A periodic singularly perturbed problem for the matrix Riccati equation (Q2577310)

The paper deals with the matrix Riccati equation \[ \mu^2\left(\dot\varrho+\varrho A(t,\mu)+A^T(t,\mu)\varrho+Q(t,\mu)\right)= \varrho B(t,\mu)R^{-1}(t,\mu)B^T(t,\mu)\varrho\tag{1} \] with the periodicity condition \[ \varrho (0,\mu)=\varrho (1,\mu)\tag{2} \] with respect to \(t\). Here, \(\mu\) is a positive small parameter, \(t\in \mathbb{R},Q(t,\mu)\) is a symmetric positive semidefinite matrix, and \(R(t,\mu)\) is a symmetric positive definite matrix, \(A(t,\mu)\), \(B(t,\mu)\), \(Q(t,\mu)\), and \(R(t,\mu)\) are 1-periodic matrices in \(t\) and sufficiently smooth with respect to \(t\) and \(\mu\) for \(t\in \mathbb{R}\) and \(\mu\in[0,\mu_0]\), \(0<\mu_0\ll 1\). A 1-periodic solution of problem (1), (2), is found, forming an extended periodic singularly perturbed problem to which the method of reduction to an initial value problem and the method of integral manifolds can be applied.