Geodesic fields for Pontryagin type $C^0$-Finsler manifolds

Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $xin M$ and $TM={(x,y);xin M;y in T_xM}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM rightarrow mathbb [0,infty)$ such that $F(x,cdot): T_xM rightarrow [0,infty)$ is an asymmetric norm. In this work we introduce the Pontryagin type $C^0$-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagins maximum principle for the problem of minimizing paths. We define the extended geodesic field $mathcal E$ on the slit cotangent bundle $T^ast Mbackslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $mathcal E$ than by a similar structure on $TM$. Finally we show that the maximum of independent Finsler structures is a Pontryagin type $C^0$-Finsler structure where $mathcal E$ is a locally Lipschitz vector field.