The solvability conditions of the Riccati algebraic matrix equation (Q2755246)

The author considers the Riccati algebraic matrix equation \(AX-XB=\Phi+\epsilon X\Psi X\), where \(A,B,\Phi,\Psi\) are known matrices; \(X\) is unknown matrix; \(\epsilon\in[0,\epsilon_0]\) is a small parameter. Necessary and sufficient solvability conditions for the considered equation in the critical and the non-critical cases are obtained. Here the critical case means that the matrices \(A\) and \(B\) have common eigenvalues, the non-critical case means that the matrices \(A\) and \(B\) have no common eigenvalues. The conditions are formulated in terms of the Jordan structure of the matrices \(A\) and \(B\).