  Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $x\in M$
and $TM=\{(x,y);x\in M;y \in T_xM\}$ 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 Pontryagin's maximum
principle for the problem of minimizing paths. We define the extended geodesic
field $\mathcal E$ on the slit cotangent bundle $T^\ast M\backslash 0$ 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.
