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 & Continuous Dynamical Systems - A, 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, (1999), 11. Google Scholar

[2]

A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory,, SIAM J. Control and Optimization, 41 (2002), 991. doi: 10.1137/S036301290138866X. Google Scholar

[3]

M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals,, Numerical Linear Algebra, 2 (2012), 511. doi: 10.3934/naco.2012.2.511. Google Scholar

[4]

V. G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method,, SIAM J. Control, 4 (1966), 326. doi: 10.1137/0304027. Google Scholar

[5]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Springer Verlag, (2003). Google Scholar

[6]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds,, Mathématiques & Applications, (2004). Google Scholar

[7]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007). Google Scholar

[8]

A. Dmitruk, Jacobi type conditions for singular extremals,, Control and Cybernetics, 37 (2008), 285. Google Scholar

[9]

U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control and Optimization, 41 (2003), 1843. doi: 10.1137/S0363012901399271. Google Scholar

[10]

U. Felgenhauer, Lipschitz stability of broken extremals in bang-bang control problems,, in: Large-Scale Scientific Computing (Sozopol 2007), (2008), 317. doi: 10.1007/978-3-540-78827-0_35. Google Scholar

[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. Google Scholar

[12]

H. Gardner-Moyer, Sufficient conditions for a strong minimum in singular control problems,, SIAM J. Control, 11 (1973), 620. doi: 10.1137/0311048. Google Scholar

[13]

M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120. Google Scholar

[14]

U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations,, Acta Applicandae Mathematicae, 135 (2015), 191. doi: 10.1007/s10440-014-9952-6. Google Scholar

[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, 19 (2014), 2657. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar

[16]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183. doi: 10.1142/S0218339002000597. Google Scholar

[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. doi: 10.3934/mbe.2013.10.803. Google Scholar

[18]

H. Maurer and N. Osmolovskii, Quadratic sufficient optimality conditions for bang-bang control problems,, Control and Cybernetics, 33 (2003), 555. Google Scholar

[19]

J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals,, J. of Mathematical Analysis and Applications, 269 (2002), 98. doi: 10.1016/S0022-247X(02)00008-2. Google Scholar

[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, (2012). doi: 10.1137/1.9781611972368. Google Scholar

[21]

B. Piccoli and H. Sussmann, Regular synthesis and sufficient conditions for optimality,, SIAM J. on Control and Optimization, 39 (2000), 359. doi: 10.1137/S0363012999322031. Google Scholar

[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. doi: 10.1137/090771405. Google Scholar

[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. doi: 10.1109/CDC.2006.376760. Google Scholar

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964). Google Scholar

[25]

A. Sarychev, Morse index and sufficient optimality conditions for bang-bang Pontryagin extremals,, in: System Modeling and Optimization, 180 (1992), 440. doi: 10.1007/BFb0113311. Google Scholar

[26]

A. Sarychev, First and second order sufficient optimality conditions for bang-bang controls,, SIAM J. on Control and Optimization, 35 (1997), 315. doi: 10.1137/S0363012993246191. Google Scholar

[27]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar

[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. doi: 10.1007/BF02459681. Google Scholar

[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. doi: 10.1137/0325025. Google Scholar

[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. doi: 10.1137/0325048. Google Scholar

[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. doi: 10.1137/0325062. Google Scholar

[32]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51. Google Scholar

[33]

A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41. doi: 10.1142/S0218339095000058. Google Scholar

[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. Google Scholar

[35]

A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell Proliferation, 29 (1996), 117. doi: 10.1046/j.1365-2184.1996.00995.x. Google Scholar

[36]

T. E. Wheldon, Mathematical Models in Cancer Research,, Boston-Philadelphia: Hilger Publishing, (1988). Google Scholar

show all references

References:
[1]

A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control,, in Differential Geometry and Control, (1999), 11. Google Scholar

[2]

A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory,, SIAM J. Control and Optimization, 41 (2002), 991. doi: 10.1137/S036301290138866X. Google Scholar

[3]

M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals,, Numerical Linear Algebra, 2 (2012), 511. doi: 10.3934/naco.2012.2.511. Google Scholar

[4]

V. G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method,, SIAM J. Control, 4 (1966), 326. doi: 10.1137/0304027. Google Scholar

[5]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Springer Verlag, (2003). Google Scholar

[6]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds,, Mathématiques & Applications, (2004). Google Scholar

[7]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007). Google Scholar

[8]

A. Dmitruk, Jacobi type conditions for singular extremals,, Control and Cybernetics, 37 (2008), 285. Google Scholar

[9]

U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control and Optimization, 41 (2003), 1843. doi: 10.1137/S0363012901399271. Google Scholar

[10]

U. Felgenhauer, Lipschitz stability of broken extremals in bang-bang control problems,, in: Large-Scale Scientific Computing (Sozopol 2007), (2008), 317. doi: 10.1007/978-3-540-78827-0_35. Google Scholar

[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. Google Scholar

[12]

H. Gardner-Moyer, Sufficient conditions for a strong minimum in singular control problems,, SIAM J. Control, 11 (1973), 620. doi: 10.1137/0311048. Google Scholar

[13]

M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120. Google Scholar

[14]

U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations,, Acta Applicandae Mathematicae, 135 (2015), 191. doi: 10.1007/s10440-014-9952-6. Google Scholar

[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, 19 (2014), 2657. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar

[16]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183. doi: 10.1142/S0218339002000597. Google Scholar

[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. doi: 10.3934/mbe.2013.10.803. Google Scholar

[18]

H. Maurer and N. Osmolovskii, Quadratic sufficient optimality conditions for bang-bang control problems,, Control and Cybernetics, 33 (2003), 555. Google Scholar

[19]

J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals,, J. of Mathematical Analysis and Applications, 269 (2002), 98. doi: 10.1016/S0022-247X(02)00008-2. Google Scholar

[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, (2012). doi: 10.1137/1.9781611972368. Google Scholar

[21]

B. Piccoli and H. Sussmann, Regular synthesis and sufficient conditions for optimality,, SIAM J. on Control and Optimization, 39 (2000), 359. doi: 10.1137/S0363012999322031. Google Scholar

[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. doi: 10.1137/090771405. Google Scholar

[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. doi: 10.1109/CDC.2006.376760. Google Scholar

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964). Google Scholar

[25]

A. Sarychev, Morse index and sufficient optimality conditions for bang-bang Pontryagin extremals,, in: System Modeling and Optimization, 180 (1992), 440. doi: 10.1007/BFb0113311. Google Scholar

[26]

A. Sarychev, First and second order sufficient optimality conditions for bang-bang controls,, SIAM J. on Control and Optimization, 35 (1997), 315. doi: 10.1137/S0363012993246191. Google Scholar

[27]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar

[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. doi: 10.1007/BF02459681. Google Scholar

[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. doi: 10.1137/0325025. Google Scholar

[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. doi: 10.1137/0325048. Google Scholar

[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. doi: 10.1137/0325062. Google Scholar

[32]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51. Google Scholar

[33]

A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41. doi: 10.1142/S0218339095000058. Google Scholar

[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. Google Scholar

[35]

A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell Proliferation, 29 (1996), 117. doi: 10.1046/j.1365-2184.1996.00995.x. Google Scholar

[36]

T. E. Wheldon, Mathematical Models in Cancer Research,, Boston-Philadelphia: Hilger Publishing, (1988). Google Scholar

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