# 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?

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.

• Yes "Obeying a Ricatti equation is not synonymous to obeying a Lyapunov equation". That is the key point. Thank you. And in the LQR case, where the cost to be minimized is quadratic, Lyapunov equation is equivalent to Ricatti equation. I have read several versions of these sentences but I failed to understand some concepts. All Ricatti equations are Lyapunov equations, but vice versa does not hold. right? – user252783 Jun 20 '16 at 6:53
• @user252783 I'm not sure what is saying "all Ricatti equations are Lyapunov equations". As I said, they have to do with totally different properties of the system--optimality vs. stability. Perhaps you can argue this must practically be the case, since a reasonable optimization functional will also force the system to be stable (again, you can think of the LQR case) but there is no mathematical reason I'm aware of that lets you make this association a priori. – JMJ Jun 20 '16 at 20:43

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.