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December  2020, 10(4): 735-759. doi: 10.3934/mcrf.2020018

## Maximal discrete sparsity in parabolic optimal control with measures

 1 Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Universitätsstraße 1, 56070 Koblenz, Germany 2 Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

* Corresponding author: Evelyn Herberg

Received  June 2018 Revised  November 2019 Published  March 2020

Fund Project: The second author acknowledges support of the priority programme 1962 (SPP 1962) funded by the Deutsche Forschungsgemeinschaft

We consider variational discretization [18] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in $L^q$ and weakly-$*$ in $\mathcal{M}$, respectively, to their smooth counterparts, where $q \in (1,\min\{2,1+2/d\}]$ is the spatial dimension. Furthermore, we compare our approach to [8], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

Citation: Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control & Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar [2] J. J. Ahlberg and E. N. Nilson, Convergence properties of the spline fit, J. Soc. Indust. Appl. Math., 11 (1963), 95-104.  doi: 10.1137/0111007.  Google Scholar [3] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. I, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. Google Scholar [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [6] E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.  Google Scholar [7] E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 51 (2013), 28-63.  doi: 10.1137/120872395.  Google Scholar [8] E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM. Control, Optimisation and Calculus of Variations, 22 (2016), 355-370.  doi: 10.1051/cocv/2015008.  Google Scholar [9] E. Casas, B. Vexler and E. Zuazua, Sparse initial data identification for parabolic PDE and its finite element approximations, Mathematical Control and Related Fields, 5 (2015), 377-399.  doi: 10.3934/mcrf.2015.5.377.  Google Scholar [10] C. Clason, Nonsmooth Analysis and Optimization, eprint, arXiv: 1708.04180. Google Scholar [11] C. Clason and K. 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Wollner, The damped Crank-Nicolson time-marching scheme for the adaptive solution of the Black-Scholes equation, Journal of Computational Finance, 18 (2015), 1-37.   Google Scholar [16] W. Gong, Error estimates for finite element approximations of parabolic equations with measure data, Mathematics of Computation, 82 (2013), 69-98.  doi: 10.1090/S0025-5718-2012-02630-5.  Google Scholar [17] W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control, SIAM Journal on Control and Optimization, 52 (2014), 97-119.  doi: 10.1137/110840133.  Google Scholar [18] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications. An International Journal, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009.  Google Scholar [20] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar [21] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 3078-3108.  doi: 10.1137/140959055.  Google Scholar [22] J. R. Munkres, Topology, Prentice Hall Inc., Upper Saddle River, NJ, 2000.  Google Scholar [23] J. A. Nitsche, $L_{\infty }$-convergence of Finite Element Approximation in Journées "Éléments Finis", Univ. Rennes, Rennes, 1975.  Google Scholar [24] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM Journal on Control and Optimization, 51 (2013), 2788-2808.  doi: 10.1137/120889137.  Google Scholar [25] R. A. Polyak, Complexity of the regularized Newton's method, Pure and Applied Functional Analysis, 3 (2018), 327-347.   Google Scholar [26] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987.  Google Scholar [27] W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar [28] G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications. An International Journal, 44 (2009), 159-181.  doi: 10.1007/s10589-007-9150-9.  Google Scholar [29] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006.  Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar [2] J. J. Ahlberg and E. N. Nilson, Convergence properties of the spline fit, J. Soc. Indust. Appl. Math., 11 (1963), 95-104.  doi: 10.1137/0111007.  Google Scholar [3] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. I, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. Google Scholar [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [6] E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.  Google Scholar [7] E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 51 (2013), 28-63.  doi: 10.1137/120872395.  Google Scholar [8] E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM. Control, Optimisation and Calculus of Variations, 22 (2016), 355-370.  doi: 10.1051/cocv/2015008.  Google Scholar [9] E. Casas, B. Vexler and E. Zuazua, Sparse initial data identification for parabolic PDE and its finite element approximations, Mathematical Control and Related Fields, 5 (2015), 377-399.  doi: 10.3934/mcrf.2015.5.377.  Google Scholar [10] C. Clason, Nonsmooth Analysis and Optimization, eprint, arXiv: 1708.04180. Google Scholar [11] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control, Optimisation and Calculus of Variations, 17 (2011), 243-266.  doi: 10.1051/cocv/2010003.  Google Scholar [12] N. von Daniels, M. Hinze and M. Vierling, Crank-Nicolson time stepping and variational discretization of control-constrained parabolic optimal control problems, SIAM Journal on Control and Optimization, 53 (2015), 1182-1198.  doi: 10.1137/14099680X.  Google Scholar [13] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [14] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.  Google Scholar [15] C. Goll, R. Rannacher and W. Wollner, The damped Crank-Nicolson time-marching scheme for the adaptive solution of the Black-Scholes equation, Journal of Computational Finance, 18 (2015), 1-37.   Google Scholar [16] W. Gong, Error estimates for finite element approximations of parabolic equations with measure data, Mathematics of Computation, 82 (2013), 69-98.  doi: 10.1090/S0025-5718-2012-02630-5.  Google Scholar [17] W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control, SIAM Journal on Control and Optimization, 52 (2014), 97-119.  doi: 10.1137/110840133.  Google Scholar [18] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications. An International Journal, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009.  Google Scholar [20] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar [21] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 3078-3108.  doi: 10.1137/140959055.  Google Scholar [22] J. R. Munkres, Topology, Prentice Hall Inc., Upper Saddle River, NJ, 2000.  Google Scholar [23] J. A. Nitsche, $L_{\infty }$-convergence of Finite Element Approximation in Journées "Éléments Finis", Univ. Rennes, Rennes, 1975.  Google Scholar [24] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM Journal on Control and Optimization, 51 (2013), 2788-2808.  doi: 10.1137/120889137.  Google Scholar [25] R. A. Polyak, Complexity of the regularized Newton's method, Pure and Applied Functional Analysis, 3 (2018), 327-347.   Google Scholar [26] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987.  Google Scholar [27] W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar [28] G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications. An International Journal, 44 (2009), 159-181.  doi: 10.1007/s10589-007-9150-9.  Google Scholar [29] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006.  Google Scholar
Numerical setup on $4 \times 12$ space-time grid with $q = \tfrac{4}{3}$. From left to right: control $u = \delta _{(1/2,1/2)}$, associated state $y(u)$ (sampled from the analytic solution with spacial Fourier modes), discrete desired state $y_{\operatorname{d}}$ and calculated controls $u_{\sigma,0}$ and $u_{\operatorname{DG},0}$ for $\alpha = 0$. Here the controls are represented by their coefficients. In the case of piecewise constant controls in time, which are used in the discontinuous Galerkin setting, this leads to coefficient values, which are scaled with $\tfrac{1}{\tau} = 8$
The dependence on the penalty parameter $\alpha$ of the measure norm of $u_{\sigma,\alpha}$ and $u_{\operatorname{DG},\alpha}$ (left) and the errors $y_{\sigma,\alpha} - y_{\operatorname{d}}$ and $y_{\operatorname{DG},\alpha} - y_{\operatorname{d}}$ in the $L^{4/3}$ norm (right)
Top row: The measure control and the optimal controls $u_{\sigma,0.456}$ and $u_{\operatorname{DG},0.456}$. Bottom row: The associated state $y(u)$ (sampled from the analytic solution with spacial Fourier modes) and the associated states $y_{\sigma,0.456}$ and $y_{\operatorname{DG},0.456}$
Numerical setup on a $4 \times 48$ space-time grid with $q = \tfrac{4}{3}$. From left to right: true control $u_{\operatorname{true}} = \delta _{(1/2,1/2)}$, interpolation of the associated state $y(u_{\operatorname{true}})$, adjoint state $w_{\bar{\alpha}}$ multiplied by $-1$ for easier visualization and desired state $y_{\operatorname{d}}$ calculated using Fenchel duality
Top row: The difference of the measure norms of the true control $u_{\operatorname{true}}$ and calculated optimal control $u_i$, $i \in \{\sigma, \operatorname{DG}\}$ for $\tau = \tfrac{h}{2}$ (left) and $\tau = \tfrac{h^2}{2}$ (right). Bottom row: The $L^{4/3}$ norm of the difference of the associated states $y_i - y_{\operatorname{true}}$, $i \in \{\sigma, \operatorname{DG}\}$ for $\tau = \tfrac{h}{2}$ (left) and $\tau = \tfrac{h^2}{2}$ (right)
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