Appl Math Optim 45:325–345 (2002)
2002 Springer-Verlag New York Inc.
Optimal Control for the Degenerate Elliptic Logistic Equation
J. A. Montero,
and A. Su´arez
Departamento Ecuaciones Diferenciales y An´alisis Num´erico,
Facultad de Matem´aticas, C/ Tarﬁa s/n, C.P. 41012,
Universidad de Sevilla, Sevilla, Spain
Departamento An´alisis Matem´atico, C.P. 18071,
Universidad de Granada, Granada, Spain
Communicated by R. Triggiani
Abstract. We consider the optimal control of harvesting the diffusive degenerate
elliptic logistic equation. Under certain assumptions, we prove the existence and
uniqueness of an optimal control. Moreover, the optimality system and a character-
ization of the optimal control are also derived. The sub-supersolution method, the
singular eigenvalue problem and differentiability with respect to the positive cone
are the techniques used to obtain our results.
KeyWords. Degenerate logistic equation, Singular eigenvalueproblems, Optimal
AMS Classiﬁcation. Primary 49J20, 49K20, 92D25, Secondary 35J65.
This work considers the optimal harvesting control of a species whose state is governed
by the degenerate (nonlinear slow diffusion) elliptic logistic equation, i.e.,
= (a − f )w − ew
w = 0on∂,
M. Delgado and A. Su´arez were partially supported by CICYT of Spain (MAR98-0486) and J. A.
Montero by “Junta de Andaluc´ıa” (FQM116) and DGESIC (PB98-1343).