Complete characterization of quadratic Lyapunov functions for planar discrete systems. (Q1434753)

The object of this paper is the switched discrete linear system \[ x(k+1) = A_{\sigma(k)}x(k) \] in dimension \(2\). Here \(\sigma(k):\{0,1,2,\dots\}\rightarrow \{1,2,\dots,N\}\) is a piecewise constant function called switching signal that makes the choice between the \(N\) subsystems defined by \(A_{\sigma(k)}\). Exponential stability of the switched system may be achieved by a common quadratic Lyapunov function. The paper deals with the complete description of this type of Lyapunov function for the dimension \(2\).