Let M be a differentiable manifold, T x its tangent space at ε and TM={(x,y);x M; y M} bundle. A C 0 -Finsler structure is continuous function F:TM → [0, ∞ ) such that F(x, · ):T an asymmetric norm. In this work we introduce the Pontryagin type structures, which are structures satisfy minimum requirements of Pontryagin's maximum principle for problem minimizing paths. We define extended geodesi...

One of the key properties of the length of a curve is its lower semicontinuity : if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of pointwise convergence suffices. There are higher-dimensional analogs of this semicontinuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, e.g., [1] and [4]) im...

This paper is concerned with a symbolic-based synthesis of feedback control policies for hybrid and continuous dynamic systems. A key step in our synthesis procedure is a new method to solve the following dynamic programming problem: ∂V ∂t (z, t) = min v ∂V ∂z (z, t)ρv(z, v)v ż = ρ(z, v), t ∈ [0, T ] (1) V (z, T ) = Ψ(z, T ) (2) Here V (z, t) is the cost-to-go function associated with a certain...

In this note based on paper [3] we deal with domains D (i.e. connected open subsets) of a Finsler manifold (M, F ). At first we carry out a comparison between different notions of convexity for their boundaries. Then a careful application of variational methods to the geodesic problem yields that the convexity of ∂D is equivalent to the existence of a minimal geodesic for each pair of points of...

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples metrics are Randers with closed [Formula: see text]-forms. Another family well-known projectively flat on text]-sphere that have constant positive curvature. In this paper, we prove some and dynamical characterizations metrics, several rigidity resul...

Every Riemannian metric or Finsler on a manifold induces spray via its geodesics. In this paper, we discuss several expressions for the [Formula: see text]-curvature of spray. We show that sprays obtained by projective deformation using S-curvature always have vanishing text]-curvature. Then establish Beltrami Theorem with text].

In this paper we study geometries on the manifold of curves. We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S → lR (n ≥ 2, S is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h : S → lR of c. We discuss Riemannian and Finsler metrics F (c, h) on this manifold M , and in particular the case of the geom...