Doubt about the extent of one condition for ODEs in order to achieve rest at equilibrium in finite time

Question

Doubt about the extent of one condition for ODEs in order to achieve rest at equilibrium in finite time

Intro

I am studying how differential equation could lead to solutions that stops moving in finite time.

For scalar autonomous ODEs of first and second order where investigated by V. T. Haimo on the papers Finite Time Differential Equations (1985) and Finite Time Controllers (1986), from where extended to its use in automatic control.

As example, some conditions required are the following:

1. The diff. eq. stands the trivial zero solution
2. The diff. eq. have at least one finite "singular point" in time $T\in(-\infty,\,\infty)$ where happens to be true $y(T)=\dot{y}(T)=0$

Here, in this singular point where $y(T)=\dot{y}(T)=0$ it is possible to stitch the traditional solution for $t<T$ with the zero solution for $t\geq T$: this is not as trivial at could be seen at first glance, since later I realized that no non-piecewise power series could reproduce this behavior since it would violate the Identity theorem, so I realized then that every differential equation I saw in engineering, since they could be represented by power series, were always of never-ending solutions in time, which looks wrong for daily life experience, at least for classic mechanics.

An example of this kind of differential equation would be: $$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,\,x(0)=1$$ Can stand the finite duration solution $$x(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$$ which have a finite ending time at $t=2$.

Question

In the paper of 1985 it is said:

"One notices immediately that finite time differential equations cannot be Lipschitz at the origin*. As all solutions reach zero in finite time, there is non-uniqueness of solutions through zero in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations."

$\mathit{(^*\!)\!\!:}$ in the paper is set the singular point at $t=T=0$.

My question is:

It is required for an ODE to be non-Lipschitz for having a finite time $0<T<\infty$ such it happen their non trivial not-always-zero solutions could become $y[T]=0 \wedge y'[T] = 0$ independently of the order of the ODE?

If you could come with a counterexample it would be great.

Answer 1

Yes. If the ODE is Lipschitz, then the solution can tend to zero only asymptotically and cannot reach zero in finite time. This is an immediate consequence of the local existence and uniqueness theorem for such ODEs (the Picard–Lindelöf theorem).

Suppose that the system reaches zero for the first time at time $T$. By the uniqueness theorem, there is an $\epsilon$ such that in the small interval from $T-\epsilon$ to $T+\epsilon$ the system has a unique solution. But one solution for this interval is that $y(t) \equiv 0$ in this interval (because $y(T)=y'(T)=0$). This is the backward time mentioned in the paper (we are running the equation backward from time $T$ to time $T-\epsilon$). Since the solution is locally unique, it follows that $y(T-\epsilon)=0$ contradicting the assumption that $T$ was the first time when zero was reached.

As for daily life experience, a system that reaches very close to zero is indistinguishable from zero. For example, if you weigh an object by putting it on one pan of a balance, you would produce a damped oscillation that decays to zero asymptotically. When you perceive that the balance is at rest, a very precise measurement might detect tiny oscillations still going on.