# American Institute of Mathematical Sciences

2017, 9(4): 459-486. doi: 10.3934/jgm.2017018

## On the relationship between the energy shaping and the Lyapunov constraint based methods

 1 Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina 2 Departamento de Matemática, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina 3 Departamento de Matemática, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina

S. Grillo and L. Salomone thank CONICET for its financial support. The authors also thank the referees and the editor for their useful remarks.

Received  December 2016 Revised  April 2017 Published  October 2017

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the general ideas around the Lyapunov constraint based (LCB) method, whose related partial differential equations (PDEs) were originally studied for underactuated systems with only one actuator, and then we study its PDEs for an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of control systems defined by simple Hamiltonian functions, that the LCB method and the Chang's version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that the Chang's improvement of the energy shaping method is precisely the LCB method. As a by-product, coordinate-free and connection-free expressions of Chang's matching conditions are obtained.

Citation: Sergio Grillo, Leandro Salomone, Marcela Zuccalli. On the relationship between the energy shaping and the Lyapunov constraint based methods. Journal of Geometric Mechanics, 2017, 9 (4) : 459-486. doi: 10.3934/jgm.2017018
##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, Advanced Book Program, Reading, Mass. , 1978. [2] S. Arimoto and F. Miyazaki, Stability and robustness of pid feedback control for robot manipulators of sensory capability, Robotics Research: 1st Internat. Symp. , M. Brady, R. P. Paul (Eds. ), Cambridge: MIT Press, (1983), 783-799. [3] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, 1978. [4] D. R. Auckly, L. V. Kapitanski and W. White, Control of nonlinear underactuated systems, Comm. Pure Appl. Math., 53 (2000), 354-369. doi: 10.1002/(SICI)1097-0312(200003)53:3<354::AID-CPA3>3.0.CO;2-U. [5] A. Bacciotti and L. Rosier, Regularity of Liapunov functions for stable systems, Systems & Control Letters, 41 (2000), 265-270. doi: 10.1016/S0167-6911(00)00062-1. [6] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756. doi: 10.1016/0005-1098(92)90034-D. [7] A. M. Bloch, N. E. Leonard and J. E. Marsden, Stabilization of mechanical systems using controlled lagrangians, Proc. of the 36th IEEE Conf. on Decision and Control, (1997), 2356-2361. doi: 10.1109/MCS.2010.939943. [8] A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems Ⅰ: The first matching theorem, IEEE Trans. Automat.Control, 45 (2000), 2253-2270. doi: 10.1109/9.895562. [9] A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems Ⅱ: Potential shaping, IEEE Trans. Automat. Control., 46 (2001), 1556-1571. doi: 10.1109/9.956051. [10] A. M. Bloch, Nonholonomic Mechanics and Control, volume 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003. [11] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, New York-London, 1975. [12] R. W. Brockett, Control theory and analytical mechanics, in 1976 Ames Research Center (NASA) Conference on Geometric Control Theory, (R. Hermann and C. Martin, eds. ), Math Sci Press, Brookline, Massachusetts, Lie Groups: History, Frontiers, and Applications, 7 (1977), 1-48. [13] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. [14] H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets J. Math. Phys. , 47 (2006), 022902, 29 pp. [15] M. Chaalal and N. Achour, Stabilization of a class of mechanical systems with impulse effects by lyapunov constraints, 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015,335-340. [16] D. E. Chang, The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control and Optimization, 48 (2010), 4821-4845. doi: 10.1137/070691310. [17] D. E. Chang, Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation, IEEE Trans. Automat. Contr., 55 (2010), 1888-1893. doi: 10.1109/TAC.2010.2049279. [18] D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, Proc. of the 18th Mediterranean Conf. on Control & Automation, (2010), 226-230. doi: 10.1109/MED.2010.5547672. [19] D. E. Chang, On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regular and Chaotic Dynamics, 19 (2014), 556-575. doi: 10.1134/S1560354714050049. [20] D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations (Special Issue Dedicated to JL Lions), 8 (2002), 393-422. [21] N. Crasta, R. Ortega, H. Pillai and J. Velazquez, The Matching Equations of Energy Shaping Controllers for Mechanical Systems are not Simplified with Generalized Forces, Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, University of Bologna, Bertinoro, Italy, 2012. [22] S. Grillo, Sistemas Noholónomos Generalizados, Ph. D. thesis, Instituto Balseiro, 2007. [23] S. Grillo, Higher order constrained Hamiltonian systems Journal of Mathematical Physics, 50 (2009), 082901, 34 pp. [24] S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, Int. Journal of Geom. Meth. in Mod. Physics, 7 (2010), 857-886. doi: 10.1142/S0219887810004580. [25] S. Grillo, J. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, Journal of Geom. Mech., 3 (2011), 145-196. doi: 10.3934/jgm.2011.3.145. [26] S. Grillo, L. Salomone and M. Zuccalli, On the asymptotic stabilizability of underactuated systems with two degrees of freedom and the Lyapunov constraint based method, preprint, arXiv: : 1604. 08475. [27] J. Hamberg, Gerneral Matching Conditions in the Theory of Controlled Lagrangians, in Proc. CDC, Phoenix, AZ, 1999. [28] C. Kellett, Classical converse theorems in Lyapunov's second method, Dyn. Syst. Ser. B, 20 (2015), 2333-2360, arXiv:1502. doi: 10.3934/dcdsb.2015.20.2333. [29] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 2002. [30] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley & Son, New York, 1996. [31] P. Krishnaprasad, Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonl. Anal. Th. Meth. and Appl., 9 (1985), 1011-1035. doi: 10.1016/0362-546X(85)90083-5. [32] C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. [33] C.-M. Marle, Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6. [34] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1999. [35] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. [36] J. L. Massera, Contributions to stability theory, Annals of Math., 64 (1956), 182-206. doi: 10.2307/1969955. [37] Erratum in Annals of Math. , 68 (1958), 202. [38] R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. Aut. Control, 47 (2002), 1218-1233. doi: 10.1109/TAC.2002.800770. [39] D. Pérez, Sistemas Noholónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos, Proyecto Integrador, Instituto Balseiro, 2006. [40] D. Pérez, Sistemas con Vínculos de Orden Superior y su Aplicación a la Teoría de Control Automático, Master thesis, Instituto Balseiro, 2007. [41] J. G. Romero, A. Donaire and R. Ortega, Robust energy shaping control of mechanical systems, Syst. Control Lett., 62 (2013), 770-780. doi: 10.1016/j.sysconle.2013.05.011. [42] A. J. van der Schaft, Hamiltonian dynamics with external forces and observations, Mathematical Systems Theory, 15 (1982), 145-168. doi: 10.1007/BF01786977. [43] A. J. van der Schaft, Stabilization of Hamiltonian systems, Nonlinear Analysis, Theory, Methods & Applications, 10 (1986), 1021-1035. doi: 10.1016/0362-546X(86)90086-6. [44] A. S. Shiriaev, J. W. Perram and C. C. Canudas, Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach, IEEE Trans. Automat. Contr., 50 (2005), 1164-1176. doi: 10.1109/TAC.2005.852568. [45] J. C. Willems, System theoretic models for the analysis of physical systems, Ricerche di Automatica, 10 (1979), 71-106. [46] C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation, European Journal of Control, 10 (2004), 478-496. doi: 10.3166/ejc.10.478-496.

show all references

##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, Advanced Book Program, Reading, Mass. , 1978. [2] S. Arimoto and F. Miyazaki, Stability and robustness of pid feedback control for robot manipulators of sensory capability, Robotics Research: 1st Internat. Symp. , M. Brady, R. P. Paul (Eds. ), Cambridge: MIT Press, (1983), 783-799. [3] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, 1978. [4] D. R. Auckly, L. V. Kapitanski and W. White, Control of nonlinear underactuated systems, Comm. Pure Appl. Math., 53 (2000), 354-369. doi: 10.1002/(SICI)1097-0312(200003)53:3<354::AID-CPA3>3.0.CO;2-U. [5] A. Bacciotti and L. Rosier, Regularity of Liapunov functions for stable systems, Systems & Control Letters, 41 (2000), 265-270. doi: 10.1016/S0167-6911(00)00062-1. [6] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756. doi: 10.1016/0005-1098(92)90034-D. [7] A. M. Bloch, N. E. Leonard and J. E. Marsden, Stabilization of mechanical systems using controlled lagrangians, Proc. of the 36th IEEE Conf. on Decision and Control, (1997), 2356-2361. doi: 10.1109/MCS.2010.939943. [8] A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems Ⅰ: The first matching theorem, IEEE Trans. Automat.Control, 45 (2000), 2253-2270. doi: 10.1109/9.895562. [9] A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems Ⅱ: Potential shaping, IEEE Trans. Automat. Control., 46 (2001), 1556-1571. doi: 10.1109/9.956051. [10] A. M. Bloch, Nonholonomic Mechanics and Control, volume 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003. [11] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, New York-London, 1975. [12] R. W. Brockett, Control theory and analytical mechanics, in 1976 Ames Research Center (NASA) Conference on Geometric Control Theory, (R. Hermann and C. Martin, eds. ), Math Sci Press, Brookline, Massachusetts, Lie Groups: History, Frontiers, and Applications, 7 (1977), 1-48. [13] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. [14] H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets J. Math. Phys. , 47 (2006), 022902, 29 pp. [15] M. Chaalal and N. Achour, Stabilization of a class of mechanical systems with impulse effects by lyapunov constraints, 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015,335-340. [16] D. E. Chang, The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control and Optimization, 48 (2010), 4821-4845. doi: 10.1137/070691310. [17] D. E. Chang, Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation, IEEE Trans. Automat. Contr., 55 (2010), 1888-1893. doi: 10.1109/TAC.2010.2049279. [18] D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, Proc. of the 18th Mediterranean Conf. on Control & Automation, (2010), 226-230. doi: 10.1109/MED.2010.5547672. [19] D. E. Chang, On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regular and Chaotic Dynamics, 19 (2014), 556-575. doi: 10.1134/S1560354714050049. [20] D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations (Special Issue Dedicated to JL Lions), 8 (2002), 393-422. [21] N. Crasta, R. Ortega, H. Pillai and J. Velazquez, The Matching Equations of Energy Shaping Controllers for Mechanical Systems are not Simplified with Generalized Forces, Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, University of Bologna, Bertinoro, Italy, 2012. [22] S. Grillo, Sistemas Noholónomos Generalizados, Ph. D. thesis, Instituto Balseiro, 2007. [23] S. Grillo, Higher order constrained Hamiltonian systems Journal of Mathematical Physics, 50 (2009), 082901, 34 pp. [24] S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, Int. Journal of Geom. Meth. in Mod. Physics, 7 (2010), 857-886. doi: 10.1142/S0219887810004580. [25] S. Grillo, J. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, Journal of Geom. Mech., 3 (2011), 145-196. doi: 10.3934/jgm.2011.3.145. [26] S. Grillo, L. Salomone and M. Zuccalli, On the asymptotic stabilizability of underactuated systems with two degrees of freedom and the Lyapunov constraint based method, preprint, arXiv: : 1604. 08475. [27] J. Hamberg, Gerneral Matching Conditions in the Theory of Controlled Lagrangians, in Proc. CDC, Phoenix, AZ, 1999. [28] C. Kellett, Classical converse theorems in Lyapunov's second method, Dyn. Syst. Ser. B, 20 (2015), 2333-2360, arXiv:1502. doi: 10.3934/dcdsb.2015.20.2333. [29] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 2002. [30] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley & Son, New York, 1996. [31] P. Krishnaprasad, Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonl. Anal. Th. Meth. and Appl., 9 (1985), 1011-1035. doi: 10.1016/0362-546X(85)90083-5. [32] C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. [33] C.-M. Marle, Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6. [34] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1999. [35] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. [36] J. L. Massera, Contributions to stability theory, Annals of Math., 64 (1956), 182-206. doi: 10.2307/1969955. [37] Erratum in Annals of Math. , 68 (1958), 202. [38] R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. Aut. Control, 47 (2002), 1218-1233. doi: 10.1109/TAC.2002.800770. [39] D. Pérez, Sistemas Noholónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos, Proyecto Integrador, Instituto Balseiro, 2006. [40] D. Pérez, Sistemas con Vínculos de Orden Superior y su Aplicación a la Teoría de Control Automático, Master thesis, Instituto Balseiro, 2007. [41] J. G. Romero, A. Donaire and R. Ortega, Robust energy shaping control of mechanical systems, Syst. Control Lett., 62 (2013), 770-780. doi: 10.1016/j.sysconle.2013.05.011. [42] A. J. van der Schaft, Hamiltonian dynamics with external forces and observations, Mathematical Systems Theory, 15 (1982), 145-168. doi: 10.1007/BF01786977. [43] A. J. van der Schaft, Stabilization of Hamiltonian systems, Nonlinear Analysis, Theory, Methods & Applications, 10 (1986), 1021-1035. doi: 10.1016/0362-546X(86)90086-6. [44] A. S. Shiriaev, J. W. Perram and C. C. Canudas, Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach, IEEE Trans. Automat. Contr., 50 (2005), 1164-1176. doi: 10.1109/TAC.2005.852568. [45] J. C. Willems, System theoretic models for the analysis of physical systems, Ricerche di Automatica, 10 (1979), 71-106. [46] C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation, European Journal of Control, 10 (2004), 478-496. doi: 10.3166/ejc.10.478-496.
 [1] Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387 [2] Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451 [3] Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995 [4] Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589 [5] Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81 [6] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [7] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579 [8] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [9] Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467 [10] Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603 [11] Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 [12] Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841 [13] Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381 [14] David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 [15] Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013 [16] Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99 [17] Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385 [18] Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990 [19] Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493 [20] Robert Jankowski, Barbara Łupińska, Magdalena Nockowska-Rosiak, Ewa Schmeidel. Monotonic solutions of a higher-order neutral difference system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 253-261. doi: 10.3934/dcdsb.2018017

2016 Impact Factor: 0.857

Article outline

[Back to Top]