Open Access
Issue |
ESAIM: COCV
Volume 31, 2025
|
|
---|---|---|
Article Number | 2 | |
Number of page(s) | 34 | |
DOI | https://doi.org/10.1051/cocv/2024085 | |
Published online | 06 January 2025 |
- G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas, R. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations. ESAIM Control Optim. Calc. Var. 23 (2017) 263–295. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- E. Casas, M. Mateos and A. Rösch, Finite element approximation of sparse parabolic control problems. Math. Control Relat. Fields 7 (2017) 393–417. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity. Comput. Optim. Appl. 70 (2018) 239–266. [CrossRef] [MathSciNet] [Google Scholar]
- R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50 (2012) 943–963. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50 (2012) 1735–1752. [Google Scholar]
- E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51 (2013) 28–63. [Google Scholar]
- E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces. ESAIM Control Optim. Calc. Var. 22 (2016) 355–370. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control Optim. 52 (2014) 3078–3108. [Google Scholar]
- K. Kunisch, P. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls. SIAM J. Control Optim. 54 (2016) 1212–1244. [CrossRef] [MathSciNet] [Google Scholar]
- P. Trautmann, B. Vexler and A. Zlotnik, Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Math. Control Relat. Fields 8 (2018) 411–449. [CrossRef] [MathSciNet] [Google Scholar]
- E. Herberg, M. Hinze and H. Schumacher, Maximal discrete sparsity in parabolic optimal control with measures. Math. Control Relat. Fields 10 (2020) 735–759. [CrossRef] [MathSciNet] [Google Scholar]
- D. Meidner and B. Vexler, A priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim. 49 (2011) 2183–2211. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas and K. Kunisch, Using sparse control methods to identify sources in linear diffusion-convection equations. Inverse Probl. 35 (2019) 114002 [CrossRef] [Google Scholar]
- E. Casas, B. Vexler and E. Zuazua, Sparse initial data identification for parabolic PDE and its finite element approximations. Math. Control Relat. Fields 5 (2015) 377–399. [CrossRef] [MathSciNet] [Google Scholar]
- E. Herberg and M. Hinze, Variational discretization approach applied to an optimal control problem with bounded measure controls, Radon Series on Computational and Applied Mathematics. Vol. 29, “Optimization and Control for Partial Differential Equations” (2022). [Google Scholar]
- B. Vexler, D. Leykekhman and D. Walter, Numerical analysis of sparse initial data identification for parabolic problems. ESAIM: M2AN 54 (2020) 1139–1180. [CrossRef] [EDP Sciences] [Google Scholar]
- A. Boulanger and P. Trautmann, Sparse optimal control of the KdV-Burgers equation on a bounded domain. SIAM J. Control Optim. 55 (2017) 3673–3706. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52 (2014) 339–364. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas and K. Kunisch, Stabilization by sparse controls for a class of semilinear parabolic equations. SIAM J. Control Optim. 55 (2017) 512–532. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas and K. Kunisch, Optimal control of the two-dimensional stationary Navier-Stokes equations with measure valued controls. SIAM J. Control Optim. 57 (2019) 1328–1354. [CrossRef] [MathSciNet] [Google Scholar]
- C. Clason and K. Kunisch, A duality approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: COCV 17 (2011) 243–266. [CrossRef] [EDP Sciences] [Google Scholar]
- C. Clason and K. Kunisch, A measure space approach to optimal source placement. Comput. Optim. Appl. 53 (2012) 155–171. [CrossRef] [MathSciNet] [Google Scholar]
- G.-M. Coclite and A. Garavello, A time-dependent optimal harvesting problem with measure-valued solutions. SIAM J. Control Optim. 55 (2017) 913–935. [CrossRef] [MathSciNet] [Google Scholar]
- D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51 (2013) 2797–2821. [CrossRef] [MathSciNet] [Google Scholar]
- D. Leykekhman and B. Vexler, A priori error estimates for three dimensional parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 54 (2016) 2403–2435. [Google Scholar]
- B. Vexler, A. Zlotnik and P. Trautmann, On a finite element method for measure-valued optimal control problems governed by the 1D generalized wave equation. C. R. Math. Acad. Sci. Paris 356 (2018) 523–531. [CrossRef] [MathSciNet] [Google Scholar]
- W. Gong and B.-Y. Li, Improved error estimates for semi-discrete finite element solutions of parabolic Dirichlet boundary control problems. IMA J. Numer. Anal. 40 (2020) 2898–2939. [CrossRef] [MathSciNet] [Google Scholar]
- D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints. SIAM J. Control Optim. 47 (2008) 1301–1329. [CrossRef] [MathSciNet] [Google Scholar]
- A. Rösch, Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwendungen 23 (2004) 353–376. [Google Scholar]
- A. Bensoussan and J.-L. Lions, Impulse Control and Quasi-Variational Inequalities, Bordas, Paris (1984). [Google Scholar]
- A.Yu. Chebotarev, Impulse control of temperature in a free convection model. Comput. Math. Math. Phys. 49 (2009) 594–601. [CrossRef] [MathSciNet] [Google Scholar]
- Y.-L. Duan and L.-J. Wang, Minimal norm control problems governed by semilinear heat equation with impulse control. J. Optim. Theory Appl. 184 (2020) 400–418. [CrossRef] [MathSciNet] [Google Scholar]
- Y. Duan, L. Wang and C. Zhang, Minimal time impulse control of an evolution equation. J. Optim. Theory Appl. 183 (2019) 902–919. [CrossRef] [MathSciNet] [Google Scholar]
- K.-D. Phung, G.-S. Wang and Y.-S. Xu, Impulse output rapid stabilization for heat equations. J. Differ. Equ. 263 (2017) 5012–5041. [CrossRef] [Google Scholar]
- S.-L. Qin and G.-S. Wang, Controllability of impulse controlled systems of heat equations coupled by constant matrices. J. Differ. Equ. 263 (2017) 6456–6493. [CrossRef] [Google Scholar]
- E. Trélat, L.-J. Wang and Y.-B. Zhang, Impulse and sampled-data optimal control of heat equations, and error estimates. SIAM J. Control Optim. 54 (2016) 2787–2819. [CrossRef] [MathSciNet] [Google Scholar]
- Q.-S. Yan, Periodic optimal control problems governed by semilinear parabolic equations with impulse control. Acta Math. Sci. 36B (2016) 847–862. [CrossRef] [MathSciNet] [Google Scholar]
- T. Yang, Impulse Control Theory. Springer-Verlag, Berlin, Heidelberg (2001). [Google Scholar]
- J. Yong and P. Zhang, Necessary conditions of optimal impulse controls for distributed parameter systems. Bull. Aust. Math. Soc. 45 (1992) 305–326. [Google Scholar]
- X. Yu, J.-F. Huang and K.-S. Liu, Finite element approximations of impulsive optimal control problems for heat equations. J. Math. Anal. Appl. 477 (2019) 250–271. [CrossRef] [MathSciNet] [Google Scholar]
- W. Gong, Error estimates for finite element approximations of parabolic equations with measure data. Math. Comput. 82 (2013) 69–98. [Google Scholar]
- A. Khapalov, Exact controllability of second-order hyperbolic equations with impulse controls. Appl. Anal. 63 (1996) 223–238. [CrossRef] [MathSciNet] [Google Scholar]
- N. von Daniels, M. Hinze and M. Vierling, Crank–Nicolson time stepping and variational discretization of control- constrained parabolic optimal control problems. SIAM J. Control Optim. 53 (2015) 1182–1198. [CrossRef] [MathSciNet] [Google Scholar]
- S. Lang, Real Analysis. 2nd edn. Addison-Wesley, Reading, MA (1983). [Google Scholar]
- J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including C1 interfaces. Interfaces Free Bound. 9 (2007) 233–252. [CrossRef] [MathSciNet] [Google Scholar]
- L.-C. Evans, Partial Differential Equations. Vol. 19. AMS (2002). [Google Scholar]
- J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin (1972). [Google Scholar]
- W. Gong, M. Hinze and Z.-J. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52 (2014) 97–119. [CrossRef] [MathSciNet] [Google Scholar]
- W. Gong, H.-P. Liu and N.-N. Yan, Adaptive finite element method for parabolic equations with Dirac measure. Comput. Methods Appl. Mech. Eng. 328 (2018) 217–241. [CrossRef] [Google Scholar]
- 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) 1961–1997. [Google Scholar]
- J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971). [CrossRef] [Google Scholar]
- V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. 2nd edn. Springer Series in Computational Mathematics, Vol. 25. Springer-Verlag, Berlin (2006). [Google Scholar]
- E. Casas, Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297–1327. [Google Scholar]
- I. Ekeland and R. Temam, Convex Analysis and Variational Problems. Classics Appl. Math. 28. SIAM, Philadelphia (1999). [Google Scholar]
- L. Meziani, On the dual space C*0(S, X). Acta Math. Univ. Comenian (N.S.) 78 (2009) 153–160. [MathSciNet] [Google Scholar]
- P.-G. Ciarlet, The Finite Element Methods for Elliptic Problems. 1st edn., Vol. 4. North-Holland (1978). [Google Scholar]
- M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–63. [CrossRef] [MathSciNet] [Google Scholar]
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