Matrix Differential Equations and Lyapunov Transformations

Abstract

The main purpose of this paper is to establish necessary conditions and sufficient conditions for the existence of a solution Q(t) of the Lyapunov matrix differential equation of the form

$${{dQ\left( t \right)} \over {dt}} = A\left( t \right)Q\left( t \right) - Q\left( t \right)B\left( t \right) + C\left( t \right),t \in \left[ {\gamma ,\delta } \right]$$

where$Q\left( t \right),{{dQ\left( t \right)} \over {dt}},{Q^{ - 1}}\left( t \right){\left[ {{{dQ\left( t \right)} \over {dt}}} \right]^{ - 1}}$ are bounded for t ∈ [γ, δ]. Such solutions are used to generate Lyapunov transformations which map one differential system to another system which has the same stability properties. Solutions of higher order Riccati matrix differential equations are used to uncouple stiff linear time-varying differential systems.