Lyapunov equation, uniqueness, stability. With respect to $\dot{x}(t)=Ax(t)+Bu(t)$, and $u=-Kx(t)$, and $K=R^{-1}B^{\rm T}P$, where P solution of $\tilde{A}^{\rm T}P+P\tilde{A}+Q+PBR^{-1}B^{\rm T}P$, where  $\tilde{A}=A-BK$.
I believe that, for a given $A, B, Q, R$,  the optimal control $u^*$, in terms of the solution $P$ of the algebraic ricati equation,  $A^{\rm T}P+PA+Q-PBR^{-1}B^{\rm T}P$ is unique.
I know that, for the given system, there exists several $K_i$, $i=1,2,\dotsc$ that stabilizes the system. My confusion is do each of all those $i$ number of $K$s also should meet a Lyapunov equation such as  $\tilde{A}_i^{\rm T}P+P\tilde{A}_i+Q+PBR^{-1}B^{\rm T}P$, where $\tilde{A}=A-BK_i$? as they are stabilizing. 
I know that $P$ matrix that is solution of Lyapunov equation stabilizes the closed-loop. Does it implies that all the $P$ matrices, that makes the system stable should meet a Lyapunov equation?
Is it that the $P$ matrix that represents optimal control is unique and meets algebraic Ricati equation and other stationarity conditions for optimality.  ?
And the $P$ matrices that stabilizes the system just meets a Lyapunov equation?
 A: Lyapunov's theorem guarantees that any full state feedback gain $K$ for which $\dot{x} = (A-BK)x$ is globally asymptotically stable must also satisfy a Lyapunov equation. E.g. see here. For a system to be GAS we must have some $x_e\in X$, where $X$ is the state space for the system $\dot{x} = f(x)$, for which $f(x_e) = 0$ and $\lim_{t\rightarrow\infty}x(t) = x_e$ for any path $x(t)$. Since the typical LTI stability criterion (e.g. $\mathfrak{Re}\{\lambda\}< 0$ for all $\lambda \in\Lambda(A-BK)$) is equivalent to GAS for LTI systems (e.g. see here), then there are unique pairs of positive, symmetric matrices $P_i,Q_i$ for which $A-BK_i$ obeys a Lyapunov equation for each $K_i$ for which $A-BK_i$ is stable in the typical LTI sense. 

I know that $P$ matrix that is solution of Lyapunov equation stabilizes the closed-loop. Does it implies that all the $P$ matrices, that makes the system stable should meet a Lyapunov equation?

Yes, but $P$ is a unique $P_i$ for each $K_i$. 

Is it that the $P$ matrix that represents optimal control is unique and meets algebraic Ricati equation and other stationarity conditions for optimality. ?

Generally obeying a Ricatti equation is not synonymous with obeying a Lyapunov equation--the first is an optimality constraint, the second is a stability constraint. In fact, they only case of equivalence I can think of is LQR/LQG problems where the Lyupinov function happens to be a quadratic form similar to the integrand of the optimization functional. 

And the $P$ matrices that stabilizes the system just meets a Lyapunov equation?

There is a unique $P_i$ which stabilizes the system which is also the unique solution to the Lyapunov equation for that system. 
A: Sorry for the short answer without details. But you will find your answer in the following slides. 
You can start reading from slide 10 of the first slide set.
http://www.dcsc.tudelft.nl/~sc4160/download/hs_slides_chapter_4.pdf
http://www.dcsc.tudelft.nl/~sc4160/download/hs_slides_chapter_5.pdf
