Conservation of a bounded solution of the Riccati equation under a small perturbation (Q909089)

The author considers the differential equation \[ (1)\quad dX/dt=XA_ 1(t)X+A_ 2(t)X+XA_ 3(t)+A_ 4(t)+\epsilon F(t,X) \] where X is an \((r_ 1\times r_ 2)\)-matrix and \(A_ i(t)\), \(i=1,...,4\), are continuous and bounded matrix functions of appropriate dimensions. He assumes, in addition, that the \(A_ i(t)\) satisfy a certain definiteness condition and that the matrix function F(t,X) is bounded and Lipschitzian for all \(t\in R\), \(\| X\| <1\). He proves that under these assumptions there exists some \({\bar \epsilon}>0\) such that for each \(\epsilon\in [-{\bar \epsilon},{\bar \epsilon}]\) equation (1) has a unique solution \(X^*(t,\epsilon)\) that satisfies the inequality \(\| X^*(t,\epsilon)\| <1\) for all \(t\in R\).