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 & 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, (2003). Google Scholar

[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. doi: 10.1137/0301011. Google Scholar

[3]

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

[4]

A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975). Google Scholar

[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. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[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, (2001), 3. Google Scholar

[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, (2001), 57. Google Scholar

[8]

N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control,, Trans. of the American Mathematical Society, 348 (1996), 3133. doi: 10.1090/S0002-9947-96-01577-2. Google Scholar

[9]

J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations,, Manuscripta Matematicae, 68 (1990), 209. doi: 10.1007/BF02568760. Google Scholar

[10]

R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar

[11]

H. K. Khalil, Nonlinear Systems,, 3rd. ed., (2002). Google Scholar

[12]

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

[13]

H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie,, Springer Verlag, (1985). doi: 10.1007/978-3-642-69884-2. Google Scholar

[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. doi: 10.1023/A:1016027113579. Google Scholar

[15]

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

[16]

U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177. doi: 10.1142/S0218339014400014. 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]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical & Computational Biology, (2007). Google Scholar

[19]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253. doi: 10.1007/BF00248267. Google Scholar

[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, 26 (2005), 129. doi: 10.1002/oca.756. Google Scholar

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

[22]

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

[23]

H. Schättler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Numerical Algebra, 2 (2012), 631. doi: 10.3934/naco.2012.2.631. Google Scholar

[24]

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

[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, (2012), 7691. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar

[30]

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

[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. doi: 10.1007/s10107-004-0559-y. Google Scholar

show all references

References:
[1]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Series: Mathematics and Applications, (2003). Google Scholar

[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. doi: 10.1137/0301011. Google Scholar

[3]

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

[4]

A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975). Google Scholar

[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. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[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, (2001), 3. Google Scholar

[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, (2001), 57. Google Scholar

[8]

N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control,, Trans. of the American Mathematical Society, 348 (1996), 3133. doi: 10.1090/S0002-9947-96-01577-2. Google Scholar

[9]

J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations,, Manuscripta Matematicae, 68 (1990), 209. doi: 10.1007/BF02568760. Google Scholar

[10]

R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar

[11]

H. K. Khalil, Nonlinear Systems,, 3rd. ed., (2002). Google Scholar

[12]

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

[13]

H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie,, Springer Verlag, (1985). doi: 10.1007/978-3-642-69884-2. Google Scholar

[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. doi: 10.1023/A:1016027113579. Google Scholar

[15]

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

[16]

U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177. doi: 10.1142/S0218339014400014. 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]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical & Computational Biology, (2007). Google Scholar

[19]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253. doi: 10.1007/BF00248267. Google Scholar

[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, 26 (2005), 129. doi: 10.1002/oca.756. Google Scholar

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

[22]

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

[23]

H. Schättler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Numerical Algebra, 2 (2012), 631. doi: 10.3934/naco.2012.2.631. Google Scholar

[24]

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

[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, (2012), 7691. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar

[30]

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

[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. doi: 10.1007/s10107-004-0559-y. Google Scholar

[1]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013
[16]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[17]

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
[18]

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
[19]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027
[20]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences