Show that $A^TW_0+W_0A=-C^TC$

Question

Let $W_0$=$\int_{0}^{\infty}e^{A^Tt}C^TCe^{At}\mathrm{d}T$,$A\in\mathbb R^{n*n}$ and all its eigenvalues are negative.

Show that $A^TW_0+W_0A=-C^TC$

Really have no idea about how to prove it.

Answer 1

I assume that they tell you that the eigenvalues of $A$ are negative in order to ensure that the integral converges. I also assume that you copied the integral incorrectly, which is to say that the second $A$ in the exponent should not be transposed.

If we know that the integral converges, then we can write $$ A^TW_0 + W_0 A = \\ A^T\left(\int_{0}^{\infty}e^{A^Tt}C^TCe^{At}dt\right) + \left( \int_{0}^{\infty}e^{A^Tt}C^TCe^{At}dt\right)A = \\ \int_{0}^{\infty}\left(A^Te^{A^Tt}C^TCe^{At} + e^{A^Tt}C^TCe^{At}A\right)dt = \\ \int_{0}^{\infty}\frac{d}{dt}\left( e^{A^Tt}C^TCe^{At}\right)dt $$ I'm sure you can take it from there.