# American Institute of Mathematical Sciences

September  2015, 35(9): 4611-4638. doi: 10.3934/dcds.2015.35.4611

## Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems

 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

Received  May 2014 Revised  October 2014 Published  April 2015

Optimal control problems with fixed terminal time are considered for multi-input bilinear systems with the control set given by a compact interval and the objective function affine in the controls. Systems of this type have been widely used in the modeling of cell-cycle specific cancer chemotherapy over a prescribed therapy horizon for both homogeneous and heterogeneous tumor populations. Necessary conditions for optimality lead to concatenations of bang and singular controls as prime candidates for optimality. In this paper, the method of characteristics will be formulated as a general procedure to embed such a controlled reference extremal into a field of broken extremals. Sufficient conditions for the strong local optimality of a controlled reference bang-bang trajectory will be formulated in terms of solutions to associated sensitivity equations. These results will be applied to a model for cell cycle specific cancer chemotherapy with cytotoxic and cytostatic agents.
Citation: Heinz Schättler, Urszula Ledzewicz. Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4611-4638. doi: 10.3934/dcds.2015.35.4611
##### References:
 [1] A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control, in Differential Geometry and Control, (G. Ferreyra, et al., Eds.), American Mathematical Society, (1999), 11-22. [2] A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014. doi: 10.1137/S036301290138866X. [3] M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals, Numerical Linear Algebra, Control and Optimization, 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511. [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] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer Verlag, Series: Mathematics and Applications, Vol. 40, 2003. [6] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, 43, Springer Verlag, Paris, 2004. [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, Mo, 2007. [8] A. Dmitruk, Jacobi type conditions for singular extremals, Control and Cybernetics, 37 (2008), 285-306. [9] U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control and Optimization, 41 (2003), 1843-1867. doi: 10.1137/S0363012901399271. [10] U. Felgenhauer, Lipschitz stability of broken extremals in bang-bang control problems, in: Large-Scale Scientific Computing (Sozopol 2007), (I. Lirkov et al., Eds.), Lecture Notes in Computer Science, vol. 4818, Springer (2008), 317-325. doi: 10.1007/978-3-540-78827-0_35. [11] U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, Control and Cybernetics, 38 (2009), 1305-1325. [12] H. Gardner-Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636. doi: 10.1137/0311048. [13] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120-130. [14] U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Mathematicae, 135 (2015), 191-207. doi: 10.1007/s10440-014-9952-6. [15] U. Ledzewicz, H. Maurer and H. Schättler, Sufficient conditions for strong local optimality in optimal control problem with $L_{2}$-type objectives and control constraints, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. [16] 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. [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] H. Maurer and N. Osmolovskii, Quadratic sufficient optimality conditions for bang-bang control problems, Control and Cybernetics, 33 (2003), 555-584. [19] J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals, J. of Mathematical Analysis and Applications, 269 (2002), 98-128. doi: 10.1016/S0022-247X(02)00008-2. [20] N. 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, Philadelphia, USA, 2012. doi: 10.1137/1.9781611972368. [21] B. Piccoli and H. Sussmann, Regular synthesis and sufficient conditions for optimality, SIAM J. on Control and Optimization, 39 (2000), 359-410. doi: 10.1137/S0363012999322031. [22] L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control and Optimization, 49 (2011), 140-161. doi: 10.1137/090771405. [23] L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-bang trajectory, in Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 6624-6629. doi: 10.1109/CDC.2006.376760. [24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [25] A. Sarychev, Morse index and sufficient optimality conditions for bang-bang Pontryagin extremals, in: System Modeling and Optimization, Lecture Notes in Control and Information Sciences, 180 (1992), 440-448. doi: 10.1007/BFb0113311. [26] A. Sarychev, First and second order sufficient optimality conditions for bang-bang controls, SIAM J. on Control and Optimization, 35 (1997), 315-340. doi: 10.1137/S0363012993246191. [27] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [28] H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. doi: 10.1007/BF02459681. [29] H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the $C^{\infty}$ nonsingular case, SIAM J. Control Optimization, 25 (1987), 433-465. doi: 10.1137/0325025. [30] H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optimization, 25 (1987), 868-904. doi: 10.1137/0325048. [31] H. J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optimization, 25 (1987), 1145-1162. doi: 10.1137/0325062. [32] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, IMACS Ann. Comput. Appl. Math., Baltzer, Basel, 5 (1989), 51-53. [33] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058. [34] 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. [35] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139. doi: 10.1046/j.1365-2184.1996.00995.x. [36] T. E. Wheldon, Mathematical Models in Cancer Research, Boston-Philadelphia: Hilger Publishing, 1988.

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##### References:
 [1] A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control, in Differential Geometry and Control, (G. Ferreyra, et al., Eds.), American Mathematical Society, (1999), 11-22. [2] A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014. doi: 10.1137/S036301290138866X. [3] M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals, Numerical Linear Algebra, Control and Optimization, 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511. [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] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer Verlag, Series: Mathematics and Applications, Vol. 40, 2003. [6] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, 43, Springer Verlag, Paris, 2004. [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, Mo, 2007. [8] A. Dmitruk, Jacobi type conditions for singular extremals, Control and Cybernetics, 37 (2008), 285-306. [9] U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control and Optimization, 41 (2003), 1843-1867. doi: 10.1137/S0363012901399271. [10] U. Felgenhauer, Lipschitz stability of broken extremals in bang-bang control problems, in: Large-Scale Scientific Computing (Sozopol 2007), (I. Lirkov et al., Eds.), Lecture Notes in Computer Science, vol. 4818, Springer (2008), 317-325. doi: 10.1007/978-3-540-78827-0_35. [11] U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, Control and Cybernetics, 38 (2009), 1305-1325. [12] H. Gardner-Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636. doi: 10.1137/0311048. [13] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120-130. [14] U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Mathematicae, 135 (2015), 191-207. doi: 10.1007/s10440-014-9952-6. [15] U. Ledzewicz, H. Maurer and H. Schättler, Sufficient conditions for strong local optimality in optimal control problem with $L_{2}$-type objectives and control constraints, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. [16] 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. [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] H. Maurer and N. Osmolovskii, Quadratic sufficient optimality conditions for bang-bang control problems, Control and Cybernetics, 33 (2003), 555-584. [19] J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals, J. of Mathematical Analysis and Applications, 269 (2002), 98-128. doi: 10.1016/S0022-247X(02)00008-2. [20] N. 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, Philadelphia, USA, 2012. doi: 10.1137/1.9781611972368. [21] B. Piccoli and H. Sussmann, Regular synthesis and sufficient conditions for optimality, SIAM J. on Control and Optimization, 39 (2000), 359-410. doi: 10.1137/S0363012999322031. [22] L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control and Optimization, 49 (2011), 140-161. doi: 10.1137/090771405. [23] L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-bang trajectory, in Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 6624-6629. doi: 10.1109/CDC.2006.376760. [24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [25] A. Sarychev, Morse index and sufficient optimality conditions for bang-bang Pontryagin extremals, in: System Modeling and Optimization, Lecture Notes in Control and Information Sciences, 180 (1992), 440-448. doi: 10.1007/BFb0113311. [26] A. Sarychev, First and second order sufficient optimality conditions for bang-bang controls, SIAM J. on Control and Optimization, 35 (1997), 315-340. doi: 10.1137/S0363012993246191. [27] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [28] H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. doi: 10.1007/BF02459681. [29] H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the $C^{\infty}$ nonsingular case, SIAM J. Control Optimization, 25 (1987), 433-465. doi: 10.1137/0325025. [30] H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optimization, 25 (1987), 868-904. doi: 10.1137/0325048. [31] H. J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optimization, 25 (1987), 1145-1162. doi: 10.1137/0325062. [32] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, IMACS Ann. Comput. Appl. Math., Baltzer, Basel, 5 (1989), 51-53. [33] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058. [34] 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. [35] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139. doi: 10.1046/j.1365-2184.1996.00995.x. [36] T. E. Wheldon, Mathematical Models in Cancer Research, Boston-Philadelphia: Hilger Publishing, 1988.
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