The inverse dynamics control in robotic applications yields the error system \begin{equation} \ddot{\mathbf{e}} + \mathbf{K}_1 \dot{\mathbf{e}} + \mathbf{K}_0 {\mathbf{e}} = \mathbf{0} \end{equation} or rewritten as ODE-system $$ \frac{d}{dt} \begin{bmatrix} \mathbf{e} \\ \dot{\mathbf{e}} \end{bmatrix} = \underbrace{ \begin{bmatrix} \mathbf{0} & \mathbf{I}\\ -\mathbf{K}_0 & -\mathbf{K}_1 \end{bmatrix} }_{\mathbf{A}} \begin{bmatrix} \mathbf{e} \\ \dot{\mathbf{e}} \end{bmatrix} \text{ .} $$ To prove asymptotic stability of the error system, A has to be a Hurwitz-Matrix.
Literature says that it is sufficient for $\mathbf{K}_0$ and $\mathbf{K}_1$ to be positive definite to guarantee asymptotic stability of the error system.
During the proof the assumption of $\mathbf{K}_0 = diag\{k_{0,1}, \dots, k_{0,n} \}$ and $\mathbf{K}_1 = diag\{k_{1,1}, \dots, k_{1,n} \}$ was made. This in return yields a decoupled system $$ \frac{d}{dt} \begin{bmatrix} {e_j} \\ \dot{{e}}_j \end{bmatrix} = \underbrace{ \begin{bmatrix} {0} & 1\\ -{k}_{0,j} & -{k}_{1,j} \end{bmatrix} }_{\mathbf{A}_j} \begin{bmatrix} {e}_j \\ \dot{{e}_j} \end{bmatrix} \text{ .} $$ for $j=1, \dots, n$ with the characteristic polynomial of $\mathbf{A}_j$ $$ p_j(s) = s^2 + k_{1,j} s + k_{0,j} $$ which is a Hurwitz polynomial for $k_{1,j} > 0$ and $k_{0,j} > 0$, hence guarantees the asymptotic stability of the decoupled system.
$\textbf{BUT}$ I have not found a proof which relies only on the assumption of $\mathbf{K}_0$ and $\mathbf{K}_1$ to be positive definite.
I tried solving the Lyapunov equation $$ \mathbf{A}^{T} \mathbf{P} + \mathbf{P} \mathbf{A} + \mathbf{Q} = \mathbf{0} $$ with the positive definite matrices $\mathbf{P}$ respectively $\mathbf{Q}$ without much luck.
How can I proof asymptotic stability of the above-mentioned error system with only the assumption of $\mathbf{K}_0$ and $\mathbf{K}_1$ to be positive definite?
Any help would be much appreciated!
A related post lead me to the answer: Help! Lyapunov proof for calculated torque control with friction term for robot
The Lyapunov function $$ V(\mathbf{e}, \dot{\mathbf{e}}) = \frac{1}{2} \dot{\mathbf{e}}^{T} \dot{\mathbf{e}} + \frac{1}{2} {\mathbf{e}}^{T} \mathbf{K}_0{\mathbf{e}} $$ yields $V(\mathbf{e}, \dot{\mathbf{e}}) > 0$ in case $\mathbf{K}_0$ is positive definite and \begin{align} \dot{V}(\mathbf{e}, \dot{\mathbf{e}}) &= \dot{\mathbf{e}}^{T} \ddot{\mathbf{e}} + \dot{\mathbf{e}}^{T} \mathbf{K}_0 \mathbf{e} \\ &= - \dot{\mathbf{e}}^{T} \mathbf{K}_1 \dot{\mathbf{e}} \end{align} yields $\dot{V}(\mathbf{e}, \dot{\mathbf{e}}) \leq 0 $ in case $\mathbf{K}_1$ is positive definite.
According to LaSalle's invariance principle the error system is asymptotic stable as the largest invariant set $\mathcal{M} \subseteq \left\{\mathbf{e}, \dot{\mathbf{e}} \in \mathbb{R}^n ~|~ \dot{V}(\mathbf{e}, \dot{\mathbf{e}}) = \mathbf{0}\right\}$ is the origin itself: $\mathcal{M} = \{\mathbf{0}\} $.