The method of rotating Lyapunov functions in the stabilization problem for two-dimensional bilinear systems (Q5942103)

Considering the system \[ \dot x= Ax+ uBx, \] where \(x\in E\), \(\dim x= 2\), \(A\) and \(B\) are \(2\times 2\) matrices and \(u\) is a control function, the stabilization by a constant \(u\) is not always possible. For these last cases the authors consider the piecewise constant (relay) control designed via the theory of variable structure systems by choosing a sliding surface \(\sigma(x)= 0\) and defining such a control \(u(x)\) so as to create a sliding mode on this line; at its turn \(\sigma(x)\) is such that the motion on the line \(\sigma= 0\) is asymptotically stable. The basic problem is to find hitting conditions; i.e. reaching \(\sigma(x)= 0\) in finite time.