Trélat, E. Some properties of the value function and its level sets for affine control systems with quadratic cost. (English) Zbl 0964.49021 J. Dyn. Control Syst. 6, No. 4, 511-541 (2000). 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. Cited in 1 ReviewCited in 23 Documents 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) Keywords:subanalyticity; regularity; endpoint mapping; optimal control; value function; abnormal minimizer; sub-Riemannian geometry PDF BibTeX XML Cite \textit{E. Trélat}, J. Dyn. Control Syst. 6, No. 4, 511--541 (2000; Zbl 0964.49021) Full Text: DOI arXiv OpenURL