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Finite time controllers. (English) Zbl 0603.93005
Necessary and sufficient conditions are given for the solution of the ordinary differential equation $$d^ 2x/dt^ 2=g(x,\dot x)$$ from an initial point $$(x_ 0,\dot x_ 0)\in R^ 2$$ to arrive at (0,0) in finite time, where $$g(0,0)=0$$, and g is C except at (0,0) where it is continuous. In particular, the following classes of second-order systems result in trajectories which reach (0,0) in finite time:

$(i)\quad d^ 2x/dt^ 2=-sgn(x)| x|^ a-sgn(\dot x)| x|^ b,\text{ where } 0<b<1\quad and\quad a>b/(2-b),$ and $(ii)\quad d^ 2x/dt^ 2=-sgn(x)| x|^ a-sgn(\dot x)| \dot x|^ b+f(x)+d(\dot x),$ where $$0<b<1,$$ $$a>b/(2-b)>0,$$ $$f(0)=d(0)=0,$$ $$O(f)>O(| x|^ a)$$ and $$O(d)>O(| \dot x|^ b).$$
Remark: In Lemmas 1 and 2, statements like ”... with $$0<S<T$$ such that $$S<t<Tx(t)\dot x(t)<0...$$” should read ”... with $$0<S<T$$ such that x(t)ẋ(t)$$<0$$ for $$S<t<T...$$.”
Reviewer: J.Gayek

##### MSC:
 93B05 Controllability 93B03 Attainable sets, reachability 93C10 Nonlinear systems in control theory 34D20 Stability of solutions to ordinary differential equations 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 34H05 Control problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
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