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Model for hepatitis C virus transmissions
Parametrization of the attainable set for a nonlinear control model of a biochemical process
1. | Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204 |
2. | Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992 |
3. | Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona |
These analytical results generalize the earlier findings, which were obtained for a trilinear reaction rate (which corresponds to the law of mass action) and reported in [18,19], to the case of a general rate of reaction. These results allow to reduce the problem of constructing the optimal control to a straightforward constrained finite dimensional optimization problem.
References:
[1] |
J.-P. Aubin and A. Cellina, "Differential Inclusions: Set-Valued Maps and Viability Theory," Springer, Berlin-New York, 1984.
doi: 10.1007/978-3-642-69512-4. |
[2] |
A. D. Bojarski, J. Rojas and T. Zhelev, Modelling and sensitivity analysis of ATAD, Computers & Chemical Engineering, 34 (2010), 802-811. |
[3] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer-Verlag, Berlin-Heidelberg-New York, 2003. |
[4] |
G. Bromström and H. Drange, On the mathematical formulation and parameter estimation of the norwegian sea plankton system, Sarsia, 85 (2000), 211-225. |
[5] |
D. Brune, Optimal control of the complete-mix activated sludge process, Environmental Technology Letters, 6 (1985), 467-476.
doi: 10.1080/09593338509384365. |
[6] |
M. Burke, M. Chapwanya, K. Doherty, I. Hewitt, A. Korobeinikov, M. Meere, S. McCarthy, M. O'Brien, V. T. N. Tuoi, H. Winstanley and T. Zhelev, Modeling of autothermal thermophylic aerobic digestion, Mathematics-in-Industry Case Studies Journal, 2 (2010), 34-63. |
[7] |
S. Busenberg, S. Kumar, P. Austin and G. Wake, The Dynamics of a model of a plankton-nutrient interaction, Bulletin of Mathematical Biology, 52 (1990), 677-696. |
[8] |
F. L. Chernous'ko and V. B. Kolmanovskii, Computational and approximate methods of optimal control, Journal of Mathematical Sciences, 12 (1979), 310-353.
doi: 10.1007/BF01098370. |
[9] |
F. L. Chernous'ko, "Ellipsoidal state Estimation for Dynamical Systems," CRS Press, Boca Raton, Florida, 1994.
doi: 10.1016/j.na.2005.01.009. |
[10] |
B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967. |
[11] |
A. V. Dmitruk, A generalized estimate on the number of zeroes for solutions of a class of linear differential equations, SIAM Journal of Control and Optimization, 30 (1992), 1087-1091.
doi: 10.1137/0330057. |
[12] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM Journal on Applied Mathematics, 67 (2007), 337-353.
doi: 10.1137/060654876. |
[13] |
M. Graells, J. Rojas and T. Zhelev, Energy efficiency optimization of wastewater treatment. Study of ATAD, Computer Aided Chemical Engineering, 28 (2010), 967-972. |
[14] |
E. N. Khailov and E. V. Grigorieva, On the attainability set for a nonlinear system in the plane, Moscow University. Computational Mathematics and Cybernetics, (2001), 27-32. |
[15] |
E. V. Grigorieva and E. N. Khailov, A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good, Dynamical Systems and Differential Equations, (2003), 359-364. |
[16] |
E. N. Khailov and E. V. Grigorieva, Description of the attainability set of a nonlinear controlled system in the plane, Moscow University. Computational Mathematics and Cybernetics, (2005), 23-28. |
[17] |
E. V. Grigorieva and E. N. Khailov, Attainable set of a nonlinear controlled microeconomic model, Journal of Dynamical and Control Systems, 11 (2005), 157-176.
doi: 10.1007/s10883-005-4168-8. |
[18] |
E. V. Grigorieva, N. V. Bondarenko, E. N. Khailov and A. Korobeinikov, Three-dimensional nonlinear control model of wastewater biotreatment, Neural, Parallel, and Scientific Computations, 20 (2012), 23-35. |
[19] |
E. V. Grigorieva, N. V. Bondarenko, E. N. Khailov and A. Korobeinikov, Finite-dimensional methods for optimal control of autothermal thermophilic aerobic digestion, in "Industrial Waste" (eds. K.-Y. Show and X. Guo), InTech, Croatia, (2012), 91-120.
doi: 10.5772/36237. |
[20] |
T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different forms of predator-prey interaction, Journal of Theoretical Biology, 227 (2004), 349-358.
doi: 10.1016/j.jtbi.2003.09.020. |
[21] |
V. I. Gurman and E. A. Trushkova, Estimates for attainability sets of control systems, Differential Equations, 45 (2009), 1636-1644.
doi: 10.1134/S0012266109110093. |
[22] |
Kh. G. Guseinov, A. N. Moiseev and V. N. Ushakov, On the approximation of reachable domains of control systems, Journal of Applied Mathematics and Mechanics, 62 (1998), 169-175.
doi: 10.1016/S0021-8928(98)00022-7. |
[23] |
P. Hartman, "Ordinary Differential Equations," Jorn Wiley & Sons, New York-London-Sydney, 1964. |
[24] |
A. N. Kolmogorov, Sulla teoria di Volterra della lotta per l'esistenza, Giorn. Ist. Ital. Attuari, 7 (1936), 74-80. |
[25] |
V. A. Komarov, Estimates of the attainable set for differential inclusions, Mathematical Notes, 37 (1985), 916-925. |
[26] |
A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Mathematical Medicine and Biology, 26 (2009), 309-321.
doi: 10.1093/imammb/dqp009. |
[27] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Mathematical Medicine and Biology, 26 (2009), 225-239. |
[28] |
A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete and Continuous Dynamical System. Ser. B, 14 (2010), 1095-1103.
doi: 10.3934/dcdsb.2010.14.1095. |
[29] |
M. A. Krasnosel'skii, "The Operator of Translation along the Trajectories of Differential Equations," American Mathematical Society, Providence, RI, 1968. |
[30] |
A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control," Birkhäuser, Boston, 1997. |
[31] |
U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.
doi: 10.1093/imammb/dqp012. |
[32] |
U. Ledzewicz, E. Kashdan and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.
doi: 10.3934/mbe.2011.8.307. |
[33] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, Journal of Mathematical Biology, 64 (2012), 557-577.
doi: 10.1007/s00285-011-0424-6. |
[34] |
E. B. Lee and L. Markus, "Foundations of Optimal Control Theory," Jorn Wiley & Sons, New York, 1967. |
[35] |
M. S. Nikol'skii, Approximation of the attainability set for a controlled process, Mat. Zametki, 41 (1987), 71-76, 121. |
[36] |
N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. Part 2: proofs, variational derivatives and representations, Control and Cybernetics, 36 (2007), 5-45. |
[37] |
A. I. Ovseevich and F. L. Chernous'ko, Two-sided estimates on the attainability domains of controlled systems, Journal of Applied Mathematics and Mechanics, 46 (1982), 590-595.
doi: 10.1016/0021-8928(82)90005-3. |
[38] |
A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mathematical Notes, 27 (1980), 429-437, 494. |
[39] |
A. I. Panasyuk, Equations of attainable set dynamics. I. Integral funnel equation, Journal of Optimization Theory and Applications, 64 (1990), 349-366.
doi: 10.1007/BF00939453. |
[40] |
A. I. Panasyuk, Equations of attainable set dynamics. II. Partial differential equations, Journal of Optimization Theory and Applications, 64 (1990), 367-377.
doi: 10.1007/BF00939454. |
[41] |
T. Partasarathy, "On Global Univalence Theorems," Springer-Verlag, Berlin-Heidelberg-New York, 1983. |
[42] |
A. Sard, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 48 (1942), 883-890.
doi: 10.1090/S0002-9904-1942-07811-6. |
[43] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control. Theory, Methods and Examples," Springer, New York-Heidelberg-Dordrecht-London, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[44] |
L. Schwartz, "Analyse Mathematique 1," Hermann, Paris, 1967. |
[45] |
G. V. Shevchenko, Numerical method for solving a nonlinear time-optimal control problem with additive control, Computational Mathematics and Mathematical Physics, 47 (2007), 1768-1778.
doi: 10.1134/S0965542507110048. |
[46] |
G. V. Shevchenko, Numerical solution of a nonlinear time optimal control problem, Computational Mathematics and Mathematical Physics, 51 (2011), 537-549.
doi: 10.1134/S0965542511040154. |
[47] |
S. Ichiraku, A note on global implicit function theorems, IEEE Transactions on Curcuits and Systems, 32 (1985), 503-505.
doi: 10.1109/TCS.1985.1085729. |
[48] |
D. Szolnoki, Set-oriented methods for computing reachable sets and control sets, Discrete and Continuous Dynamical Systems. Ser. B, 3 (2003), 361-382.
doi: 10.3934/dcdsb.2003.3.361. |
[49] |
Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model, SIAM Journal on Applied Mathematics, 61 (2000), 803-833.
doi: 10.1137/S0036139998347834. |
[50] |
H. R. Thieme and Z. Feng, Endemic models with arbitrarily distributed periods of infection II: Fast disease dynamics and permanent recovery, SIAM Journal on Applied Mathematics, 61 (2000), 983-1012.
doi: 10.1137/S0036139998347846. |
[51] |
A. N. Tikhonov, A. B. Vasil'eva and A. G. Sveshnikov, "Differential Equations," Springer-Verlag, Berlin-Heidelberg-New York, 1985.
doi: 10.1007/978-3-642-82175-2. |
[52] |
A. I. Tyatyushkin and O. V. Morzhin, Constructive methods of control optimization in nonlinear systems, Automation and Remote Control, 70 (2009), 772-786.
doi: 10.1134/S0005117909050063. |
[53] |
A. I. Tyatyushkin and O. V. Morzhin, Numerical investigation of attainability sets of nonlinear controlled differential systems, Automation and Remote Control, 72 (2011), 1291-1300.
doi: 10.1134/S0005117911060178. |
[54] |
P. Varaiya and A. B. Kurzhanski, Ellipsoidal methods for dynamics and control. Part I, Journal of Mathematical Sciences, 139 (2006), 6863-6901.
doi: 10.1007/s10958-006-0397-y. |
[55] |
O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov, "Elementary Topology: Problem Textbook," AMS, 2008. |
show all references
References:
[1] |
J.-P. Aubin and A. Cellina, "Differential Inclusions: Set-Valued Maps and Viability Theory," Springer, Berlin-New York, 1984.
doi: 10.1007/978-3-642-69512-4. |
[2] |
A. D. Bojarski, J. Rojas and T. Zhelev, Modelling and sensitivity analysis of ATAD, Computers & Chemical Engineering, 34 (2010), 802-811. |
[3] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer-Verlag, Berlin-Heidelberg-New York, 2003. |
[4] |
G. Bromström and H. Drange, On the mathematical formulation and parameter estimation of the norwegian sea plankton system, Sarsia, 85 (2000), 211-225. |
[5] |
D. Brune, Optimal control of the complete-mix activated sludge process, Environmental Technology Letters, 6 (1985), 467-476.
doi: 10.1080/09593338509384365. |
[6] |
M. Burke, M. Chapwanya, K. Doherty, I. Hewitt, A. Korobeinikov, M. Meere, S. McCarthy, M. O'Brien, V. T. N. Tuoi, H. Winstanley and T. Zhelev, Modeling of autothermal thermophylic aerobic digestion, Mathematics-in-Industry Case Studies Journal, 2 (2010), 34-63. |
[7] |
S. Busenberg, S. Kumar, P. Austin and G. Wake, The Dynamics of a model of a plankton-nutrient interaction, Bulletin of Mathematical Biology, 52 (1990), 677-696. |
[8] |
F. L. Chernous'ko and V. B. Kolmanovskii, Computational and approximate methods of optimal control, Journal of Mathematical Sciences, 12 (1979), 310-353.
doi: 10.1007/BF01098370. |
[9] |
F. L. Chernous'ko, "Ellipsoidal state Estimation for Dynamical Systems," CRS Press, Boca Raton, Florida, 1994.
doi: 10.1016/j.na.2005.01.009. |
[10] |
B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967. |
[11] |
A. V. Dmitruk, A generalized estimate on the number of zeroes for solutions of a class of linear differential equations, SIAM Journal of Control and Optimization, 30 (1992), 1087-1091.
doi: 10.1137/0330057. |
[12] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM Journal on Applied Mathematics, 67 (2007), 337-353.
doi: 10.1137/060654876. |
[13] |
M. Graells, J. Rojas and T. Zhelev, Energy efficiency optimization of wastewater treatment. Study of ATAD, Computer Aided Chemical Engineering, 28 (2010), 967-972. |
[14] |
E. N. Khailov and E. V. Grigorieva, On the attainability set for a nonlinear system in the plane, Moscow University. Computational Mathematics and Cybernetics, (2001), 27-32. |
[15] |
E. V. Grigorieva and E. N. Khailov, A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good, Dynamical Systems and Differential Equations, (2003), 359-364. |
[16] |
E. N. Khailov and E. V. Grigorieva, Description of the attainability set of a nonlinear controlled system in the plane, Moscow University. Computational Mathematics and Cybernetics, (2005), 23-28. |
[17] |
E. V. Grigorieva and E. N. Khailov, Attainable set of a nonlinear controlled microeconomic model, Journal of Dynamical and Control Systems, 11 (2005), 157-176.
doi: 10.1007/s10883-005-4168-8. |
[18] |
E. V. Grigorieva, N. V. Bondarenko, E. N. Khailov and A. Korobeinikov, Three-dimensional nonlinear control model of wastewater biotreatment, Neural, Parallel, and Scientific Computations, 20 (2012), 23-35. |
[19] |
E. V. Grigorieva, N. V. Bondarenko, E. N. Khailov and A. Korobeinikov, Finite-dimensional methods for optimal control of autothermal thermophilic aerobic digestion, in "Industrial Waste" (eds. K.-Y. Show and X. Guo), InTech, Croatia, (2012), 91-120.
doi: 10.5772/36237. |
[20] |
T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different forms of predator-prey interaction, Journal of Theoretical Biology, 227 (2004), 349-358.
doi: 10.1016/j.jtbi.2003.09.020. |
[21] |
V. I. Gurman and E. A. Trushkova, Estimates for attainability sets of control systems, Differential Equations, 45 (2009), 1636-1644.
doi: 10.1134/S0012266109110093. |
[22] |
Kh. G. Guseinov, A. N. Moiseev and V. N. Ushakov, On the approximation of reachable domains of control systems, Journal of Applied Mathematics and Mechanics, 62 (1998), 169-175.
doi: 10.1016/S0021-8928(98)00022-7. |
[23] |
P. Hartman, "Ordinary Differential Equations," Jorn Wiley & Sons, New York-London-Sydney, 1964. |
[24] |
A. N. Kolmogorov, Sulla teoria di Volterra della lotta per l'esistenza, Giorn. Ist. Ital. Attuari, 7 (1936), 74-80. |
[25] |
V. A. Komarov, Estimates of the attainable set for differential inclusions, Mathematical Notes, 37 (1985), 916-925. |
[26] |
A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Mathematical Medicine and Biology, 26 (2009), 309-321.
doi: 10.1093/imammb/dqp009. |
[27] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Mathematical Medicine and Biology, 26 (2009), 225-239. |
[28] |
A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete and Continuous Dynamical System. Ser. B, 14 (2010), 1095-1103.
doi: 10.3934/dcdsb.2010.14.1095. |
[29] |
M. A. Krasnosel'skii, "The Operator of Translation along the Trajectories of Differential Equations," American Mathematical Society, Providence, RI, 1968. |
[30] |
A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control," Birkhäuser, Boston, 1997. |
[31] |
U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.
doi: 10.1093/imammb/dqp012. |
[32] |
U. Ledzewicz, E. Kashdan and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.
doi: 10.3934/mbe.2011.8.307. |
[33] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, Journal of Mathematical Biology, 64 (2012), 557-577.
doi: 10.1007/s00285-011-0424-6. |
[34] |
E. B. Lee and L. Markus, "Foundations of Optimal Control Theory," Jorn Wiley & Sons, New York, 1967. |
[35] |
M. S. Nikol'skii, Approximation of the attainability set for a controlled process, Mat. Zametki, 41 (1987), 71-76, 121. |
[36] |
N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. Part 2: proofs, variational derivatives and representations, Control and Cybernetics, 36 (2007), 5-45. |
[37] |
A. I. Ovseevich and F. L. Chernous'ko, Two-sided estimates on the attainability domains of controlled systems, Journal of Applied Mathematics and Mechanics, 46 (1982), 590-595.
doi: 10.1016/0021-8928(82)90005-3. |
[38] |
A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mathematical Notes, 27 (1980), 429-437, 494. |
[39] |
A. I. Panasyuk, Equations of attainable set dynamics. I. Integral funnel equation, Journal of Optimization Theory and Applications, 64 (1990), 349-366.
doi: 10.1007/BF00939453. |
[40] |
A. I. Panasyuk, Equations of attainable set dynamics. II. Partial differential equations, Journal of Optimization Theory and Applications, 64 (1990), 367-377.
doi: 10.1007/BF00939454. |
[41] |
T. Partasarathy, "On Global Univalence Theorems," Springer-Verlag, Berlin-Heidelberg-New York, 1983. |
[42] |
A. Sard, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 48 (1942), 883-890.
doi: 10.1090/S0002-9904-1942-07811-6. |
[43] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control. Theory, Methods and Examples," Springer, New York-Heidelberg-Dordrecht-London, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[44] |
L. Schwartz, "Analyse Mathematique 1," Hermann, Paris, 1967. |
[45] |
G. V. Shevchenko, Numerical method for solving a nonlinear time-optimal control problem with additive control, Computational Mathematics and Mathematical Physics, 47 (2007), 1768-1778.
doi: 10.1134/S0965542507110048. |
[46] |
G. V. Shevchenko, Numerical solution of a nonlinear time optimal control problem, Computational Mathematics and Mathematical Physics, 51 (2011), 537-549.
doi: 10.1134/S0965542511040154. |
[47] |
S. Ichiraku, A note on global implicit function theorems, IEEE Transactions on Curcuits and Systems, 32 (1985), 503-505.
doi: 10.1109/TCS.1985.1085729. |
[48] |
D. Szolnoki, Set-oriented methods for computing reachable sets and control sets, Discrete and Continuous Dynamical Systems. Ser. B, 3 (2003), 361-382.
doi: 10.3934/dcdsb.2003.3.361. |
[49] |
Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model, SIAM Journal on Applied Mathematics, 61 (2000), 803-833.
doi: 10.1137/S0036139998347834. |
[50] |
H. R. Thieme and Z. Feng, Endemic models with arbitrarily distributed periods of infection II: Fast disease dynamics and permanent recovery, SIAM Journal on Applied Mathematics, 61 (2000), 983-1012.
doi: 10.1137/S0036139998347846. |
[51] |
A. N. Tikhonov, A. B. Vasil'eva and A. G. Sveshnikov, "Differential Equations," Springer-Verlag, Berlin-Heidelberg-New York, 1985.
doi: 10.1007/978-3-642-82175-2. |
[52] |
A. I. Tyatyushkin and O. V. Morzhin, Constructive methods of control optimization in nonlinear systems, Automation and Remote Control, 70 (2009), 772-786.
doi: 10.1134/S0005117909050063. |
[53] |
A. I. Tyatyushkin and O. V. Morzhin, Numerical investigation of attainability sets of nonlinear controlled differential systems, Automation and Remote Control, 72 (2011), 1291-1300.
doi: 10.1134/S0005117911060178. |
[54] |
P. Varaiya and A. B. Kurzhanski, Ellipsoidal methods for dynamics and control. Part I, Journal of Mathematical Sciences, 139 (2006), 6863-6901.
doi: 10.1007/s10958-006-0397-y. |
[55] |
O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov, "Elementary Topology: Problem Textbook," AMS, 2008. |
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