I faced some nonlinearity in my problems. I need to check a matrix inequality condition in order to check the feasibility of designed controller through a continuous design problem. My problem is that I need an LMI to be checked but my developed formula in NMI in a bad style, could anybody help me with transforming this NMI to LMI: \ $x$: decision variables(unknown) $P$: symmetric positive definite matrix(unknown) The original NMI condition: $$ A_0(x)PA_1(x)^\top+A_1(x)PA_0(x)^\top +a A_0(x)PA_0(x) <0 $$ but after using Schur complement I developed following inequality: $$ M:=\left( \begin{array}{cc} A_0(x)PA_1(x)^\top+A_1(x)PA_0(x)^\top & A_0(x)^\top \\ A_0(x) & -aP^{-1} \end{array} \right)<0, $$ meaning $M$ is negative definite.
another transformation of this MI is this with is some how relaxed but one nonlinear term is still left: $$ N:=\left( \begin{array}{cc} 0 & A_0(x)^\top/a+A_1(x)^\top \\ A_0(x)/a+A_1(x) & -aP^{-1} \end{array} \right)<0, $$ this yields to this inequality: $$ A_0(x)PA_1(x)^\top+A_1(x)PA_0(x)^\top +a^{(-1)} A_0(x)PA_0(x) + a A_1(x)PA_1(x)<0 $$