arXiv subjects

Vaibhav Kumar Jena

Publications and source records attributed to Vaibhav Kumar Jena.

7 recordsLinked to original sources

arXiv2026

Observability for wave equations with critically singular potentials

In this article, we prove boundary observability estimates for wave equations on bounded domains $Ω\subseteq \mathbb{R}^n$, with a critically singular potential that diverges as the inverse square distance to $\partial Ω$. The result is applicable in all dimensions and for wave equations with general time-dependent lower order terms. The key geometric assumption is a convexity condition on $Ω$. The main tool for our result is a global Carleman estimate that is carefully adapted to the critical singular nature of the potential.

math.AP↗

arXiv2026

Control and homogenization of a coupled hyperbolic system with an oscillating coefficient

We study the control and homogenization properties of a coupled hyperbolic system characterised by a rapidly oscillating coefficient present in the principal term. Due to the high-frequency microstructural variations of the density parameter $\varepsilon$, classical uniform controllability results across all initial data fail under boundary controls alone. To overcome this limitation, we divide the spectrum into low and high-frequency components. Utilizing Ingham-Beurling-type inequalities combined with spectral gap analysis, we demonstrate the uniform null controllability of the projected low-frequency and high-frequency states. Furthermore, we analyse the structural limits as $\varepsilon \to 0$ and prove that the associated sequence of boundary controls converge weakly to a null control for the underlying homogenized system. Finally, we show that by using an additional interior feedback control strategy, full space controllability can be achieved for all initial data in the energy space.

math.AP↗

arXiv2024

Uniqueness of solution for ultrahyperbolic equations with lower order terms

In this article, we prove a variety of uniqueness results for ultrahyperbolic equations with general space and time dependent lower order terms. We address the problem of determining uniqueness of solutions from boundary data as well as when the data is prescribed on an interior subset. Furthermore, we also present the case when the domain may change with respect to any one time component and obtain analogous results. Our main tool for this purpose is Carleman estimate. We obtain different uniqueness results depending on the location of the reference point for the Carleman estimate relative to the domain.

math.AP↗

arXiv2024

Control and homogenization of a system of coupled parabolic equations with an oscillating coefficient

In this article, we study the uniform null controllability problem for a system of coupled parabolic equations with an oscillating coefficient. This is done in three steps -- first, we study the spectral properties of an elliptic operator; second, we allow the system to evolve freely and obtain the required decay; third, we use a Carleman estimate to prove a suitable observability result. This uniform null controllability property is then used to homogenize the associated coupled parabolic system.

math.AP↗

arXiv2022

Interior control of waves on time dependent domains

We obtain a novel interior control result for wave equations on time dependent domains. This is done by deriving a suitable Carleman estimate and proving the corresponding observability inequality. We consider the wave equation with time dependent lower order coefficients, without any time analyticity assumptions. Moreover, we obtain improved control regions when compared with standard Carleman methods.

math.AP↗

arXiv2021

Control of waves on Lorentzian manifolds with curvature bounds

We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds.

math.AP↗

arXiv2021

Carleman Estimate for Ultrahyperbolic Operators and Improved Interior Control for Wave Equations

In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ \mathbb{R}^m_t \times \mathbb{R}^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with time-dependent lower order terms. The key improvements are: (1) we obtain smaller observation regions compared to standard Carleman estimate results, and (2) we also prove observability when the observation point lies inside the domain. Finally, as a corollary of the observability result, we obtain improved interior controllability for the wave equation.

math.AP↗