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Optimal control of a non-smooth quasilinear elliptic equation
September  2021, 11(3): 555-578. doi: 10.3934/mcrf.2021012

## Optimal control of ODEs with state suprema

 1 Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany 2 Brandenburgische Technische Universität Cottbus-Senftenberg, Institute of Mathematics, 03046 Cottbus, Germany

* Corresponding author: D. Wachsmuth

Received  May 2019 Revised  November 2019 Published  September 2021 Early access  March 2021

Fund Project: The second and third author were supported by DFG grants within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged

We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.

Citation: Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control and Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012
##### References:
 [1] V. Azhmyakov, A. Ahmed and E. I. Verriest, On the optimal control of systems evolving with state suprema, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, 3617–3623. doi: 10.1109/CDC.2016.7798813. [2] D. D. Bainov and S. G. Hristova, Differential Equations with Maxima, Boca Raton, FL: CRC Press, 2011. [3] H. T. Banks, Necessary conditions for control problems with variable time lags, SIAM Journal on Control, 6 (1968), 9–47. doi: 10.1137/0306002. [4] H. T. Banks, Variational problems involving functional differential equations, SIAM Journal on Control, 7 (1969), 1-17.  doi: 10.1137/0307001. [5] P. Blanchard, D. J. Higham and N. J. Higham, Accurately computing the log-sum-exp and softmax functions, IMA Journal of Numerical Analysis, 2020. doi: 10.1093/imanum/draa038. [6] H. Cartan, Calcul Différentiel, Hermann, Paris, 1967. [7] C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation, Math. Control Relat. Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011. [8] F. H. Clarke and P. R. Wolenski, Necessary conditions for functional differential inclusions, Applied Mathematics & Optimization, 34 (1996), 51–78. doi: 10.1007/BF01182473. [9] S. Dashkovskiy, S. G. Hristova, O. Kichmarenko and K. Sapozhnikova, Behavior of solutions to systems with maximum, IFAC-PapersOnLine, 50 (2017), 12925–12930, 20th IFAC World Congress. doi: 10.1016/j.ifacol.2017.08.1790. [10] J. Diestel and J. J. Uhl, Vector Measures, Mathematical surveys, American Mathematical Society, Providence, RI, 1977. [11] R. E. Edwards, Functional Analysis: Theory and Applications, Dover books on mathematics, Dover Publications, 1995. [12] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, vol. 28 of Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088. [13] I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT Press, 2016, URL http://www.deeplearningbook.org. [14] F. Kruse and M. Ulbrich, A self-concordant interior point approach for optimal control with state constraints, SIAM Journal on Optimization, 25 (2015), 770–806. doi: 10.1137/130936671. [15] N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, vol. 19 of Advances in Mechanics and Mathematics, Springer, New York, 2009. doi: 10.1007/b120946. [16] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J. 1970. [17] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [18] W. Rudin, Real and Complex Analysis, Second edition. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. [19] E. I. Verriest and V. Azhmyakov, Advances in optimal control of differential systems with the state suprema, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), 2017,739–744. doi: 10.1109/CDC.2017.8263748. [20] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer New York, 1996. doi: 10.1007/978-1-4612-4050-1.

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##### References:
 [1] V. Azhmyakov, A. Ahmed and E. I. Verriest, On the optimal control of systems evolving with state suprema, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, 3617–3623. doi: 10.1109/CDC.2016.7798813. [2] D. D. Bainov and S. G. Hristova, Differential Equations with Maxima, Boca Raton, FL: CRC Press, 2011. [3] H. T. Banks, Necessary conditions for control problems with variable time lags, SIAM Journal on Control, 6 (1968), 9–47. doi: 10.1137/0306002. [4] H. T. Banks, Variational problems involving functional differential equations, SIAM Journal on Control, 7 (1969), 1-17.  doi: 10.1137/0307001. [5] P. Blanchard, D. J. Higham and N. J. Higham, Accurately computing the log-sum-exp and softmax functions, IMA Journal of Numerical Analysis, 2020. doi: 10.1093/imanum/draa038. [6] H. Cartan, Calcul Différentiel, Hermann, Paris, 1967. [7] C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation, Math. Control Relat. Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011. [8] F. H. Clarke and P. R. Wolenski, Necessary conditions for functional differential inclusions, Applied Mathematics & Optimization, 34 (1996), 51–78. doi: 10.1007/BF01182473. [9] S. Dashkovskiy, S. G. Hristova, O. Kichmarenko and K. Sapozhnikova, Behavior of solutions to systems with maximum, IFAC-PapersOnLine, 50 (2017), 12925–12930, 20th IFAC World Congress. doi: 10.1016/j.ifacol.2017.08.1790. [10] J. Diestel and J. J. Uhl, Vector Measures, Mathematical surveys, American Mathematical Society, Providence, RI, 1977. [11] R. E. Edwards, Functional Analysis: Theory and Applications, Dover books on mathematics, Dover Publications, 1995. [12] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, vol. 28 of Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088. [13] I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT Press, 2016, URL http://www.deeplearningbook.org. [14] F. Kruse and M. Ulbrich, A self-concordant interior point approach for optimal control with state constraints, SIAM Journal on Optimization, 25 (2015), 770–806. doi: 10.1137/130936671. [15] N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, vol. 19 of Advances in Mechanics and Mathematics, Springer, New York, 2009. doi: 10.1007/b120946. [16] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J. 1970. [17] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [18] W. Rudin, Real and Complex Analysis, Second edition. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. [19] E. I. Verriest and V. Azhmyakov, Advances in optimal control of differential systems with the state suprema, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), 2017,739–744. doi: 10.1109/CDC.2017.8263748. [20] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer New York, 1996. doi: 10.1007/978-1-4612-4050-1.
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