Optimal control for the degenerate elliptic logistic equation
type: Item
instance of: scholarly article
title: Optimal control for the degenerate elliptic logistic equation
zbMATH Open document ID: 1017.49006
author: R. Smith
published in: Applied Mathematics and Optimization
publication date: 2002-06-27T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/40111
review text: The paper considers the problem of maximizing the functional \[ I(z,u)= \int_\Omega [\lambda h(u(x)) z(x)- k(u(x))] dx \] subject to the state equation \[ \Delta z(x)+ (a(x)- u(x)) z^\alpha(x)- e(x) z^\beta(x)= 0\quad\text{in }\Omega,\;z= 0\quad\text{on }\partial\Omega.\tag{1} \] Here \(\Omega\subset\mathbb{R}^n\) is a bounded regular domain and \(u\), \(0< u(x)\leq a(x)\) a.e. \(x\in\Omega\), plays the role of control. The existence and dependence on \(u\) of positive solutions of (1) are investigated. After that the existence, uniqueness and smoothness properties of the optimal control are derived.
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B37
zbMATH DE Number: 1761093
zbMATH Keywords: elliptic equation
zbMATH Keywords: optimality system
zbMATH Keywords: properties of solution
zbMATH Keywords: optimal control
reviewed by: Uldis Raitums
MaRDI profile type: Publication
OpenAlex ID: W1978700117
DOI: 10.1007/S00245-001-0039-1
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description: scientific article; zbMATH DE number 1761093