Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the control-to-observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). Combining a conforming discretization of the measure space with a semismooth Newton method allows the numerical solution of the optimal control problem.

Accepted:

DOI: 10.1051/cocv/2015046

Keywords: Optimal control, measure control, control constraints, Fenchel duality, unbounded operators

^{1}; Schiela, Anton

^{2}

@article{COCV_2017__23_1_217_0, author = {Clason, Christian and Schiela, Anton}, title = {Optimal control of elliptic equations with positive measures}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {217--240}, publisher = {EDP-Sciences}, volume = {23}, number = {1}, year = {2017}, doi = {10.1051/cocv/2015046}, mrnumber = {3601022}, zbl = {1365.49005}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015046/} }

TY - JOUR AU - Clason, Christian AU - Schiela, Anton TI - Optimal control of elliptic equations with positive measures JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 217 EP - 240 VL - 23 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015046/ DO - 10.1051/cocv/2015046 LA - en ID - COCV_2017__23_1_217_0 ER -

%0 Journal Article %A Clason, Christian %A Schiela, Anton %T Optimal control of elliptic equations with positive measures %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 217-240 %V 23 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015046/ %R 10.1051/cocv/2015046 %G en %F COCV_2017__23_1_217_0

Clason, Christian; Schiela, Anton. Optimal control of elliptic equations with positive measures. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 1, pp. 217-240. doi : 10.1051/cocv/2015046. http://archive.numdam.org/articles/10.1051/cocv/2015046/

H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). | MR | Zbl

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010). | MR | Zbl

L${}^{2}$ estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47 (1985) 627–632. | DOI | MR | Zbl

,Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50 (2012) 1735–1752. | DOI | MR | Zbl

, and ,Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52 (2014) 339–364. | DOI | MR | Zbl

and ,A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: COCV 17 (2011) 243–266. | Numdam | MR | Zbl

and ,A measure space approach to optimal source placement. Comput. Optim. Appl. 53 (2011) 155–171. | DOI | MR | Zbl

and ,A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. 51 (2012) 931–939. | DOI | MR | Zbl

and ,I. Ekeland and R. Témam, Convex Analysis and Variational Problems. Vol. 28 of Classics Appl. Math. SIAM, Philadelphia (1999). | MR | Zbl

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (1977). | MR

S. Goldberg, Unbounded Linear Operators. Dover Publications Inc., Mineola, NY (2006). | MR

Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60 (2009) 397–428. | DOI | MR | Zbl

, , and ,A variational discretization concept in control constrained optimization$:$ The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. | DOI | MR | Zbl

,K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. Vol. 15 of Adv. Design Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008). | MR | Zbl

Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control. Numer. Funct. Anal. Optim. 32 (2011) 983–1007. | DOI | MR | Zbl

, and ,H.H. Schaefer and M.P. Wolff, Topological Vector Spaces. Vol. 3 of Grad. Texts Math. 2nd edition. Springer-Verlag, New York (1999). | MR | Zbl

State constrained optimal control problems with states of low regularity. SIAM J. Control Optim. 48 (2009) 2407–2432. | DOI | MR | Zbl

,Elliptic optimal control problems with ${L}^{1}$-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. | DOI | MR | Zbl

,Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. | DOI | Numdam | MR | Zbl

,G.M. Troianiello. Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics. Plenum Press (1987). | MR | Zbl

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, methods and applications. Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). | MR | Zbl

Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2003) 805–842 (2002). | DOI | MR | Zbl

,Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858–886. | Numdam | MR | Zbl

and ,Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Mat. Pura Appl. 117 (1978) 173–206. | DOI | MR | Zbl

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