Given the linear system $\dot x = Ax + bu$, $x(0) = x_0 \in \mathbb R^n$ and $A \in \mathbb R^{n\times n}, b \in \mathbb R^n$.
The functional to be minimized w.r.t the system is given as $$J(u;x_0) = \int_0^\infty \gamma u^2 - \frac{1}{\gamma}x^\top Q x \hspace{1mm}dt,$$ where $\gamma >0$ and $Q\succeq 0.\\$
Let $P \in \mathbb R^{n\times n}$ be the positive semi-definite solution of $$PA+A^\top P + \frac{1}{\gamma}Q+ \frac{1}{\gamma}Pbb^\top P = 0.$$ Show that $J(u; x_0) \geq -x_0^\top Px_0 $ for abitrary choice of $u$.
My thoughts:
With a change of sign in the Riccatti equation and the cost functional, this problem is equivalent to the standard LQR problem, for which $J \geq x_0^\top P x_0$. Is this enough to conclude the lower bound for $J$ here? How would I prove this in good fashion using standard LQR theory (i.e. no dynamic programming etc.)?