Some properties of the value function and its level sets for affine control systems with quadratic cost

Details Some properties of the value function and its level sets for affine control systems with quadratic cost Some properties of the value function and its level sets for affine control systems with quadratic cost

2006-07-18

arxiv.org/abs/math/0607424v1

Journal of Dynamical and Control Systems 6 (4) (2000) 511--541

Mathematics

Optimization and Control

Scientific paper

Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems: $\ds{\dot{x}=f\_0(x)+\sum\_{i=1}^m u\_if\_i(x)}$, with a cost of the form: $\ds{C(u)=\int\_0^T \sum\_{i=1}^m u\_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.

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