(This question was originally asked at Math.SE, where it didn't receive any answers.)

Consider the linear time-varying system $$ \dot{x} = A(t) x, $$

where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \rightarrow \mathbb{R}^{n\times n}$ is continuous.

It is known (see for instance, [1, Example 4.21]) that if there exists a positive definite matrix valued function $P:[0,+\infty) \rightarrow \mathbb{R}^{n\times n}$ that is continuously differentiable, bounded and satisfying the matrix differential equation

$$ - \dot{P}(t) = P(t)A(t) + A^{T}(t)P(t) + Q(t),$$

where $Q$ is a continuous, symmetric and positive definite matrix valued function, then the origin of the system is exponentially stable.

Supposing that $Q$ is also bounded, then the converse is valid (see [1, Theorem 4.12]). It is known any counter-example for the case that $Q$ is not bounded?

(1): Hassan K. Khalil (2002). *Nonlinear Systems*. Prentice Hall.