We have developed a method and algorithms for solving the generalized Lyapunov
equation for a wide class of continuous time-varying bilinear systems based on the Gramian
method and an iterative solution construction method proposed earlier for such equations. The
approach consists in diagonalizing the original system, obtaining a separable spectral decomposition
of the Gramian of the time-invariant linear part in terms of the combination spectrum of
the dynamics matrix of the linear part, applying the spectral decomposition of the kernel matrix
of the solution obtained at the previous step at each iteration step, and then aggregating the
matrix entries. A spectral decomposition of the Gramians of controllability and observability of
a time-varying bilinear system is obtained as the sum of sub-Gramian matrices corresponding to
pair combinations of the eigenvalues of the dynamics matrix of the linear part. A new method
and algorithm for entry-by-entry calculation of matrices for solving the generalized Lyapunov
equation for bilinear systems has been developed. The fundamental novelty of the approach lies
in the transfer of calculations from the solution matrix to the calculation of the sequence of its
entries at each iteration step.