# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021028
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## On the nonuniqueness and instability of solutions of tracking-type optimal control problems

 1 Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany

*Corresponding author: Constantin Christof

Received  July 2020 Revised  March 2021 Early access May 2021

Fund Project: This research was conducted within the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754.

We study tracking-type optimal control problems that involve a non-affine, weak-to-weak continuous control-to-state mapping, a desired state $y_d$, and a desired control $u_d$. It is proved that such problems are always nonuniquely solvable for certain choices of the tuple $(y_d, u_d)$ and instable in the sense that the set of solutions (interpreted as a multivalued function of $(y_d, u_d)$) does not admit a continuous selection.

Citation: Constantin Christof, Dominik Hafemeyer. On the nonuniqueness and instability of solutions of tracking-type optimal control problems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021028
##### References:
 [1] A. Ahmad Ali, K. Deckelnick and M. Hinze, Global minima for optimal control of the obstacle problem, ESAIM Control, Optimisation and Calculus of Variations, 26 (2020), Paper No. 64, 22 pp. doi: 10.1051/cocv/2019039.  Google Scholar [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM, Philadelphia, PA, 2006.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, vol. 100, Pitman, Boston, MA, 1984.  Google Scholar [4] T. Betz, C. Meyer, A. Rademacher and K. Rosin, Adaptive optimal control of elastoplastic contact problems, Ergebnisberichte des Instituts für Angewandte Mathematik, TU Dortmund, Nr. 496, 2014, 10 pp. http://www.mathematik.tu-dortmund.de/papers/BetzMeyerRademacherRosin2014.pdf Google Scholar [5] J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counter examples, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139087322.  Google Scholar [6] A. L. Brown, Set valued mappings, continuous selections, and metric projections, Journal of Approximation Theory, 57 (1989), 48-68.  doi: 10.1016/0021-9045(89)90083-X.  Google Scholar [7] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM Journal on Control and Optimization, 50 (2012), 2355-2372.  doi: 10.1137/120862892.  Google Scholar [8] C. Christof, Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities, SIAM Journal on Control and Optimization, 57 (2019), 192-218.  doi: 10.1137/18M1183662.  Google Scholar [9] C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation, Mathematical Control and Related Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011.  Google Scholar [10] C. Christof and B. Vexler, New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints, ESAIM Control, Optimisation and Calculus of Variations, 27 (2021), Paper No. 4, 39 pp. doi: 10.1051/cocv/2020059.  Google Scholar [11] C. Christof and G. Wachsmuth, On second-order optimality conditions for optimal control problems governed by the obstacle problem, Optimization, (2020). doi: 10.1080/02331934.2020.1778686.  Google Scholar [12] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society, 40 (1936), 396-414.  doi: 10.1090/S0002-9947-1936-1501880-4.  Google Scholar [13] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Mathematics, vol. 1543 Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0084195.  Google Scholar [14] J. Fletcher and W. B. 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Wollner, OPTPDE - A collection of problems in PDE-constrained optimization, http://www.optpde.net Google Scholar [20] R. Herzog, A. Rösch, S. Ulbrich and W. Wollner, OPTPDE: A collection of problems in PDE-constrained optimization, in Trends in PDE Constrained Optimization, International Series of Numerical Mathematics, vol. 165, Birkhäuser/Springer, Cham, 2014,539–543. doi: 10.1007/978-3-319-05083-6_34.  Google Scholar [21] P. C. Kainen, V. Kůrková and A. Vogt, Geometry and topology of continuous best and near best approximations, Journal of Approximation Theory, 105 (2000), 252-262.  doi: 10.1006/jath.2000.3467.  Google Scholar [22] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, vol. 31, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar [23] V. Klee, Convexity of Chebyshev sets, Mathematische Annalen, 142 (1961), 292-304.  doi: 10.1007/BF01353420.  Google Scholar [24] K. Kunisch and D. Wachsmuth, Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control, Optimisation and Calculus of Variations, 18 (2012), 520-547.  doi: 10.1051/cocv/2011105.  Google Scholar [25] J. Lohéac, E. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1587-1644.  doi: 10.1142/S0218202517500270.  Google Scholar [26] R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, NY, 1998. doi: 10.1007/978-1-4612-0603-3.  Google Scholar [27] E. Muselli, Affinity and well-posedness for optimal control problems in Hilbert spaces, Journal of Convex Analysis, 14 (2007), 767-784.   Google Scholar [28] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, 2012. Google Scholar [29] B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Mathematical Journal, 5 (1939), 249-253.  doi: 10.1215/S0012-7094-39-00522-3.  Google Scholar [30] D. Pighin, Nonuniqueness of minimizers for semilinear optimal control problems, preprint, 2020, arXiv: 2002.04485. Google Scholar [31] B. Schweizer, Partielle Differentialgleichungen, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-642-40638-6.  Google Scholar [32] U. Westphal and J. Frerking, On a property of metric projections onto closed subsets of Hilbert spaces, Proceedings of the American Mathematical Society, 105 (1989), 644-651.  doi: 10.1090/S0002-9939-1989-0946636-6.  Google Scholar [33] K. Yosida, Functional Analysis, 6$^th$ edition, Springer-Verlag, Berlin-New York, 1980.  Google Scholar [34] T. Zolezzi, A characterization of well-posed optimal control systems, SIAM Journal on Control and Optimization, 19 (1981), 604-616.  doi: 10.1137/0319038.  Google Scholar [35] E. Zuazua, Some results and open problems on the controllability of linear and semilinear heat equations, in Carleman Estimates and Applications to Uniqueness and Control Theory, Birkhäuser Boston, Boston, MA, 2001,191–211.  Google Scholar

show all references

##### References:
 [1] A. Ahmad Ali, K. Deckelnick and M. Hinze, Global minima for optimal control of the obstacle problem, ESAIM Control, Optimisation and Calculus of Variations, 26 (2020), Paper No. 64, 22 pp. doi: 10.1051/cocv/2019039.  Google Scholar [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM, Philadelphia, PA, 2006.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, vol. 100, Pitman, Boston, MA, 1984.  Google Scholar [4] T. Betz, C. Meyer, A. Rademacher and K. Rosin, Adaptive optimal control of elastoplastic contact problems, Ergebnisberichte des Instituts für Angewandte Mathematik, TU Dortmund, Nr. 496, 2014, 10 pp. http://www.mathematik.tu-dortmund.de/papers/BetzMeyerRademacherRosin2014.pdf Google Scholar [5] J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counter examples, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139087322.  Google Scholar [6] A. L. Brown, Set valued mappings, continuous selections, and metric projections, Journal of Approximation Theory, 57 (1989), 48-68.  doi: 10.1016/0021-9045(89)90083-X.  Google Scholar [7] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM Journal on Control and Optimization, 50 (2012), 2355-2372.  doi: 10.1137/120862892.  Google Scholar [8] C. Christof, Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities, SIAM Journal on Control and Optimization, 57 (2019), 192-218.  doi: 10.1137/18M1183662.  Google Scholar [9] C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation, Mathematical Control and Related Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011.  Google Scholar [10] C. Christof and B. Vexler, New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints, ESAIM Control, Optimisation and Calculus of Variations, 27 (2021), Paper No. 4, 39 pp. doi: 10.1051/cocv/2020059.  Google Scholar [11] C. Christof and G. Wachsmuth, On second-order optimality conditions for optimal control problems governed by the obstacle problem, Optimization, (2020). doi: 10.1080/02331934.2020.1778686.  Google Scholar [12] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society, 40 (1936), 396-414.  doi: 10.1090/S0002-9947-1936-1501880-4.  Google Scholar [13] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Mathematics, vol. 1543 Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0084195.  Google Scholar [14] J. Fletcher and W. B. Moors, Chebyshev sets, Journal of the Australian Mathematical Society, 98 (2015), 161-231.  doi: 10.1017/S1446788714000561.  Google Scholar [15] M. Gugat, G. Leugering and G. Sklyar, Lp-optimal boundary control for the wave equation, SIAM Journal on Control and Optimization, 44 (2005), 49-74.  doi: 10.1137/S0363012903419212.  Google Scholar [16] D. Hafemeyer, Optimal Control of the Parabolic Obstacle Problem, PhD thesis, Technische Universität München, 2020. Google Scholar [17] J. Heinonen, P. Koselka, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, New Mathematical Monographs, vol. 27, Cambridge University Press, Cambridge, 2015.  doi: 10.1017/CBO9781316135914.  Google Scholar [18] M. Herty, R. Pinnau and M. Seaïd, Optimal control in radiative transfer, Optimization Methods & Software, 22 (2007), 917-936.  doi: 10.1080/10556780701405783.  Google Scholar [19] R. Herzog, A. Rösch, S. Ulbrich and W. Wollner, OPTPDE - A collection of problems in PDE-constrained optimization, http://www.optpde.net Google Scholar [20] R. Herzog, A. Rösch, S. Ulbrich and W. Wollner, OPTPDE: A collection of problems in PDE-constrained optimization, in Trends in PDE Constrained Optimization, International Series of Numerical Mathematics, vol. 165, Birkhäuser/Springer, Cham, 2014,539–543. doi: 10.1007/978-3-319-05083-6_34.  Google Scholar [21] P. C. Kainen, V. Kůrková and A. Vogt, Geometry and topology of continuous best and near best approximations, Journal of Approximation Theory, 105 (2000), 252-262.  doi: 10.1006/jath.2000.3467.  Google Scholar [22] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, vol. 31, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar [23] V. Klee, Convexity of Chebyshev sets, Mathematische Annalen, 142 (1961), 292-304.  doi: 10.1007/BF01353420.  Google Scholar [24] K. Kunisch and D. Wachsmuth, Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control, Optimisation and Calculus of Variations, 18 (2012), 520-547.  doi: 10.1051/cocv/2011105.  Google Scholar [25] J. Lohéac, E. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1587-1644.  doi: 10.1142/S0218202517500270.  Google Scholar [26] R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, NY, 1998. doi: 10.1007/978-1-4612-0603-3.  Google Scholar [27] E. Muselli, Affinity and well-posedness for optimal control problems in Hilbert spaces, Journal of Convex Analysis, 14 (2007), 767-784.   Google Scholar [28] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, 2012. Google Scholar [29] B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Mathematical Journal, 5 (1939), 249-253.  doi: 10.1215/S0012-7094-39-00522-3.  Google Scholar [30] D. Pighin, Nonuniqueness of minimizers for semilinear optimal control problems, preprint, 2020, arXiv: 2002.04485. Google Scholar [31] B. Schweizer, Partielle Differentialgleichungen, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-642-40638-6.  Google Scholar [32] U. Westphal and J. Frerking, On a property of metric projections onto closed subsets of Hilbert spaces, Proceedings of the American Mathematical Society, 105 (1989), 644-651.  doi: 10.1090/S0002-9939-1989-0946636-6.  Google Scholar [33] K. Yosida, Functional Analysis, 6$^th$ edition, Springer-Verlag, Berlin-New York, 1980.  Google Scholar [34] T. Zolezzi, A characterization of well-posed optimal control systems, SIAM Journal on Control and Optimization, 19 (1981), 604-616.  doi: 10.1137/0319038.  Google Scholar [35] E. Zuazua, Some results and open problems on the controllability of linear and semilinear heat equations, in Carleman Estimates and Applications to Uniqueness and Control Theory, Birkhäuser Boston, Boston, MA, 2001,191–211.  Google Scholar
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