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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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