Common quadratic Lyapunov function with all convex combinations Hurwitz

Question

I have two $n\times n$ Hurwitz stable matrices, so $A_i\in H_n$ for $i=1,2$. I also know that every possible convex combination of these two matrices is Hurwitz, i.e.,

\begin{equation} C(A_1,A_2) = cA_1+(1-c)A_2 \in H_n \hspace{0.5cm} \forall c\in[0,1]. \end{equation}

I would like to determine tractable conditions under which a common quadratic Lyapunov function exists for the unforced systems $\dot{x}(t)=A_i x(t)$ such that

\begin{equation} A_iP+PA_i^T = -Q_i \hspace{0.5cm} i=1,2 \end{equation}

with $P=P^T>0$ and $Q_i=Q_i^T>0$.

I am familiar with some results that might help here, but so far nothing is applicable. The matrices do not commute. I cannot establish that they are simultaneously triangularizable. They are generated from nonminimal systems and it is necessary that they are not expressed in companion form. They are of order $n$ which will certainly be greater than 2. Perhaps this is not possible. However, I was hoping that I could exploit the known stability of all convex combinations somehow.

I also have a related formulation that might be more feasible. Say the problem is now framed as an unforced time-varying system $\dot{x}(t)=A(t)x(t)$ with

\begin{equation} A(t) = c(t)A_1 +(1-c(t))A_2 \hspace{0.5cm} c(t)\in[0,1]. \end{equation}

The eigenvalues of $A(t)$ are always in the open LHP and $c(t)$ behaves nice enough to ensure that the system is exponentially stable. I would like to determine conditions under which a constant Lyapunov matrix exists \begin{equation} A(t)P+PA^T(t) = -Q(t). \end{equation} I view this as somewhat akin to finding a common Lyapunov function for a switched system that can switch between an infinite number of modes. Maybe I could restrict the time variation of $c(t)$ to help make this possible?

Edit: Looking at this from the time-varying system perspective, I think a converse Lyapunov theorem could help. If the system is exponentially stable for all permissible $c(t)$ trajectories then can I claim there exists a Lyapunov function satisfying \begin{equation} c_1\|x\|^2 \le V(x,t) \le c_2 \|x\|^2 \end{equation} \begin{equation} \dot{V}(x,t) \le -c_3 \|x\|^2 \end{equation} with $c_1,c_2,c_3>0$? Could I further claim that the form of such a function is \begin{equation} V(x,t) = x^TP(t)x \end{equation} with a smooth $P(t)=P^T(t)>0$ for an appropriate $c(t)$? Would there be any loss of generality in assuming this $P(t)$ to be a specific $P(t)$, i.e. one satisfying properties of another result I wish to use?

Edit 2: Is there any tractable way to relate the state transition matrices for $A_1$, $A_2$, and $A(t)$? I believe that I would have to work directly from the $\dot{\Phi} = A(t)\Phi$ statement to determine its transition matrix which seems daunting.

Answer 1

See

R. Shorten and K. Narendra, “Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems,” in Proc. Amer. Control Conf. , 1999, pp. 1410–1414.

and

R. Shorten, K. Narendra, and O. Mason, “A result on common quadratic Lyapunov functions,”IEEE Trans. Automat. Control , vol. 48, no. 1, pp. 110–113, Jan. 2003

The results are summarized in Theorem 1 of

Lin, H. and Antsaklis, P.J. (2009). Stability and stabiliz-ability of switched linear systems: A survey of recent re-sults.IEEE Transactions on Automatic Control , 54(2), 308–322. doi:10.1109/TAC.2008.2012009.