American Institute of Mathematical Sciences

Advanced
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

Heinz Schättler 1, , Urszula Ledzewicz 2, and  Helmut Maurer 3,

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.
Keywords: sufficient conditions for local optimality, cancer chemotherapy., Optimal control, Riccati equation.
Mathematics Subject Classification: Primary: 49K15; Secondary: 49L99, 93C1.
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.

show all references

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.

[1]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129
[2]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017
[3]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial and Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329
[4]

Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803
[5]

Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040
[6]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[7]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[8]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
[9]

J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431
[10]

Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100
[11]

Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks and Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007
[12]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[13]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[14]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[15]

Vladimir Srochko, Vladimir Antonik, Elena Aksenyushkina. Sufficient optimality conditions for extremal controls based on functional increment formulas. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013
[16]

Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114
[17]

Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561
[18]

Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544
[19]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[20]

Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (139)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy