2024/01/14 by Mohan Mallick, Mallick, Mohan, A N Ardra +3
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2401.07264
openalex publication_date 2024/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider semi-linear elliptic equations of the following form: \ \beginaligned -Δu amp;= λ[u-\dfracu2K-c \dfracu21+u2-h(x) u]=:λfh(u), amp;amp; x ∈ Ω, (∂ u)/(∂ η)amp;+qu = 0, amp;amp; x∈∂Ω, \endaligned . where, h∈ U=\h∈ L2(Ω): 0≤ h(x)≤ H\. We prove the existence and uniqueness of the positive solution for large λ. Further, we establish the existence of an optimal control h∈ U that maximizes the functional J(h)=∫Ωh(x)uh(x)~\rmdx-∫Ω(B1+B2 h(x))h(x)~\rmdx over U, where uh is the unique positive solution of the above problem associated with h, B1>0 is the cost per unit effort when the level of effort is low and B2>0 represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.