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

Optimal bounded controls problem for bilinear systems

El Hassan Zerrik 1, and  Nihale El Boukhari 1,

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.
Keywords: Bilinear systems, cancer chemotherapy., bounded controls, Pontryagin principle.
Mathematics Subject Classification: Primary: 49K15, 49K30; Secondary: 92C5.
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

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