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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

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