I have the following problem. Consider a system \dot x1 = \mu f1(x1,x2,x3), \dot x2=\mu f2(x1,x2,x3), \dot x3 = f3(x1,x2,x3), where 1 \le \mu \le M and f_i's are polynomial. If I find a Lyapunov function in the vertex \mu = 1 and a Lyapunov function in the vertex \mu = M, can I conclude that the system is asymptotically stable for all 1 \le \mu \le M?
I couldn't read what you have written very well. I think if the expressions are affine with respect to \mu, you can use the vertices of the polytope instead of the whole polytope.
It depends. The reason is this: The fundamental assumption is that your system \dot{X} = f(X) where X represents the variables x1, x2 and x3, has a fixed point X*. Your Lyapunov function is a continuously differentiable real-valued function V(X) such that: (i) V(X)>0 for all X \neq X* and V(X*)=0, and (ii) \dot{V} < 0 for all X \neq X*; only when these two conditions are met, the fixed point X* is globally asymptotically stable. Hence the "depends" in the opening sentence; your parameter \mu could change the stability of your system. Therefore, provided that there are no bifurcations of f(X) for 1 \le \mu \le M, then the system is globally asymptotically stable at a fixed point X*. A final comment: you can claim that your system is asymptotically stable around a given fixed point; this precision is important because you could have multiple fixed points, and not all are necessarily asymptotically stable even if you find a Lyapunov function that works in one of them.
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