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IMA Journal of Mathematical Control and Information 1999 16(1):43-58; doi:10.1093/imamci/16.1.43
© 1999 by Institute of Mathematics and its Applications
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Existence and optimal control for periodic parabolic equations with nonlocal terms

W. ALLEGRETTO{dagger} and P. NISTRI{ddagger}

Department of Mathematical Sciences, University of Alberta Edmonton, Alberta, Canada T6G 2G1
Dipartimento di Sistemi ed Informatica, Universita di Firenze Via di S. Marta 3, 50139 Firenze, Italy
{dagger} E-mail: retl{at}retl.math.ualberta.ca
{ddagger} E-mail: pnistri{at}ingfil.ing.unifi.it

Motivated by models which have been proposed for some problems in mathematical biology and fisheries management and elsewhere, we consider a nonlinear periodic parabolic problem and an associated cost functional J. A key feature of our problem is the presence of a nonlocal term which—as we show by direct example—renders the standard mono-tonicity methods invalid. We therefore employ topological methods to deal both with existence of solutions and of minima of J over the Control set. Some considerations are also presented on related systems and on the question of uniqueness.


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