Abstract
Let \(T >0\) be fixed. We consider the optimal control problem for analytic affine systems: \(\dot x = f_0 (x) + \sum\limits_{i = 1}^m {u_i f_i (x)} \), with a cost of the form: \(C(u) = \int\limits_0^T {\;\sum\limits_{i = 1}^m {\;u_i^2 (t)\;dt} } \). 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.
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Trélat, E. Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost. Journal of Dynamical and Control Systems 6, 511–541 (2000). https://doi.org/10.1023/A:1009552511132
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DOI: https://doi.org/10.1023/A:1009552511132