## Time optimal control of a second order nonlinear plant.(English)Zbl 0547.93040

A model of a nonlinear object is defined as $(1)\quad\ddot e+a\dot e|\dot e| +be=-Ku.$ It is assumed that the time-optimal control is of ”bang-bang” type. For $$u=\pm 1$$, trajectories of the solution of (1) are defined in the state plane $$(\dot e,e)$$. It is shown that, for $$u=\pm 1$$, the control system is a) totally stable at the origin if $$a/bK>0,$$ b) stable in some neighbourhood of the origin if $$b<0$$ and $$a/bK>0.$$ c) totally unstable at the origin if $$b=0$$ and $$a/K<0$$. The assumption that the time-optimal control is of ”bang-bang” type hasn’t been justified in the paper. It results from the author’s conviction that time-optimal control of second-order, nonlinear systems (shown in quoted literature) is of ”bang-bang” type. The analysis of the transition of state trajectories in $$E^ 2$$, performed for three values of the parameters a and b, is quite rudimentary. The paper must be considered as an elementary excercise of shape examination of trajectories in $$E^ 2$$, under the simple assumption that the control is of ”bang-bang” type.
Reviewer: W.Hejmo

### MSC:

 93C10 Nonlinear systems in control theory 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 93D99 Stability of control systems 93B03 Attainable sets, reachability 93C15 Control/observation systems governed by ordinary differential equations

### Keywords:

time-optimal control; second-order; nonlinear systems
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### References:

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