Integral Formulae Associated with the Euler-Lagrange Operators of Multiple Integral Problems in the Calculus of Variations. (Short Communication).
Integral Formulae Associated with the Euler-Lagrange Operators of Multiple Integral Problems in the Calculus of Variations.
Interior sphere property for level sets of the value function of an exit time problem
We consider an optimal control problem for a system of the form = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.
Invariance of global solutions of the Hamilton-Jacobi equation
We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.