Abstract

In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The two-dimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797--2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish ${\mathcal O}(\sqrt{k}+h)$ order of convergence rate for the control in the $L^2$ norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97--119] and in addition does not require any relationship between the time step $k$ and the mesh size $h$. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the $L^\infty(\Omega;L^2(I))$ norm.

Keywords

  1. optimal control
  2. pointwise control
  3. parabolic problems
  4. finite elements
  5. discontinuous Galerkin
  6. error estimates
  7. pointwise error estimates

MSC codes

  1. 49M25
  2. 49K20
  3. 65N30
  4. 65N15

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
H. W. Alt, Linear Functional Analysis. An Application Oriented Introduction, 6th ed., Springer, Berlin, 2011.
2.
H. Amann, Nonautonomous parabolic equations involving measures, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 306 (2003), pp. 16--52.
3.
M. Amouroux and J.-P. Babary, On the optimal pointwise control and parametric optimization of distributed parameter systems, Internat. J. Control, 28 (1978), pp. 789--807.
4.
T. Apel, O. Benedix, D. Sirch, and B. Vexler, A priori mesh grading for an elliptic problem with Dirac right-hand side, SIAM J. Numer. Anal., 49 (2011), pp. 992--1005.
5.
A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Oper. Theory Adv. Appl. 69, Birkhäuser Verlag, Basel, 1994.
6.
I. Babuška, Error-bounds for finite element method, Numer. Math., 16 (1970/1971), pp. 322--333.
7.
E. Casas, $L^2$ estimates for the finite element method for the Dirichlet problem with singular data, Numer. Math., 47 (1985), pp. 627--632.
8.
E. Casas, C. Clason, and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), pp. 28--63.
9.
E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), pp. 355--370.
10.
E. Casas, B. Vexler, and E. Zuazua, Sparse initial data identification for parabolic PDE and it's finite element approximations, Math. Control Relat. Fields, 5 (2015), pp. 377--399.
11.
E. Casas and E. Zuazua, Spike controls for elliptic and parabolic PDEs, Systems Control Lett., 62 (2013), pp. 311--318.
12.
I. Chryssoverghi, Approximate methods for optimal pointwise control of parabolic systems, Systems Control Lett., 1 (1981), pp. 216--219.
13.
A. Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc., 360 (2008), pp. 4335--4347.
14.
E. J. Dean and P. Gubernatis, Pointwise control of Burgers' equation---a numerical approach, Comput. Math. Appl., 22 (1991), pp. 93--100.
15.
K. Disser, J. Rehberg, and A. ter Elst, Hölder estimates for parabolic operators on domains with rough boundary, preprint, arXiv:1503.07035, 2015.
16.
J. Droniou and J.-P. Raymond, Optimal pointwise control of semilinear parabolic equations, Nonlinear Anal., 39 (2000), pp. 135--156.
17.
J. Elschner, J. Rehberg, and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces Free Bound., 9 (2007), pp. 233--252.
18.
K. Eriksson, Finite element methods of optimal order for problems with singular data, Math. Comp., 44 (1985), pp. 345--360.
19.
K. Eriksson, C. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, ESAIM Math. Model. Numer. Anal., 19 (1985), pp. 611--643.
20.
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: Optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM J. Numer. Anal., 32 (1995), pp. 706--740.
21.
S. J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc., 119 (1993), pp. 225--233.
22.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Classics Math., Springer, Berlin, 2001.
23.
W. Gong, Error estimates for finite element approximations of parabolic equations with measure data, Math. Comp., 82 (2013), pp. 69--98.
24.
W. Gong, M. Hinze, and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control, SIAM J. Control Optim., 52 (2014), pp. 97--119.
25.
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), pp. 45--61.
26.
T. Köppl and B. Wohlmuth, Optimal a priori error estimates for an elliptic problem with Dirac right-hand side, SIAM J. Numer. Anal., 52 (2014), pp. 1753--1769.
27.
K. Krumbiegel and J. Rehberg, Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints, SIAM J. Control Optim., 51 (2013), pp. 304--331.
28.
K. Kunisch, K. Pieper, and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), pp. 3078--3108.
29.
D. Leykekhman, D. Meidner, and B. Vexler, Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints, Comput. Optim. Appl., 55 (2013), pp. 769--802.
30.
D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control, SIAM J. Numer. Anal., 51 (2013), pp. 2797--2821.
31.
D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, published online, http://dx.doi.org/10.1007/s00211-016-0821-2
32.
D. Leykekhman and B. Vexler, Finite element pointwise results on convex polyhedral domains, SIAM J. Numer. Anal., 54 (2016), pp. 561--587.
33.
D. Leykekhman and B. Vexler, Pointwise best approximation results for Galerkin finite element solutions of parabolic problems, SIAM J. Numer. Anal., 54 (2016), pp. 1365--1384.
34.
J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, Reg. Conf. Ser. Appl. Math. 6, SIAM, Philadelphia, 1972.
35.
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. II, Grundlehren Math. Wiss. 182, Springer, New York, 1972.
36.
D. Meidner, R. Rannacher, and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), pp. 1961--1997.
37.
D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretisation of parabolic optimal control problems Part I: Problems without control constraints, SIAM J. Control Optim., 47 (2008), pp. 1150--1177.
38.
D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretisation of parabolic optimal control problems Part II: Problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 1301--1329.
39.
A. M. Ramos, R. Glowinski, and J. Periaux, Pointwise control of the Burgers equation and related Nash equilibrium problems: Computational approach, J. Optim. Theory Appl., 112 (2002), pp. 499--516.
40.
R. Rannacher, $L^\infty$-stability estimates and asymptotic error expansion for parabolic finite element equations, in Extrapolation and Defect Correction (1990), Bonner Math. Schriften 228, University of Bonn, Bonn, 1991, pp. 74--94.
41.
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 31 (1977), pp. 414--442.
42.
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp., 64 (1995), pp. 907--928.
43.
R. Scott, Finite element convergence for singular data, Numer. Math., 21 (1973), pp. 317--327.
44.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd ed.,Johann Ambrosius Barth, Heidelberg, 1995.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2403 - 2435
ISSN (online): 1095-7138

History

Submitted: 26 June 2015
Accepted: 20 June 2016
Published online: 21 September 2016

Keywords

  1. optimal control
  2. pointwise control
  3. parabolic problems
  4. finite elements
  5. discontinuous Galerkin
  6. error estimates
  7. pointwise error estimates

MSC codes

  1. 49M25
  2. 49K20
  3. 65N30
  4. 65N15

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.