Research

A general theory that allows to determine whether a given solution is stable or not was developed by Jacobi. He introduced the notion of conjugate points and later Morse gave a functional analytic interpretation of his result -- each time we cross a conjugate point an eigenvalue of the Hessian crosses zero and becomes negative. Such results are known as Morse theorems. The only problem with Jacobi's and Morse's original approach is that it requires the extremals to be smooth. But now we have plenty of natural examples from optimal control, where optimal solutions are not regular enough. So in a joint work with A. Agrachev, we extended the theory to cover those cases as well. There is little resemblance to the classical theory learned in standard courses of calculus of variations, but all the classical results can be derived as special cases. We saw that the information about the Hessian along an extremal curve (of any type: normal, abnormal, bang-bang, their concatenations etc) can be encoded in a curve in the Lagrangian Grassmanian. In order to find the negative inertia index, one has to compute its symplectic invariant, known as the Maslov index. Check out this article and this one for more information or the CIME notes of A. Agrachev.

In a work with A. Agrachev and S. Baranzini we have obtained Morse index formulas for graph parametrised optimal control problems. They can be used to study of fine properties of ground state of the nonlinear Schroedinger equation on metric graphs or the stability of various structures in civil engineering.