COMPARISON-THEOREMS FOR THE MATRIX RICCATI EQUATION

Abstract

We derive comparison theorems for the matrix-valued Riccati equations of the form R(i)'(z) = B(i)(z) + A(i)(z)R(i)(z) + R(i)(z)D(i)(z) + R(i)(z)C(i)(z)R(i)(z), R(i)(0) = R0,i, i = 1, 2. Such equations arise in transport problems and other applications. Sufficient conditions under which R1(z) - R2(z) greater-than-or-equal-to 0 and R1(z) - R2(z) > 0 (in the componentwise sense) for all z are given, respectively. The comparison theorems in which the positivity is defined in terms of positive semidefiniteness were obtained by Royden. Because of the intrinsic disparity in the nature of positivity structure, the techniques developed there cannot be applied to our problem.