Consider the following autonomous system of differential equations:

$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$

where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\mathbf v(\mathbf x)$ is Lipschitz continuous (therefore the solution is unique given initial conditions). Suppose that $\mathbf x^*$ is a fixed point, i.e., $\mathbf v(\mathbf x^*) = 0$. I need to characterize the stability of $\mathbf x^*$. However, $\mathbf v(\mathbf x)$ may not be differentiable (it is only Lipschitz continuous). Therefore the straightforward method of looking at the real parts of the eigenvalues of the Jacobian does not apply. I'm trying to find alternative characterizations.

Can someone point me to references in the literature or books about the stability of nonsmooth differential equations, where the right-hand side is Lipschitz continuous (like the system above)? Are there any helpful theorems in this topic?


I suggest looking at Lyapunov's direct method and possibly LaSalle's principle and their nonsmooth extensions.




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