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Optimal bounded controls problem for bilinear systems
1. | MACS Team, Faculty of Sciences, University Moulay Ismail, Meknes, Morocco, Morocco |
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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