Asymptotic stability of a nonlinear, damped, mass spring system

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\begin{example}[Nonlinear spring mass system with damper] \index{spring mass system} \action{KJA}{Relabel as nonlinear oscillator? here and in other chapters. [later]} Consider a nonlinear, damped spring mass system with dynamics \begin{displaymath}

 \aligned 
   \fder{x_1}{t} &= x_2 \\
   \fder{x_2}{t} &= -b(x_2) - k(x_1)
 \endaligned

\end{displaymath} Here $b$ and $k$ are smooth functions modeling the friction in the damper and restoring force of the spring, respectively. We will assume that $b, k$ have the property \index{passivity} \begin{displaymath}

 \aligned
   \sigma b (\sigma) & \ge 0 \quad \forall \sigma \in [ - \sigma_0 ,
   \sigma_0] \\
   \sigma k (\sigma) & \ge 0 \quad \forall \sigma \in [ - \sigma_0,
   \sigma_0],

\endaligned \end{displaymath} where equality is only achieved when $\sigma = 0$.

Consider the Lyapunov function candidate \begin{displaymath}

 V(x) = \frac{x_2^2}{2} + \int_0^{x_1} k (\sigma) \, d \sigma,

\end{displaymath} which is positive definite and gives \begin{displaymath}

 \fder{V(x)}{t} = - x_2 b (x_2) .

\end{displaymath} Choosing $c = \hbox{min} ( V ( - \sigma_0, 0) , V (\sigma_0, 0))$ so as to apply the Krasovskii-Lasalle principle, we see that \begin{displaymath}

 \fder{V(x)}{t}  \le 0 \quad \mbox{for} \; x \in \Omega_r := \{ x : V(x)
 \le c \}. 

\end{displaymath} As a consequence of the Krasovskii-Lasalle principle, the trajectory enters the largest invariant set in $\Omega_r \cap \{ x_1, x_2 : \dot V = 0 \} = \Omega_r \cap \{ x_1 , 0 \}$. To obtain the largest invariant set in this region, note that \begin{displaymath}

 x_2(t) \equiv 0 \quad\implies\quad
 x_1(t) \equiv x_{10} \quad\implies\quad
 \dot x_2(t) = 0 = - b(0) - k(x_{10}),

\end{displaymath} where $x_{10}$ is some constant. Consequently, we have that \begin{displaymath}

 k(x_{10}) = 0 \quad\implies\quad x_{10} = 0.  

\end{displaymath} Thus, the largest invariant set inside $\Omega_r \cap \{ x_1, x_2 : \dot V = 0 \}$ is the origin and, by the Krasovskii-Lasalle principle, the origin is locally asymptotically stable. \end{example}