Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost

AbstractLet $$T >0$$ 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.