Global asymptotic stability of generalized Liénard equation (Q1898423): Difference between revisions

The authors present new sufficient conditions for global asymptotic stability of the zero solution of the system \(\dot x= \varphi(y)- F(x)\), \(\dot y= -g(x)\), which is equivalent to the generalized Liénard equation. The crucial conditions read \[ \limsup_{x\to +\infty} \Biggl(F(x)+ \int^x_0 {g(x)\over 1+ F_-(x)} dx\Biggr)= +\infty, \] \[ \limsup_{x\to -\infty} \Biggl(- F(x)+ \int^x_0 {g(x)\over 1+ F_+(x)} dx\Biggr)= +\infty, \] where \(F_+(x)= \max\{0, F(x)\}\), \(F_-(x)= \max\{0, - F(x)\}\). Proofs are based on the Lyapunov function \(V(x, y)= G(x)+ \int^y_0 \varphi(s) ds\), where \(G(x)= \int^s_0 g(s)ds\), and on discussion of properties of the trajectories in the \((z, y)\) phase plane obtained by the Filippov transformation \(z= G(x)\).