On the solution of one matrix differential equation

Question

We have the next equation $$\dot K=AK+KA^T+HH^T,$$ where $K(t),A(t),H \in R^{n \times n}, K-$is a symmetric matrix. How one can prove that the solution of this equation is as follows: $$K(t)=\int_0^te^{As}HH^T e^{A^Ts}ds$$ on $ [0,t]$, provideed that $K(0)=\boldsymbol{0}$?

Answer 1

This is a continuous-time Lyapunov differential equation (cf. this post and references therein [1]), which unique solution is given by $$ K(t) = \int_0^t \Phi_{A^\top}^\top(t,s)\, H H^\top \Phi_{A^\top}(t,s) \,\text d s $$ where $\Phi_{A^\top}$ is the transition matrix for the system $\dot X = A^\top X$, i.e., the matrix $\Phi_{A^\top}(t,s)$ solves $$ \dot X(t) = A^\top(t)\, X(t), \qquad X(s) = I . $$

---

[1] J.M. Davis, I.A. Gravagne, R.J. Marks II, A.A. Ramos, "Algebraic and dynamic Lyapunov equations on time scales", In 42nd Southeastern Symposium on System Theory (SSST), 2010, arXiv:0910.1895v1