Research
There are two central concepts in the theory of control of Partial differential equations: Controllability and Stabilization. I am working on these two notions for various coupled PDEs such as the ODE-parabolic model, ODE-hyperbolic model, two-parabolic model, hyperbolic-parabolic model, hyperbolic-elliptic system, parabolic-elliptic system, dispersive model etc. To explore the controllability and stabilization of the above system, we utilized some methods: The backstepping method, Gramian approach, Carleman estimate, Compactness-uniqueness method, Moment method, method of Characteristics, Source term method, Fixed point arguments etc.
The stabilizability of the partial differential equation is one of the fundamental studies of the theory of the control of PDEs. I am interested in different methods that can be implemented to study the infinite time behaviour of a solution of some coupled PDE in some suitably defined Sobolev spaces. Feedback stabilizability is an interesting topic of study for control systems. We have studied the feedback stabilization of some linear and nonlinear ODE-Parabolic (using the method of backstepping) and a linear first-order hyperbolic-elliptic mixed-class coupled system (using the Gramian-based method).
On the other hand, Control of partial differential equations is a highly fascinating topic in analysis. I am interested in the theory and method which can be explored to study this type of problem. We have studied the controllability issues for various linear and nonlinear models like linearized compressible Navier-Stokes equation in the case of creeping flow, linearized compressible Navier-Stokes equation with Maxwell's law, linear stabilized Kuramoto-Sivashinsky (KS in short) equation, coupled transport-KS-KdV, coupled KS-KdV elliptic, coupled Schr\"odinger equations. We have utilized the method of moments, Hilbert uniqueness method, Carleman approach, and Compactness-uniqueness approach, Ingham Inequality to prove the controllability of the aforementioned coupled systems.
Publications & Preprints
Stackelberg-Nash exact controllability for the Benney-Lin type equation with mixed boundary conditions, submitted 2024, (with Manish Kumar).
Insensitizing control problem for the Kawahara equation, submitted, 2024 (with Manish Kumar).
Controllability and Stabilizability of the linearized compressible Navier-Stokes system with Maxwell's law, submitted, 2023 (with Sakil Ahamed).
Stabilization of Kawahara equation with input saturation and saturated boundary feedback, submitted, 2023 (with Hugo Parada).
On the controllability of a system coupling Kuramoto-Sivashinsky-Korteweg-de Vries and transport equations, accepted in Math. Control Signals Systems, 2024 ( with Manish Kumar).
Coupled linear Schr\"odinger equations: Control and stabilization results, Zeitschrift Angew. Math. Phys. (75) 2024, (with Kuntal Bhandari, Roberto de A. Capistrano-Filho and Thiago Yukio Tanaka ).
Local exponential stabilization of Rogers-McCulloch and FitzHugh-Nagumo equation by the method of backstepping, ESAIM Control Optim. Calc. Var, (30) 2024, ( with Shirshendu Chowdhury and Rajib Dutta).
Local null-controllability of a two-parabolic nonlinear system with coupled boundary conditions by a single Neumann control, Evol. Equ. Control Theory 13 (2024), pp. 587-615, (with Kuntal Bhandari and Jiten Kumbhakar).
Local null controllability of the stabilized Kuramoto-Sivashinsky system using moment method, Adv. Differential Equations 29 (3/4) 223 - 290, March/April 2024, https://doi.org/10.57262/ade029-0304-223 ( with Manish Kumar ).
Asymptotic behavior of the linearized compressible barotropic Navier-Stokes system with a time varying delay term in the boundary or internal feedback, Math. Meth. Appl. Sci. 46 (2023), 17288–17312, DOI 10.1002/mma.9500.
Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations, J. Math. Anal. Appl 525 (2023) 127213. ( with Kuntal Bhandari).
Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system, Evol. Equ. Control Theory 12 (2023), pp. 907-943 ( with Shirshendu Chowdhury and Rajib Dutta).
Boundary stabilizability of the linearized compressible Navier-Stokes equation in one dimension by backstepping approach, SIAM J. Control Optim 59 (2021), pp. 2147-2173. ( with Shirshendu Chowdhury and Rajib Dutta).