In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.

No abstract

In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on a manifold M and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of an accessory problem on the tangent space to M at the final point of the trajectory. Two examples are proposed.

We consider a functional of the type F(u) = Ω F (x, u, . . . , D k u) dx, where Ω is an open bounded set of R n and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of F with respect to the weak topology of W k,p (Ω; R m ) under p-growth conditions for the integrand F .

When the plane is pie-sliced in n ≤ 4 parts (with nonempty interior and common vertex at the origin) our main result provides sufficient conditions for any map L, that is continuous and piecewise linear relatively to this slicing, to be invertible. This result cannot be plainly extended to a greater number of slices. Also, some examples show that the assumptions cannot be relaxed too much. Our result is proved by a combination of linear algebra and topological arguments.