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

Jésus Ildefonso Díaz; Tommaso Mingazzini; Ángel Manuel Ramos; Jésus Ildefonso Díaz; Tommaso Mingazzini; Ángel Manuel Ramos

doi:10.3934/nhm.2012.7.605

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 605-615. doi: 10.3934/nhm.2012.7.605

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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

Received: 01 January 2012 Revised: 01 September 2012
Primary: 58F15, 58F17; Secondary: 53C35.

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.
- Optimal control,
- semilinear equations,
- free boundary
Citation: Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with costdepending on the free boundary[J]. Networks and Heterogeneous Media, 2012, 7(4): 605-615. doi: 10.3934/nhm.2012.7.605

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Abstract

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.

References

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