Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners.

Keywords: bilateral control-state constraints, heat equation, mesh independence, optimal control, PDE-constrained optimization, semismooth Newton method

@article{COCV_2009__15_3_626_0, author = {Hinterm\"uller, Michael and Kopacka, Ian and Volkwein, Stefan}, title = {Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {626--652}, publisher = {EDP-Sciences}, volume = {15}, number = {3}, year = {2009}, doi = {10.1051/cocv:2008042}, mrnumber = {2542576}, zbl = {1167.49027}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008042/} }

TY - JOUR AU - Hintermüller, Michael AU - Kopacka, Ian AU - Volkwein, Stefan TI - Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 626 EP - 652 VL - 15 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008042/ DO - 10.1051/cocv:2008042 LA - en ID - COCV_2009__15_3_626_0 ER -

%0 Journal Article %A Hintermüller, Michael %A Kopacka, Ian %A Volkwein, Stefan %T Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 626-652 %V 15 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008042/ %R 10.1051/cocv:2008042 %G en %F COCV_2009__15_3_626_0

Hintermüller, Michael; Kopacka, Ian; Volkwein, Stefan. Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, pp. 626-652. doi : 10.1051/cocv:2008042. http://archive.numdam.org/articles/10.1051/cocv:2008042/

[1] Sobolev Spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). | MR | Zbl

,[2] Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems, in Control and estimation of distributed parameter systems (Vorau, 1996), Internat. Ser. Numer. Math. 126 (1998) 15-32. | MR | Zbl

and ,[3] Block preconditioners for KKT systems in PDE-governed optimal control problems, in Fast solution of discretized optimization problems (Berlin, 2000), Internat. Ser. Numer. Math. 138 (2001) 1-18. | MR | Zbl

and ,[4] Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. I. The Krylov-Schur solver. SIAM J. Sci. Comput. 27 (2005) 687-713. | MR | Zbl

and ,[5] Evolution Problems I, Mathematical Analysis and Numerical Methods for Science and Technology 5. Springer-Verlag, Berlin (1992). | MR

and ,[6] Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, Rhode Island (1998). | MR | Zbl

,[7] Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Verlag, Berlin (2002). | Zbl

and ,[8] Optimal ${H}^{p,p/2}$ error estimates for a parabolic Galerkin method. SIAM J. Numer. Anal. 18 (1981) 681-692. | MR | Zbl

,[9] Mesh-independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems. ANZIAM Journal 49 (2007) 1-38. | MR | Zbl

,[10] A SQP-semismooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints. SIAM J. Opt. 16 (2006) 1177-1200. | MR | Zbl

and ,[11] A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B 101 (2004) 151-184. | MR | Zbl

and ,[12] The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Opt. 13 (2003) 865-888. | MR | Zbl

, and ,[13] Fast solution techniques in constrained optimal boundary control of the semilinear heat equation. Internat. Ser. Numer. Math. 155 (2007) 119-147. | MR | Zbl

, and ,[14] Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin (1971). | MR | Zbl

,[15] Convergence of approximations versus regularity of solutions for convex, control-constrained optimal control problems. Appl. Math. Optim. 8 (1981) 69-95. | MR | Zbl

,[16] Iterative Solution of Nonlinear Equations in several Variables, Computer Science and Applied Mathematics. Academic Press, New York (1970). | MR | Zbl

and ,[17] The Method of Discretization in Time and Partial Differential Equations, Mathematics and Applications 4. D. Reichel Publishing Company, Boston-Dordrecht-London (1982). | MR | Zbl

,[18] Navier-Stokes Equations, Studies in Mathematics and its Applications. North-Holland, Amsterdam (1979). | MR | Zbl

,[19] Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam (1978). | MR | Zbl

,[20] Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Opt. 15 (2005) 616-634. | MR | Zbl

,[21] Optimale Steuerung partieller Differentialgleichungen. Vieweg Verlag, Wiesbaden (2005). | Zbl

,*Cited by Sources: *