# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022016
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## Pitfalls in applying optimal control to dynamical systems: An overview and editorial perspective

 1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 2 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il, 62026-1653, USA 3 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA

* Corresponding author: Urszula Ledzewicz

Received  September 2021 Early access February 2022

In recent years, an increasing number of papers have been published (and many more submitted for publication) in which optimal control theory is superficially applied to specific problems, especially from the biological and health sciences, but also many other fields. A lack of understanding of what it actually means to solve an optimal control problem—complex infinite-dimensional optimization problems—often leads to heavily overblown claims about optimality of solutions. In this editorial, a critical assessment of these efforts is given.

Citation: Urszula Ledzewicz, Heinz Schättler. Pitfalls in applying optimal control to dynamical systems: An overview and editorial perspective. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022016
##### References:
 [1] L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, 1974. [2] S. Bhan and H. Schättler, A variational approach to perturbation feedback control for optimal control problems with terminal constraints and free terminal time, Set-Valued Var. Anal., 27 (2019), 309-330.  doi: 10.1007/s11228-018-0486-3. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062. [4] V. G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method, SIAM J. Control, 4 (1966), 326-361.  doi: 10.1137/0304027. [5] V. G. Boltyansky, Mathematical Methods of Optimal Control, Holt, Rinehart and Winston, Inc., 1971. [6] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. [7] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, Vol. 43, Springer-Verlag, Berlin, 2004. [8] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [9] A. E. Bryson Jr. and Y. C. Ho, Applied Optimal Control, Revised Printing, Hemisphere Publishing Company, New York, 1975. [10] C. Byrnes and H. Frankowska, Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton–Jacobi–Bellman et de Riccati, C. R. Acad. Sci. Paris, 315 (1992), 427-431. [11] C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control, in: Systems, Models and Feedback: Theory and Applications, (A. Isidori and T. J. Tarn, eds.), Birkhäuser, (1992), 211–227. [12] I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Vol. 67, Springer-Verlag, Berlin-New York, 1972. [13] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979. [14] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley–Interscience, 1972. [15] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. Optim. Theory Appl., 114 (2002), 609-637.  doi: 10.1023/A:1016027113579. [16] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079.  doi: 10.1137/060665294. [17] U. Ledzewicz and H. Schättler, Combination of antiangiogenic treatment with chemotherapy as a multi-input optimal control problem, Math. Methods in the Applied Sciences, publ. online. doi: 10.1002/mma.7977. [18] D. Liberzon, Calculus of Variations and Optimal Control, Princeton University Press, Princeton, 2012. [19] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [20] H. G. Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636.  doi: 10.1137/0311048. [21] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficient conditions for optimality, SIAM J. Control Optim., 39 (2000), 359-410.  doi: 10.1137/S0363012999322031. [22] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. [23] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, Vol. 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [24] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, Vol. 42, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. [25] H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L_2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657. [26] H. J. Sussmann and J. C. Willems, 300 years of optimal control: From the brachistochrone to the maximum principle, IEEE Control Systems, 17 (1997), 32-44.  doi: 10.1109/37.588098. [27] G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984. [28] G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P. [29] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 3 (1995), 41-54.

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##### References:
 [1] L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, 1974. [2] S. Bhan and H. Schättler, A variational approach to perturbation feedback control for optimal control problems with terminal constraints and free terminal time, Set-Valued Var. Anal., 27 (2019), 309-330.  doi: 10.1007/s11228-018-0486-3. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062. [4] V. G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method, SIAM J. Control, 4 (1966), 326-361.  doi: 10.1137/0304027. [5] V. G. Boltyansky, Mathematical Methods of Optimal Control, Holt, Rinehart and Winston, Inc., 1971. [6] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. [7] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, Vol. 43, Springer-Verlag, Berlin, 2004. [8] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [9] A. E. Bryson Jr. and Y. C. Ho, Applied Optimal Control, Revised Printing, Hemisphere Publishing Company, New York, 1975. [10] C. Byrnes and H. Frankowska, Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton–Jacobi–Bellman et de Riccati, C. R. Acad. Sci. Paris, 315 (1992), 427-431. [11] C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control, in: Systems, Models and Feedback: Theory and Applications, (A. Isidori and T. J. Tarn, eds.), Birkhäuser, (1992), 211–227. [12] I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Vol. 67, Springer-Verlag, Berlin-New York, 1972. [13] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979. [14] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley–Interscience, 1972. [15] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. Optim. Theory Appl., 114 (2002), 609-637.  doi: 10.1023/A:1016027113579. [16] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079.  doi: 10.1137/060665294. [17] U. Ledzewicz and H. Schättler, Combination of antiangiogenic treatment with chemotherapy as a multi-input optimal control problem, Math. Methods in the Applied Sciences, publ. online. doi: 10.1002/mma.7977. [18] D. Liberzon, Calculus of Variations and Optimal Control, Princeton University Press, Princeton, 2012. [19] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [20] H. G. Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636.  doi: 10.1137/0311048. [21] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficient conditions for optimality, SIAM J. Control Optim., 39 (2000), 359-410.  doi: 10.1137/S0363012999322031. [22] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. [23] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, Vol. 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [24] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, Vol. 42, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. [25] H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L_2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657. [26] H. J. Sussmann and J. C. Willems, 300 years of optimal control: From the brachistochrone to the maximum principle, IEEE Control Systems, 17 (1997), 32-44.  doi: 10.1109/37.588098. [27] G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984. [28] G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P. [29] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 3 (1995), 41-54.
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