Regarding the Conditions given in a Stability Analysis of Differential Equations

Question

Came across the following in the text "Nonlinear Systems" by Khalil. It is an example about the stability analysis of a second-order system where the stability is determined by a Lyapunov function employing the "variable gradient method". Below are images of the entire example. My question is, what is the implication of the condition $yh(y)>0$ for all $y\neq 0$?

Answer 1

This condition implies that

• the value of the term $\delta\int_0^{x_1} h(y)\, dy$ of the Lyapunov function $V(x)$ is positive for all $x_1\ne 0$;
• the value of the term $-\gamma x_1 h(x_1)$ of the derivative $\dot V(x)$ is negative for all $x_1\ne 0$.

Indeed, $\gamma>0$, $\delta>0$, the integral $\int_0^{x_1} h(y)\, dy$ is positive for $x_1\ne 0$ since $y$ in the integrand and, thus, $h(y)$, has the same sign as $x_1$.