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Some properties of the value function and its level sets for affine control systems with quadratic cost. (English) Zbl 0964.49021
Summary: Let $$T>0$$ be fixed. We consider the optimal control problem for analytic affine systems: $$\dot x= f_0(x)+ \sum^m_{i=1} u_if_i(x)$$, with a cost of the form: $$C(u)= \int^T_0 \sum^m_{i=1} u^2_i(t) dt$$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $$S$$ is subanalytic. Second, we prove that if there exists an abnormal minimizer of corank 1, then the set of endpoints of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.

##### MSC:
 49N60 Regularity of solutions in optimal control 53C17 Sub-Riemannian geometry 49K40 Sensitivity, stability, well-posedness 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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