- #1
Liferider
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Homework Statement
State the strongest stability property of the system (stable, asymptotically/exponentially):
\begin{align}
\dot{x_1} &= x_2 \\
\dot{x_2} &= -x_1 e^{x_1 x_2}
\end{align}
Homework Equations
With the Lyapunov function candidate:
\begin{equation}
V(x) = \frac{1}{2}(x_1^2 + x_2^2)
\end{equation}
The Attempt at a Solution
V(x) is pos. def. and continously differentiable.
\begin{align}
\dot{V}(x) &= x_1 \dot{x_1} + x_2 \dot{x_2} \\
&= x_1 x_2 - x_1 x_2 e^{x_1 x_2} \\
&= x_1 x_2 (1 - e^{x_1 x_2})
\end{align}
Here, i feel like I must state that we can not conclude on the stability of the system since V-dot is not negative definite or semi-definite.
The answer is however, that it is negative semi-definite... I can not see how that is possible. Any help here please?