A. S. Shamaev

On the fundamental solution of the Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators.

On a complex fundamental solution of the Schrödinger equation

A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function.

Exact bounded boundary controllability of vibrations of a two-dimensional membrane

The boundary control of vibrations of a plane membrane is considered. A constraint is imposed on the absolute value of the control function. The goal of the control is to drive the membrane to rest. The proof technique used in this paper can be applied to a membrane of any dimension, but the two-dimensional case is considered for simplicity and illustrative purposes.