American Institute of Mathematical Sciences

Advanced
2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067

Parametrization of the attainable set for a nonlinear control model of a biochemical process

Ellina Grigorieva 1, , Evgenii Khailov 2, and  Andrei Korobeinikov 3,

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

Received  October 2012 Revised  April 2013 Published  June 2013

In this paper, we study a three-dimensional nonlinear model of a controllable reaction $ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecified nonlinear function. A model of this type describes a variety of real-life processes in chemical kinetics and biology; in this paper our particular interests is in its application to waste water biotreatment. For this control model, we analytically study the corresponding attainable set and parameterize it by the moments of switching of piecewise constant control functions. This allows us to visualize the attainable sets using a numerical procedure.
    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.
Keywords: waste water biotreatment, Environmental problem, nonlinear reaction rate, attainable set..
Mathematics Subject Classification: Primary: 49J15, 49N90; Secondary: 93C10, 93C9.
Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067
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.  Google Scholar

[2]

A. D. Bojarski, J. Rojas and T. Zhelev, Modelling and sensitivity analysis of ATAD, Computers & Chemical Engineering, 34 (2010), 802-811. Google Scholar

[3]

B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer-Verlag, Berlin-Heidelberg-New York, 2003.  Google Scholar

[4]

G. Bromström and H. Drange, On the mathematical formulation and parameter estimation of the norwegian sea plankton system, Sarsia, 85 (2000), 211-225. Google Scholar

[5]

D. Brune, Optimal control of the complete-mix activated sludge process, Environmental Technology Letters, 6 (1985), 467-476. doi: 10.1080/09593338509384365.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. Hartman, "Ordinary Differential Equations," Jorn Wiley & Sons, New York-London-Sydney, 1964.  Google Scholar

[24]

A. N. Kolmogorov, Sulla teoria di Volterra della lotta per l'esistenza, Giorn. Ist. Ital. Attuari, 7 (1936), 74-80. Google Scholar

[25]

V. A. Komarov, Estimates of the attainable set for differential inclusions, Mathematical Notes, 37 (1985), 916-925.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[29]

M. A. Krasnosel'skii, "The Operator of Translation along the Trajectories of Differential Equations," American Mathematical Society, Providence, RI, 1968.  Google Scholar

[30]

A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control," Birkhäuser, Boston, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory," Jorn Wiley & Sons, New York, 1967.  Google Scholar

[35]

M. S. Nikol'skii, Approximation of the attainability set for a controlled process, Mat. Zametki, 41 (1987), 71-76, 121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mathematical Notes, 27 (1980), 429-437, 494.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

T. Partasarathy, "On Global Univalence Theorems," Springer-Verlag, Berlin-Heidelberg-New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

L. Schwartz, "Analyse Mathematique 1," Hermann, Paris, 1967. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov, "Elementary Topology: Problem Textbook," AMS, 2008.  Google Scholar

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.  Google Scholar

[2]

A. D. Bojarski, J. Rojas and T. Zhelev, Modelling and sensitivity analysis of ATAD, Computers & Chemical Engineering, 34 (2010), 802-811. Google Scholar

[3]

B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer-Verlag, Berlin-Heidelberg-New York, 2003.  Google Scholar

[4]

G. Bromström and H. Drange, On the mathematical formulation and parameter estimation of the norwegian sea plankton system, Sarsia, 85 (2000), 211-225. Google Scholar

[5]

D. Brune, Optimal control of the complete-mix activated sludge process, Environmental Technology Letters, 6 (1985), 467-476. doi: 10.1080/09593338509384365.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. Hartman, "Ordinary Differential Equations," Jorn Wiley & Sons, New York-London-Sydney, 1964.  Google Scholar

[24]

A. N. Kolmogorov, Sulla teoria di Volterra della lotta per l'esistenza, Giorn. Ist. Ital. Attuari, 7 (1936), 74-80. Google Scholar

[25]

V. A. Komarov, Estimates of the attainable set for differential inclusions, Mathematical Notes, 37 (1985), 916-925.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[29]

M. A. Krasnosel'skii, "The Operator of Translation along the Trajectories of Differential Equations," American Mathematical Society, Providence, RI, 1968.  Google Scholar

[30]

A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control," Birkhäuser, Boston, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory," Jorn Wiley & Sons, New York, 1967.  Google Scholar

[35]

M. S. Nikol'skii, Approximation of the attainability set for a controlled process, Mat. Zametki, 41 (1987), 71-76, 121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mathematical Notes, 27 (1980), 429-437, 494.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

T. Partasarathy, "On Global Univalence Theorems," Springer-Verlag, Berlin-Heidelberg-New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

L. Schwartz, "Analyse Mathematique 1," Hermann, Paris, 1967. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov, "Elementary Topology: Problem Textbook," AMS, 2008.  Google Scholar

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