Comparison theorems for the matrix Riccati equation (Q1316197)

Riccati differential equations of the form \[ R_ i'(z)= B_ i(z)+ A_ i(z) R_ i(z)+ R_ i(z) D_ i(z)+ R_ i(z) C_ i(z) R_ i(z),\quad R_ i(0)= R_{o,i},\;i=1,2, \] are considered, where \(A_ i\), \(B_ i\), \(C_ i\) and \(D_ i\) are matrices of appropriate dimensions and piecewise continuous on some interval \([0,a)\); \(A_ i(z)\) and \(D_ i(z)\) have nonnegative entries except possibly along their diagonals and \(B_ i(z)\), \(C_ i(z)\) are nonnegative matrices. Sufficient conditions on \(A_ i\), \(B_ i\), \(C_ i\), \(D_ i\) and \(R_{o,i}\) are given under which \(R_ 1(z)\geq R_ 2(z)\) or \(R_ 1(z)> R_ 2(z)\) for all \(z\) (in the componentwise sense) respectively.