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Abnormal minimizers. (English) Zbl 0816.49019
The main purpose of this paper is to construct an example of a singular, abnormal minimizer for an optimal control system of linear controls, with $$k$$ controls, $$n$$ states, and a running cost function that is quadratic positive-definite in the controls. In this example given by Theorem 1, $$k= 2$$, $$n= 3$$ and the considered system is completely controllable. The given example is stable, and its importance is due, in part, to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature, which has been claimed that all minimizers are normal Pontryagin extremals. By this sample it follows that there exist minimizing sub-Riemannian geodesics that do not satisfy the sub-Riemannian geodesic equations.

##### MSC:
 49K40 Sensitivity, stability, well-posedness 49J15 Existence theories for optimal control problems involving ordinary differential equations 49Q20 Variational problems in a geometric measure-theoretic setting 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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