Questions tagged [finite-duration]

This tag is for questions of Finite-Duration Solutions to Differential Equations, which after an ending time by itself becomes zero forever after. For ordinary functions which have a starting and ending time, see [tag:piecewise-continuity], and if time is not the involved variable, search for [tag:compact-support]. Finite-Duration solution cannot be solutions of Linear ODE, since they fail uniqueness. Synonyms: [tag:finite-time], [tag:time-limited]

This tag is for questions of Finite Duration Solutions to Differential Equations, which after an ending time by itself becomes zero forever after. For ordinary functions which have a starting and ending time, see (piecewise-continuity), and if time is not the involved variable, search for functions of compact-support. Finite-Duration solution cannot be solutions of Linear ODEs, since as solutions they don´t fulfill Uniqueness. Synonyms: (finite-time), (time-limited), (finite-time-convergence), (finite-extinction-time), (sublinear damping), (endiness constraint)

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

Since uniqueness is not hold, the equation must have a point where is non-Lipschitz, so solutions cannot be analytic in the whole domain, but piecewise combinations of solutions with the zero function could be done, see Singular solutions.

For example, If the differential equation fulfills the following conditions it could stand finite duration solutions:

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

These conditions are sufficient for 1st and 2nd order autonomous scalar ODEs, but not nesessarilly required.

As example, the differential equation $$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,\,x(0)=1$$ 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 $t=2$.

As another example, the simplest system made by a brick sliding on an horizontal plane after an initial push, if the friction is modeled as the Coulomb damping, it will experience a uniform acceleration until it stop moving. The Newton's 2nd law that rules this system takes the form: $$x'' = -k\cdot g\cdot \text{sgn}(x')$$ where $k = \frac{\mu_k}{m} >0$, with $m$ the mass and $\mu_k$ the friction coefficient, and $g = 9.8\,\frac{m}{s^2}$ the Earth's gravity acceleration constant.

As is explained in detail here, it is possible to verify that under the endiness constraint of having a finite extinction time $T>t_0$ ($t_0$ the initial time) such as the solution fulfills $x(t)=0,\,\forall\,t\geq T$, then the system have the closed-form particular solutions: $$x(t) = \pm \,\frac{k\cdot g}{2}\cdot\left(T-t\right)^2\cdot\theta(T-t) \equiv \pm\,T^2\cdot\frac{k \cdot g}{8}\cdot\left(1-\frac{t}{T}+\left|1-\frac{t}{T}\right|\right)^2$$

Here is a list of examples of their use on physics:

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