Optimal Control of an Ill Posed Elliptic Semilinear Equation with an Exponential Non Linearity

We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity such that we cannot expect to have a solution of the state equation for any given control We then have to speak of pairs control state After having de ned a suit able functional class in which we look for solutions we prove existence of an optimal pair for a large class of cost functions using a non standard compactness argument Then we derive a rst order optimality system assuming the optimal pair is slightly more regular Introduction In this paper we are concerned with the optimal control of the following semilinear elliptic equation y ey u in y on where R n is a bounded open set being the boundary which is assumed to be Lipschitz The function u is the control that will be taken in some space Lp and y denotes the state in our control problem The equation appears in several contexts we refer for instance to D A Franck Kamenetskii for combustion theory in chemical reactors S Chandrasekhar in the study of stellar structures The equation is ill posed in the sense that there is no solution for some controls u and many solutions can be found for some others see for instance I M Gelfand M G Crandall and P H Rabinowitz F Mignot and J P Puel Th Gallou et F Mignot and J P Puel Because of the term ey we need E Casas Dpt Matem atica Aplicada Y Ciencias de la Computaci on E T S I I y T Univiersidad de Cantabria Av Los Castros s n Santander Spain E mail casas etsiso macc unican es O Kavian Universit e de Versailles Saint Quentin et Centre de Math ema tiques Appliqu ees Ecole Polytechnique Palaiseau Cedex France E mail kavian math uvsq fr J P Puel Universit e de Versailles Saint Quentin et Centre de Math ema tiques Appliqu ees Ecole Polytechnique Palaiseau Cedex France E mail jppuel cmapx polytechnique fr This research was partially supported by the European Union under HCM Project number ERBCHRXCT The rst author was also partially supported by Direcci on General de Investigaci on Cient ca y T ecnica Spain Received by the journal October Revised July Accepted for publi cation July c Soci et e de Math ematiques Appliqu ees et Industrielles Typeset by LTEX E CASAS O KAVIAN AND J P PUEL to explain what we mean by a solution of We will say that y is a solution of if it belongs to the class of functions Y fy H e y L g and it satis es the equation in the distribution sense Then the optimal control problem will be formulated in the following terms P Minimize J y u Z L x y x dx N p Z ju x jdx y u Y K satis es where K is a nonempty closed convex subset of Lp p N and L R R is a Carath eodory function of class C with respect to the second variable and satisfying appropriate growth conditions which will be shown to be the following L y x y a x jy j e L x y a x jy j e for some a a L np n p if p n and if p n and p For instance yd Y being given a typical functional J would be J y u Z jy x yd x j dx N p Z ju x jdx We should emphasize on the fact that one of the main di culties of the problem is to choose an appropriate class of solutions such that P has a solution in that class The plan of the paper is as follows In Section we will analyze the state equation and we will establish the necessary background to study the control problem The existence of a solution for P is studied in Sections and for the cases p and p respectively The case p presents some di culties and we will be able to prove the existence of an optimal control under some additional assumption on the function L We should note that as it seems to us the case p cannot be treated with the techniques we use in this paper Finally in Section the optimality conditions will be investigated Analysis of the State Equation We start this section by establishing that any solution of in the sense de ned in Section is a solution in the variational sense in H this requires to prove some regularity of the term e Lemma Let y Y be a solution of then ey H eyz L for every z H and Z ry x rz x dx Z e x u x z x dx ESAIM Cocv November Vol OPTIMAL CONTROL OF AN ILL POSED ELLIPTIC SEMILINEAR EQUATION Proof By the de nition of a solution of for all z C c we have Z ry x rz x dx Z e x u x z x dx Given z L H we can take a sequence fzkg k C c with kzkk kzk for every k N and zk z in H and zk z in L w Then for all k we can replace z by zk in and pass to the limit when k to obtain that the identity in is also true for any z L H Let us take now z H such that z and set Tk z x k if z x k z x if z x k Then Tk z L H and Z ry x rTk z x dx Z e x u x Tk z x dx k N Since Tk z z in H the only trouble to pass to the limit in this identity comes from the term ezk As Tk z and e y then from the monotone convergence theorem taking into account that Tk z x z x for almost all x we deduce Z e x z x dx lim k Z e x Tk z x dx lim k Z ry x rTk z x dx Z u x Tk z x dx Z ry x rz x dx Z u x z x dx Next for a general z H we notice that z z z with z z H This proves that is satis ed Moreover e y H and shows that for all z H