# American Institute of Mathematical Sciences

December  2012, 7(4): 605-615. doi: 10.3934/nhm.2012.7.605

## On the optimal control for a semilinear equation with cost depending on the free boundary

 1 Dept. de Matemática Aplicada, Fac. de Matemáticas, Univ. Complutense de Madrid, Madrid, 28040, Spain, Spain, Spain

Received  January 2012 Revised  September 2012 Published  December 2012

We study an optimal control problem for a semilinear elliptic boundary value problem giving rise to a free boundary. Here, the free boundary is generated due to the fact that the nonlinear term of the state equation is not differentiable. The new aspect considered in this paper, with respect to other control problems involving free boundaries, is that here the cost functional explicitly depends on the location of the free boundary. The main difficulty is to show the continuous dependence (in measure) of the free boundary with respect to the control function. The crucial tool to get such continuous dependence is to know how behaves the state solution near the free boundary, as in previous works by L.A. Caffarelli and D. Phillips among other authors. Here we improved previous results in the literature thanks to a suitable application of the Fleming-Rishel formula.
Citation: Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605
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