The focus of the proposed research is to analyze global controllability and stabilizability properties of several different types of physical processes described by nonlinear partial differential equations (PDEs). The control acts through actuators located on the boundary or concentrated in some subdomain. The primary goals are to establish the theoretical possibility of global controllability, and to obtain an explicit construction of the control strategy. The latter will be in the form of asymptotic formulae or by reduction of original nonlinear problem to a much simpler controllability problem for linear PDEs or integral equations capable of numerical solution. For example, if a physical process is described by an integrable nonlinear PDE , we reduce the question of controllability to the analysis of a controllability problem for the Gelfand-Levitan-Marchenko integral equation and then try to solve this problem asymptotically. For a physical process described by a nonintegrable PDE we plan to adapt Coron's return method, which allows construction of asymptotic solutions to a large class of controllability problems with control distributed over the whole boundary or just a part of the boundary. The main analytical tools will be the inverse scattering technique, microlocal analysis, the Carleman type estimates and soliton theory.
Understanding the control and stabilization of physical processes described by nonlinear partial differential equations is becoming increasingly important due to the rapid advances in material science, physics, aircraft design, fiber-optic communication systems. The nonlinear partial differential equations used as models are mathematically challenging and widely used in physics and engineering. The global controllability problems for these equations are motivated by practical real-world applications which include suppression of turbulence, control of waves in channels, control of plasma flow, design of a new generation of amplifiers which will allow the preservation of the shape of optical pulse in fiber-optical communication systems.
