Lyapunov Test for Controllability

If the eigenvalues of $A$ have strictly negative real parts, then the system is controllable if and only if the matrix $W$ is a Positive-Definite Matrix

Positive-Definite Matrix

A symmetric matrix with all positive eigenvalues. NOTE: All symmetric matrices have only real eigenvalues

For a matrix, $M$, to be positive-semidefinite the following has to hold

Criterion

$x^{T}Mx>0,\forall x\in {\mathbb{R}}^n\backslash \{0\}$ Given that $M$ is a $n\times n$ symmetric real matrix

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Can be written generally as

$$M=\left[\begin{array}{cc}{m}_1& m_2\\ m_2& m_3\end{array}\right]$$Link to original

$$AW+W{A}^T=-B{B}^T$$

Where $W$ is a Symmetric Matrix