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September  2015, 5(3): 377-399. doi: 10.3934/mcrf.2015.5.377

Sparse initial data identification for parabolic PDE and its finite element approximations

1. 

Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain

2. 

Centre for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, D-85747 Garching b. München, Germany

3. 

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country

Received  August 2014 Revised  November 2014 Published  July 2015

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
Citation: Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control & Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377
References:
[1]

E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487.   Google Scholar

[2]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, Springer-Verlag, (2008).  doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations,, SIAM J. Control Optim., 35 (1997), 1297.  doi: 10.1137/S0363012995283637.  Google Scholar

[4]

E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions,, SIAM J. Control Optim., 50 (2012), 1735.  doi: 10.1137/110843216.  Google Scholar

[5]

________, Parabolic control problems in measure spaces with sparse solutions,, SIAM J. Control Optim., 51 (2013), 28.  doi: 10.1137/120872395.  Google Scholar

[6]

E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional,, SIAM J. Optim., 22 (2012), 795.  doi: 10.1137/110834366.  Google Scholar

[7]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces,, SIAM J. Control Optim., 52 (2014), 339.  doi: 10.1137/13092188X.  Google Scholar

[8]

________, Parabolic control problems in space-time measure spaces,, To appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[9]

E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls,, SIAM J. Control Optim., 52 (2014), 1010.  doi: 10.1137/130917314.  Google Scholar

[10]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic pde,, Systems Control Lett., 62 (2013), 311.  doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[11]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces,, ESAIM Control Optim. Calc. Var., 17 (2011), 243.  doi: 10.1051/cocv/2010003.  Google Scholar

[12]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, North-Holland-Elsevier, (1976).   Google Scholar

[13]

C. Fabre, J. P. Puel and E. Zuazua, On the density of the range of the semigroup for semilinear heat equations,, Control and optimal design of distributed parameter systems (Minneapolis, 70 (1995), 73.  doi: 10.1007/978-1-4613-8460-1_4.  Google Scholar

[14]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Advances in Differential Equations, 5 (2000), 465.   Google Scholar

[15]

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), 97.  doi: 10.1137/110840133.  Google Scholar

[16]

J. A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differential Equations, 12 (2007), 1031.   Google Scholar

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).   Google Scholar

[18]

A. Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems,, BIT, 42 (2002), 351.  doi: 10.1023/A:1021903109720.  Google Scholar

[19]

R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations,, SIAM J. Control Optim., 50 (2012), 943.  doi: 10.1137/100815037.  Google Scholar

[20]

J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients,, Proc. Amer. Math. Soc., 130 (2002), 1055.  doi: 10.1090/S0002-9939-01-06163-9.  Google Scholar

[21]

K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems,, SIAM J. Control Optim., 52 (2014), 3078.  doi: 10.1137/140959055.  Google Scholar

[22]

D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control,, SIAM J. Numer. Anal., 51 (2013), 2797.  doi: 10.1137/120885772.  Google Scholar

[23]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $l_1$ constrained minimization,, Inverse Probl. and Imaging, 8 (2014), 199.  doi: 10.3934/ipi.2014.8.199.  Google Scholar

[24]

J.-L. Lions and B. Malgrange, Sur l'unicité rétrograde des équations paraboliques,, Math. Scand., 8 (1960), 277.   Google Scholar

[25]

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), 1961.  doi: 10.1137/100793888.  Google Scholar

[26]

K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space,, SIAM J. Control Optim., 51 (2013), 2788.  doi: 10.1137/120889137.  Google Scholar

[27]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations,, App. Math. Optim., 39 (1999), 143.  doi: 10.1007/s002459900102.  Google Scholar

[28]

W. Rudin, Real and Complex Analysis,, McGraw-Hill, (1970).   Google Scholar

[29]

A. H. Schatz, V. C. Thomée and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations,, Comm. Pure Appl. Math., 33 (1980), 265.  doi: 10.1002/cpa.3160330305.  Google Scholar

[30]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices,, Comput. Optim. Appl., 44 (2009), 159.  doi: 10.1007/s10589-007-9150-9.  Google Scholar

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems,, Second edition, (2006).   Google Scholar

[32]

G. Wachsmuth and D. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional,, ESAIM Control Optim. Calc. Var., 17 (2011), 858.  doi: 10.1051/cocv/2010027.  Google Scholar

show all references

References:
[1]

E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487.   Google Scholar

[2]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, Springer-Verlag, (2008).  doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations,, SIAM J. Control Optim., 35 (1997), 1297.  doi: 10.1137/S0363012995283637.  Google Scholar

[4]

E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions,, SIAM J. Control Optim., 50 (2012), 1735.  doi: 10.1137/110843216.  Google Scholar

[5]

________, Parabolic control problems in measure spaces with sparse solutions,, SIAM J. Control Optim., 51 (2013), 28.  doi: 10.1137/120872395.  Google Scholar

[6]

E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional,, SIAM J. Optim., 22 (2012), 795.  doi: 10.1137/110834366.  Google Scholar

[7]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces,, SIAM J. Control Optim., 52 (2014), 339.  doi: 10.1137/13092188X.  Google Scholar

[8]

________, Parabolic control problems in space-time measure spaces,, To appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[9]

E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls,, SIAM J. Control Optim., 52 (2014), 1010.  doi: 10.1137/130917314.  Google Scholar

[10]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic pde,, Systems Control Lett., 62 (2013), 311.  doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[11]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces,, ESAIM Control Optim. Calc. Var., 17 (2011), 243.  doi: 10.1051/cocv/2010003.  Google Scholar

[12]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, North-Holland-Elsevier, (1976).   Google Scholar

[13]

C. Fabre, J. P. Puel and E. Zuazua, On the density of the range of the semigroup for semilinear heat equations,, Control and optimal design of distributed parameter systems (Minneapolis, 70 (1995), 73.  doi: 10.1007/978-1-4613-8460-1_4.  Google Scholar

[14]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Advances in Differential Equations, 5 (2000), 465.   Google Scholar

[15]

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), 97.  doi: 10.1137/110840133.  Google Scholar

[16]

J. A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differential Equations, 12 (2007), 1031.   Google Scholar

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).   Google Scholar

[18]

A. Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems,, BIT, 42 (2002), 351.  doi: 10.1023/A:1021903109720.  Google Scholar

[19]

R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations,, SIAM J. Control Optim., 50 (2012), 943.  doi: 10.1137/100815037.  Google Scholar

[20]

J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients,, Proc. Amer. Math. Soc., 130 (2002), 1055.  doi: 10.1090/S0002-9939-01-06163-9.  Google Scholar

[21]

K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems,, SIAM J. Control Optim., 52 (2014), 3078.  doi: 10.1137/140959055.  Google Scholar

[22]

D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control,, SIAM J. Numer. Anal., 51 (2013), 2797.  doi: 10.1137/120885772.  Google Scholar

[23]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $l_1$ constrained minimization,, Inverse Probl. and Imaging, 8 (2014), 199.  doi: 10.3934/ipi.2014.8.199.  Google Scholar

[24]

J.-L. Lions and B. Malgrange, Sur l'unicité rétrograde des équations paraboliques,, Math. Scand., 8 (1960), 277.   Google Scholar

[25]

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), 1961.  doi: 10.1137/100793888.  Google Scholar

[26]

K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space,, SIAM J. Control Optim., 51 (2013), 2788.  doi: 10.1137/120889137.  Google Scholar

[27]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations,, App. Math. Optim., 39 (1999), 143.  doi: 10.1007/s002459900102.  Google Scholar

[28]

W. Rudin, Real and Complex Analysis,, McGraw-Hill, (1970).   Google Scholar

[29]

A. H. Schatz, V. C. Thomée and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations,, Comm. Pure Appl. Math., 33 (1980), 265.  doi: 10.1002/cpa.3160330305.  Google Scholar

[30]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices,, Comput. Optim. Appl., 44 (2009), 159.  doi: 10.1007/s10589-007-9150-9.  Google Scholar

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems,, Second edition, (2006).   Google Scholar

[32]

G. Wachsmuth and D. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional,, ESAIM Control Optim. Calc. Var., 17 (2011), 858.  doi: 10.1051/cocv/2010027.  Google Scholar

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