arXiv subjects

Kedarnath Buda

Publications and source records attributed to Kedarnath Buda.

4 recordsLinked to original sources

arXiv2025

Finite Element Analysis of Nash Equilibrium of Bi-objective Optimal Control Problem Governed by Stokes Equation with $L^2$-norm State-Constraints

This paper investigates the Nash equilibrium of a bi-objective optimal control problem governed by the Stokes equations. A multi-objective Nash strategy is formulated, and fundamental theoretical results are established, including the existence, uniqueness, and analytical characterization of the equilibrium. A finite element framework is developed to approximate the coupled optimal control system, and the corresponding optimality conditions for both the continuous and discrete formulations are rigorously derived and analyzed. Furthermore, \textit{a priori} finite element error estimates are obtained for the discrete problem, ensuring convergence and stability of the proposed method. The theoretical results are corroborated by numerical experiments, which demonstrate the accuracy and computational efficiency of the finite element approach.

math.OC↗

arXiv2025

Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE

This work investigates the existence and uniqueness of the Nash equilibrium (solutions to competitive problems in which individual controls aim at separate desired states) for a bi-objective optimal control problem governed by a fractional space-time parabolic partial differential equation. The governing equation involves a Caputo fractional derivative with respect to time of order $\gamma$ in (0,1) and a fractional Laplacian in the spatial variables of order $s$ in (0,1). The system is associated with two independent controls, each aiming at different targets. The problem is formulated as a distributed optimal control system with quadratic cost functionals. Existence and uniqueness of the Nash equilibrium are established under convexity and coercivity assumptions. The solution is computed using conjugate gradient algorithms applied iteratively to the discretized optimal control problems. The numerical experiments agree with the theoretical estimates and demonstrate the efficiency of the proposed scheme.

math.OC↗

arXiv2025

Optimal control of fractional Poisson equation from non-local to local

In this article, the limiting behavior of the solution $\bar u_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-\Delta)^s u_s(x)=f_s(x), \quad x\in \Omega$$ defined on domain $\Omega$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0, \quad x \in \Omega^c $ is established. We will prove that $\lim_{s\to 1^-} \bar u_s= \bar u$, where $\bar u$ is a solution of the optimal control problem subjected to classical Poisson equation $-\Delta u(x)=f(x), \quad x \in \Omega$ and $u(x)=0, \quad x\in \partial \Omega.$

math.NA↗

arXiv2025

Adaptive SIPG method for approximations of boundary control problems governed by parabolic PDEs

This study presents an aposteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator effectively captured control approximation errors, it had limitations in guiding refinement localization in critical cases. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach's effectiveness in achieving accurate solutions while optimizing computational efficiency. numerical experiment showcases the effectiveness of the derived error estimators.

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