June  2015, 4(2): 221-232. doi: 10.3934/eect.2015.4.221

Optimal bounded controls problem for bilinear systems

1. 

MACS Team, Faculty of Sciences, University Moulay Ismail, Meknes, Morocco, Morocco

Received  April 2014 Revised  September 2014 Published  May 2015

The aim of this paper is to study the optimal control problem for finite dimensional bilinear systems with bounded controls. We characterize an optimal control that minimizes a quadratic cost functional using Pontryagin's minimum principle, we derive sufficient conditions of uniqueness from the fixed point theorem, and we develop an algorithm that allows to compute the optimal control and the associated states. Our approach is applied to a cancer treatment by chemotherapy in order to determine the optimal dose of a killing agent.
Citation: El Hassan Zerrik, Nihale El Boukhari. Optimal bounded controls problem for bilinear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 221-232. doi: 10.3934/eect.2015.4.221
References:
[1]

Z. Aganovic and Z. Gagic, The successive approximation procedure for finite-time optimal control of bilinear systems, IEEE Trans. Automat. Control, 39 (1994), 1932-1935. doi: 10.1109/9.317128.  Google Scholar

[2]

R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton New Jersey, 1957.  Google Scholar

[3]

R. W. Brockett, Lie theory and control systems defined on spheres, SIAM J. Appl. Math., 25 (1973), 213-225. doi: 10.1137/0125025.  Google Scholar

[4]

L. G. De Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J. Theoritical Medicine, 3 (2001), 79-100. Google Scholar

[5]

H. Hermes, On local and global controllability, SIAM J. Control Opt., 12 (1974), 252-261. doi: 10.1137/0312019.  Google Scholar

[6]

E. Hofer and B. Tibken, An iterative method for the finite-time bilinear quadratic control problem, J. Optim. Theory Applications, 57 (1988), 411-427. doi: 10.1007/BF02346161.  Google Scholar

[7]

V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997.  Google Scholar

[8]

R. E. Kalman, The theory of optimal control and the calculus of variations, in Mathematical Optimization Techniques (ed., R. Bellman), Univ. of California Press, 1963, 309-331.  Google Scholar

[9]

R. E. Kalman, Y. C. Ho and K. S. Narendra, Mathematical description of linear dynamical systems, SIAM J. Control, 1 (1963), 152-192.  Google Scholar

[10]

K. Kassara and A. Moustafid, Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method, Mathematical Biosciences, 231 (2011), 135-143. doi: 10.1016/j.mbs.2011.02.010.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-53393-8.  Google Scholar

[12]

J. Kučera, On accessibility of bilinear systems, Czechoslovak Math. J., 20 (1970), 160-168.  Google Scholar

[13]

C. Lobry, Contrôlabilité des systèmes non linéaires, SIAM J. Control Opt., 8 (1970), 573-605.  Google Scholar

[14]

R. R. Mohler and R. E. Rink, Multivariable bilinear system control, in Control Systems (ed., C. T. Leondes), Academic Press, New York, 2, 1966. Google Scholar

[15]

R. R. Mohler, Bilinear Control Processes, Academic, New York, 1973.  Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes, Wiley (Inter-science), New York, 1962.  Google Scholar

[17]

H. Ramezanpour, S. Setayeshi, H. Arabalibeik and A. Jajrami, An iterative procedure for optimal control of bilinear systems, IJICS, 2 (2012), 1-10. doi: 10.5121/ijics.2012.2101.  Google Scholar

[18]

H. J. Sussmann and V. Jurdjevic, Controllability of non linear systems, J. Diff. Equations, 12 (1972), 95-116. doi: 10.1016/0022-0396(72)90007-1.  Google Scholar

[19]

A. Swierniak and Z. Duda, Singularity of optimal control in some problems related to optimal chemotherapy, Math. Comput. Modelling, 19 (1994), 255-262. doi: 10.1016/0895-7177(94)90197-X.  Google Scholar

[20]

A. Swierniak, U. Ledzewics and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Appl. Math. Coput. Sci., 13 (2003), 357-368.  Google Scholar

show all references

References:
[1]

Z. Aganovic and Z. Gagic, The successive approximation procedure for finite-time optimal control of bilinear systems, IEEE Trans. Automat. Control, 39 (1994), 1932-1935. doi: 10.1109/9.317128.  Google Scholar

[2]

R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton New Jersey, 1957.  Google Scholar

[3]

R. W. Brockett, Lie theory and control systems defined on spheres, SIAM J. Appl. Math., 25 (1973), 213-225. doi: 10.1137/0125025.  Google Scholar

[4]

L. G. De Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J. Theoritical Medicine, 3 (2001), 79-100. Google Scholar

[5]

H. Hermes, On local and global controllability, SIAM J. Control Opt., 12 (1974), 252-261. doi: 10.1137/0312019.  Google Scholar

[6]

E. Hofer and B. Tibken, An iterative method for the finite-time bilinear quadratic control problem, J. Optim. Theory Applications, 57 (1988), 411-427. doi: 10.1007/BF02346161.  Google Scholar

[7]

V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997.  Google Scholar

[8]

R. E. Kalman, The theory of optimal control and the calculus of variations, in Mathematical Optimization Techniques (ed., R. Bellman), Univ. of California Press, 1963, 309-331.  Google Scholar

[9]

R. E. Kalman, Y. C. Ho and K. S. Narendra, Mathematical description of linear dynamical systems, SIAM J. Control, 1 (1963), 152-192.  Google Scholar

[10]

K. Kassara and A. Moustafid, Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method, Mathematical Biosciences, 231 (2011), 135-143. doi: 10.1016/j.mbs.2011.02.010.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-53393-8.  Google Scholar

[12]

J. Kučera, On accessibility of bilinear systems, Czechoslovak Math. J., 20 (1970), 160-168.  Google Scholar

[13]

C. Lobry, Contrôlabilité des systèmes non linéaires, SIAM J. Control Opt., 8 (1970), 573-605.  Google Scholar

[14]

R. R. Mohler and R. E. Rink, Multivariable bilinear system control, in Control Systems (ed., C. T. Leondes), Academic Press, New York, 2, 1966. Google Scholar

[15]

R. R. Mohler, Bilinear Control Processes, Academic, New York, 1973.  Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes, Wiley (Inter-science), New York, 1962.  Google Scholar

[17]

H. Ramezanpour, S. Setayeshi, H. Arabalibeik and A. Jajrami, An iterative procedure for optimal control of bilinear systems, IJICS, 2 (2012), 1-10. doi: 10.5121/ijics.2012.2101.  Google Scholar

[18]

H. J. Sussmann and V. Jurdjevic, Controllability of non linear systems, J. Diff. Equations, 12 (1972), 95-116. doi: 10.1016/0022-0396(72)90007-1.  Google Scholar

[19]

A. Swierniak and Z. Duda, Singularity of optimal control in some problems related to optimal chemotherapy, Math. Comput. Modelling, 19 (1994), 255-262. doi: 10.1016/0895-7177(94)90197-X.  Google Scholar

[20]

A. Swierniak, U. Ledzewics and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Appl. Math. Coput. Sci., 13 (2003), 357-368.  Google Scholar

[1]

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

[2]

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

[3]

Luis A. Fernández, Cecilia Pola. Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1563-1588. doi: 10.3934/dcdsb.2014.19.1563

[4]

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

[5]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95

[6]

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

[7]

Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083

[8]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[9]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[10]

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

[11]

Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279

[12]

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

[13]

Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Corrigendum to "Singularity of controls in a simple model of acquired chemotherapy resistance". Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 2021-2021. doi: 10.3934/dcdsb.2020105

[14]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3641-3657. doi: 10.3934/dcdss.2020434

[15]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[16]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[17]

Alexis B. Cook, Daniel R. Ziazadeh, Jianfeng Lu, Trachette L. Jackson. An integrated cellular and sub-cellular model of cancer chemotherapy and therapies that target cell survival. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1219-1235. doi: 10.3934/mbe.2015.12.1219

[18]

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

[19]

Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021140

[20]

Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (124)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]