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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 and 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

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