My research lies in the area of control theory of PDEs and in fluid-structure interaction problems.Â
(1) Control Theory: The main problem considered in control theory is the following: Given a physical system, is it possible to drive the system from any given initial state to any prescribed final state, using a suitable (control) function? For instance, suppose one is riding a bicycle and wants to go from point A to B. Then, this goal can be achieved using the pedal (to push forward) and handle (to steer)---hence, this system is controllable. This basic idea is used to formulate and solve several complex problems, and has a wide variety of applications. Indeed, one can find uses of control theory in engineering, medical sciences, finance, computer sciences, etc.
In particular, I study control of geometric wave equations in general Lorentzian geometry settings. Working in this area involves using ideas from a variety of areas in Mathematics---differential geometry, functional analysis, and PDE theory. Furthermore, some of these techniques and results are also relevant to the area of inverse problems. Hence, I am also curious about related inverse problems for waves in geometric settings.
Recently I have also been interested in parabolic equations and studying their controllability problems.
(2) Fluid-Structure Interaction: We study the behaviour of a system in which a rigid object is immersed in a fluid. Depending on the conditions, the fluid can be inviscid, viscous, incompressible, or compressible, with each configuration modelled by a different PDE system. Such problems have applications in a wide variety of subjects--physics, engineering, biology, etc.