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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

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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