Solving a class of generalized Lyapunov operator differential equations without the exponential operator function (Q923196)

A method for solving operator differential equations of the type \[ (*)\quad X'(t)=A+BX(t)+X(t)D,\quad X(0)=C_ 0, \] avoiding the exponential operator function, is given. Section 1 contains an algebraic result that provides a finite algebraic expression of the solution of generalized algebraic Lyapunov operator equation, under certain uniqueness hypothesis. Section 2 concerns with the resolution of problem (*). Section 3 provides an explicit solution for a class of generalized Riccati operator differential equations in terms of a solution of certain generalized Lyapunov equation associated to the problem.