A(0), B-0 are m x m and n x n matrices, respectively. Further, C-1 and C-2 are m x r(1), D-1 and D-2 are n x r(2) and X, E are m x n matrices, while upsilon is real parameter. For optimization we use the following two optimization criteria: Tr(X(upsilon)) -> min and parallel to X(upsilon)parallel to(F) -> min. We present an efficient algorithm based on derived formulas for the trace and for the Frobenius norm of the solution X as functions upsilon -> Tr(X(upsilon)) and upsilon -> parallel to X(upsilon)parallel to(F) as well as for derivatives of these functions. That ensures fast optimization of these functions via standard optimization methods like Newton's method. A special case of this problem is a very important problem of damper viscosity optimization in mechanical systems. (C) 2012 Elsevier B.V. All rights reserved.