Does the solution exist for the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$?

Question

I am studying the consensus problem of a linear multi-agent system, and a control gain needs to be designed to reach the consensus property. But the design procedure depends on the solution of the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$, and $A$ is not Hurwitz. Does the linear matrix inequality $A^TP+PA-Q<0$ have a solution $P$?