Optimal boundary control of the heat equation with target function at terminal time Guangcao Ji a , Clyde Martin a,b, * a Department of Mathematics, Institute for the Mathematics of the Life Sciences, Texas Tech University, Box 41042, Lubbock, TX 79409, USA b Department of Mathematics and Statistics, Institute of Environmental and Human Health, Texas Tech University, Lubbock, TX 79409, USA Abstract We consider the problem of optimally fitting a solution of the heat equation to a prescribed target function. We assume boundary control of the heat equation on a bounded domain. The problem arises in constructing the response surface to a two- dimensional set of experimental data. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Splines; Boundary data; Surface fitting; Prior data 1. The toxiological problem The problem we study in this paper arises in experiments designed to measure the toxicity of chemical mixtures on cells and animals and, ultimately, toassesstherisktohumans.Measuresoftoxicityareknownindetailformany single chemicals. For example, there is a great deal known about the levels of mercury needed to cause mental impairment and likewise there is a great deal known about the levels of lead that cause learning disability in children. What is not known is how ‘‘mixtures’’ of lead and mercury affect learning or mental disability.Thisistrueformostcombinationsofchemicalcontaminantsandyet most of the time when humans are exposed to chemical contaminants in their environment they are exposed to multiple chemicals. Applied Mathematics and Computation 127 (2002) 335–345 www.elsevier.com/locate/amc * Corresponding author. 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0096-3003(01)00011-X