# American Institute of Mathematical Sciences

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

 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.
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. [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.
 [1] Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 [2] Jonas Lampart. A remark on the attainable set of the Schrödinger equation. Evolution Equations and Control Theory, 2021, 10 (3) : 461-469. doi: 10.3934/eect.2020075 [3] Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799 [4] Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170 [5] Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589 [6] Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [7] M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure and Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495 [8] David Henry. Energy considerations for nonlinear equatorial water waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2337-2356. doi: 10.3934/cpaa.2022057 [9] Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997 [10] Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure and Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031 [11] Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022042 [12] Zhong Wan, Jingjing Liu, Jing Zhang. Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2117-2139. doi: 10.3934/jimo.2019046 [13] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [14] Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060 [15] Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Dirichlet problems with a crossing reaction. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2749-2766. doi: 10.3934/cpaa.2014.13.2749 [16] Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure and Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795 [17] Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281 [18] R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497 [19] Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065 [20] Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593

2018 Impact Factor: 1.313