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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 2 x/dt 2 =g(x,x ˙) from an initial point (x 0 ,x ˙ 0 )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)d 2 x/dt 2 =-sgn(x)|x| a -sgn(x ˙)|x| b ,where0<b<1anda>b/(2-b),

and

(ii)d 2 x/dt 2 =-sgn(x)|x| a -sgn(x ˙)|x ˙| b +f(x)+d(x ˙),

where 0<b<1, a>b/(2-b)>0, f(0)=d(0)=0, O(f)>O(|x| a ) and O(d)>O(|x ˙| b )·

Remark: In Lemmas 1 and 2, statements like ”... with 0<S<T such that S<t<Tx(t)x ˙(t)<0···” should read ”... with 0<S<T such that x(t)ẋ(t)<0 for S<t<T···.”

Reviewer: J.Gayek
MSC:
93B05Controllability
93B03Attainable sets
93C10Nonlinear control systems
34D20Stability of ODE
93D05Lyapunov and other classical stabilities of control systems
34H05ODE in connection with control problems
93C15Control systems governed by ODE