I am trying to establish stability of linear dynamical system ($\dot{x}=Ax$) with non-hyperbolic equilibrium point. While soling linear matrix inequality ($A^\top P+PA \prec 0$) based on a suitable quadratic Lyapunov function candidate ($x^\top P x$), the solution happens to be a singular positive semidefinite matrix. My quation is how to get a positive matrix as a solution for the corresponding LMI?
More explicitly, consider a linear system $\dot{x}_1=x_2, \ \dot{x}_2=-x_1$ and corresponding Lyapunov function candidate $V(x)=x^\top Px, P\succ 0.$ Then, how to find a positive definite solution of the linear matrix inequality $A^\top P+PA \prec 0$?