arXiv subjects

Shri Lal Raghudev Ram Singh

Publications and source records attributed to Shri Lal Raghudev Ram Singh.

5 recordsLinked to original sources

arXiv2026

Quasi-compactness and uniform stabilization on general Banach spaces under $θ$-subordinate perturbations of semigroup generators

We prove that the quasi-compactness of an analytic semigroup is preserved under $θ$-subordinate perturbations of its generator on general Banach spaces, provided the perturbations are compact along trajectories. This allows us to prove the permanence of uniform exponential stability for an analytic semigroup under similar perturbations of its generator, provided the perturbed generator generates a strongly stable semigroup. We then show how the analyticity assumption can be relaxed to the class of Crandall-Pazy semigroups under a smaller class of $θ$-subordinate perturbations, and to immediately differentiable semigroups under bounded perturbations. As an application of the stabilization result, we consider the one-dimensional Neumann Laplacian on a non-reflexive Banach space. Finally, we present counterexamples that demonstrate the lack of uniform stabilization under the larger class of $A$-bounded perturbations.

math.FA↗

arXiv2026

Uniform stabilization for relatively bounded perturbations of generators of semigroup

In this paper, we study the robustness of exponential stability for semigroups generated by linear operators under perturbations. Extending a classical result of Gibson's Stability Theorem, we show that if the generator of an analytic exponentially stable semigroup is perturbed by a class of relatively bounded operators satisfying certain assumptions, then exponential stability is preserved, provided the perturbed semigroup is strongly stable. We also show that, for a restricted class of perturbations, the analyticity requirement can be relaxed to Gevrey regularity. Moreover, we present applications to uniformly parabolic equations, degenerate/singular parabolic equations, coupled hyperbolic plate systems, and generalized coupled systems of Kirchhoff-Love plates and a membrane-like electric network.

math.AP↗

arXiv2025

Which Consciousness Can Be Artificialized? Local Percept-Perceiver Phenomenon for the Existence of Machine Consciousness

This paper presents a novel paradigm of the local percept-perceiver phenomenon to formalize certain observations in neuroscientific theories of consciousness. Using this model, a set-theoretic formalism is developed for artificial systems, and the existence of machine consciousness is proved by invoking Zermelo-Fraenkel set theory. The article argues for the possibility of a reductionist form of epistemic consciousness within machines.

q-bio.NC↗

arXiv2024

Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies

A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=νu_{xx}-μu_{xxx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), \ x\in[0,1], \ t>0,$$ where $ν,μ,α,β>0,$ $δ\in[1,\infty)$, $γ\in(0,1)$ subject to Neumann boundary conditions is considered in this work. We first establish the well-posedness of the Neumann boundary value problem by an application of monotonicity arguments, the Hartman-Stampacchia theorem, the Minty-Browder theorem, and the Crandall-Liggett theorem. The additional difficulties caused by the third order linear term is successfully handled by proving a proper version of the Minty-Browder theorem. By using suitable feedback boundary controls, we demonstrate $\mathrm{L}^2$- and $\mathrm{H}^1$-stability properties of the closed-loop system for sufficiently large $ν>0$. The analytical conclusions from this work are supported and validated by numerical investigations.

math.AP↗

arXiv2023

Analysis and Numerical Study of Boundary control of generalized Burgers-Huxley equation

In this work, a boundary control problem for the following generalized Burgers-Huxley (GBH) equation: $$u_t=νu_{xx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), $$ where $ν,α,β>0,$ $1\leqδ<\infty$, $γ\in(0,1)$ subject to Neumann boundary conditions is analyzed. Using the Minty-Browder theorem, standard elliptic partial differential equations theory, the maximum principle and the Crandall-Liggett theorem, we first address the global existence of a unique strong solution to GBH equation. Then, for the boundary control problem, we prove that the controlled GBH equation (that is, the closed loop system) is exponentially stable in the $\mathrm{H}^1$-norm (hence pointwise) when the viscosity $ν$ is known (non-adaptive control). Moreover, we show that a damped version of GBH equation is globally asymptotically stable (in the $\mathrm{L}^2$-norm), when $ν$ is unknown (adaptive control). Using the Chebychev collocation method with the backward Euler method as a temporal scheme, numerical findings are reported for both the non-adaptive and adaptive situations, supporting and confirming the analytical results of both the controlled and uncontrolled systems.

math.AP↗