# American Institute of Mathematical Sciences

October  2014, 19(8): 2657-2679. doi: 10.3934/dcdsb.2014.19.2657

## Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

 1 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899 2 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653 3 Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, D-48149 Münster, Germany

Received  November 2013 Revised  January 2014 Published  August 2014

We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.
Citation: Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657
##### References:
 [1] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Series: Mathematics and Applications, Vol. 40, Springer Verlag, 2003. [2] J. V. Breakwell, J. L. Speyer and A. E. Bryson, jr., Optimization and control of nonlinear systems using the second variation, SIAM J. Control, 1 (1963), 193-223. doi: 10.1137/0301011. [3] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [4] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. [5] C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and realtime control, J. of Computational and Applied Mathematics, 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8. [6] C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in Online Optimization of Large Scale Systems (eds. M. Grötschel, S.O. Krumke, J. Rambau), Springer-Verlag, Berlin, 2001, 3-16. [7] C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods, in Online Optimization of Large Scale Systems (eds. M. Grötschel, S.O. Krumke, J. Rambau), Springer-Verlag, Berlin, 2001, 57-68. [8] N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control, Trans. of the American Mathematical Society, 348 (1996), 3133-3153. doi: 10.1090/S0002-9947-96-01577-2. [9] J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations, Manuscripta Matematicae, 68 (1990), 209-214. doi: 10.1007/BF02568760. [10] R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993. [11] H. K. Khalil, Nonlinear Systems, 3rd. ed., Prentice Hall, 2002. [12] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130. [13] H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, Springer Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69884-2. [14] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. doi: 10.1023/A:1016027113579. [15] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. [16] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197. doi: 10.1142/S0218339014400014. [17] U. Ledzewicz, H. Schättler, M. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering (MBE), 10 (2013), 803-819. doi: 10.3934/mbe.2013.10.803. [18] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical & Computational Biology, 2007. [19] K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283. doi: 10.1007/BF00248267. [20] H. Maurer, C. Büskens, J. H. Kim and Y. Kaja, Optimization techniques for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control, Applications and Methods, 26 (2005), 129-156. doi: 10.1002/oca.756. [21] N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. [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, Perturbation feedback control: A geometric interpretation, Numerical Algebra, Control and Optimization, 2 (2012), 631-654. doi: 10.3934/naco.2012.2.631. [24] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [25] H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, in Proc. of the 51st IEEE Conference on Decision and Control (Maui, Hawaii), IEEE, 2012, 7691-7696. [26] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, IMACS Ann. Comput. Appl. Math., 5 (1989), 51-53. [27] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058. [28] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. [29] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2001), 375-386. doi: 10.1016/S0362-546X(01)00184-5. [30] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell prolif., 29 (1996), 117-139. doi: 10.1046/j.1365-2184.1996.00995.x. [31] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

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##### References:
 [1] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Series: Mathematics and Applications, Vol. 40, Springer Verlag, 2003. [2] J. V. Breakwell, J. L. Speyer and A. E. Bryson, jr., Optimization and control of nonlinear systems using the second variation, SIAM J. Control, 1 (1963), 193-223. doi: 10.1137/0301011. [3] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [4] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. [5] C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and realtime control, J. of Computational and Applied Mathematics, 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8. [6] C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in Online Optimization of Large Scale Systems (eds. M. Grötschel, S.O. Krumke, J. Rambau), Springer-Verlag, Berlin, 2001, 3-16. [7] C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods, in Online Optimization of Large Scale Systems (eds. M. Grötschel, S.O. Krumke, J. Rambau), Springer-Verlag, Berlin, 2001, 57-68. [8] N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control, Trans. of the American Mathematical Society, 348 (1996), 3133-3153. doi: 10.1090/S0002-9947-96-01577-2. [9] J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations, Manuscripta Matematicae, 68 (1990), 209-214. doi: 10.1007/BF02568760. [10] R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993. [11] H. K. Khalil, Nonlinear Systems, 3rd. ed., Prentice Hall, 2002. [12] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130. [13] H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, Springer Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69884-2. [14] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. doi: 10.1023/A:1016027113579. [15] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. [16] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197. doi: 10.1142/S0218339014400014. [17] U. Ledzewicz, H. Schättler, M. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering (MBE), 10 (2013), 803-819. doi: 10.3934/mbe.2013.10.803. [18] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical & Computational Biology, 2007. [19] K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283. doi: 10.1007/BF00248267. [20] H. Maurer, C. Büskens, J. H. Kim and Y. Kaja, Optimization techniques for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control, Applications and Methods, 26 (2005), 129-156. doi: 10.1002/oca.756. [21] N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. [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, Perturbation feedback control: A geometric interpretation, Numerical Algebra, Control and Optimization, 2 (2012), 631-654. doi: 10.3934/naco.2012.2.631. [24] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [25] H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, in Proc. of the 51st IEEE Conference on Decision and Control (Maui, Hawaii), IEEE, 2012, 7691-7696. [26] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, IMACS Ann. Comput. Appl. Math., 5 (1989), 51-53. [27] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058. [28] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. [29] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2001), 375-386. doi: 10.1016/S0362-546X(01)00184-5. [30] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell prolif., 29 (1996), 117-139. doi: 10.1046/j.1365-2184.1996.00995.x. [31] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.
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