How to derive the Lyapunov Equation?

Question

I have a linear dynamical system: $\dfrac{dx}{dt} = Ax$ with $A \in \mathbb{R}^{n\times n}$ and $x \in \mathbb{R}^{n\times1}$.

I consider a quadratic function $V(x) = x^T P x$ where $P \in \mathbb{R} ^{n\times n}$ is a symmetric matrix, i.e., $P^T = P$.

I want to show that:

$\dot{V} = \dfrac{\partial V}{\partial x} \dfrac{dx}{dt} = x^T(A^TP + PA)x$.

Which rules do I need and how do I do it?

P.S.: I tried to use the findings in this post, but I can not reach the final equation: How to take the gradient of the quadratic form?

Yes it is a substep of the so called "Lyapunov Equation". This final form I have given is from a control theory book. So yes, this is the final result — OnurA, May 16 '20 at 16:06

score 1 · Accepted Answer · answered Jun 30 '20 at 14:07

1

$V'=x'^TPx+x^TPx'=(Ax)^TPx+x^TPAx=x^T(A^TP+PA)x$.

answered Jun 30 '20 at 14:07

1 Answers1