On solution bounds for three types of Lyapunov matrix equations: continuous, discrete and unified equations

An approach is proposed to obtain a series of solution bounds for the unified algebraic Lyapunov matrix equation by making use of its feature, i.e., a mediator between the continuous-time and discrete-time Lyapunov equations. Due to the approach, every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation. The approach is endorsed by this simple fact: the three types of Lyapunov equations share the solution, provided certain relationships are maintained among corresponding coefficient matrices of the equations. Some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds.