Solving the matrix differential equation $\dot \Delta P(t) = (A + P(t)C^{T}R^{-1}C)\Delta P(t) + \Delta P(t)(A^{T} + C^{T}R^{-1}CP(t))$

Question

Here $P, \Delta P \in \mathbb{R}^{N X N}$ The initial condition $\Delta P(0)$ is given and the dynamics of $P(t)$ is known. $ A,C,Q,R$ are constant matrices of compatible dimensions. Since it is a linear expression in $\Delta P$, can I express the solution in terms of a state transition matrix ? ( Are state transition matrices even defined for matrix differential equations ? ) I have never dealt with matrix differential equations like these and am clueless on how to proceed.

Answer 1

I'm assuming $P(t)$ and $\Delta P(t)$ are unrelated, since you said that equation is linear. So consider

$$ \dot{X} (t) = A(t) X(t) + X(t) B^T(t) $$

where $A(\cdot), B(\cdot), X(\cdot) \in \mathbb{R}^{n \times n}$. Then a solution to this equation is

$$ X(t) = \Phi(t) \Psi^T(t) $$

where $\Phi(t)$ is a solution to the equation $\dot{\Phi}(t) = A(t) \Phi(t)$ and $\Psi(t)$ is a solution to the equation $\dot{\Psi}(t) = B(t) \Psi(t)$.