September  2016, 6(3): 447-466. doi: 10.3934/mcrf.2016011

Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions

1. 

Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan

2. 

Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502

Received  June 2014 Revised  April 2016 Published  August 2016

For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
Citation: Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011
References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control,, Consultants Bureau, (1987).  doi: 10.1007/978-1-4615-7551-1.  Google Scholar

[2]

D. Angeli and A. Astolfi, A tight small gain theorem for not necessarily ISS systems,, Syst. Control Lett., 56 (2007), 87.  doi: 10.1016/j.sysconle.2006.08.003.  Google Scholar

[3]

D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Transactions on Automatic Control, 45 (2000), 1082.  doi: 10.1109/9.863594.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction To Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[6]

S. Dashkovskiy and A. Mironchenko, On the uniform input-to-state stability of reaction-diffusion systems,, in Proc. of the 49th IEEE Conference on Decision and Control (CDC), (2010), 6547.  doi: 10.1109/CDC.2010.5717779.  Google Scholar

[7]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems,, Mathematics of Control, 25 (2013), 1.  doi: 10.1007/s00498-012-0090-2.  Google Scholar

[8]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems,, SIAM Journal on Control and Optimization, 51 (2013), 1962.  doi: 10.1137/120881993.  Google Scholar

[9]

S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks,, Mathematics of Control, 19 (2007), 93.  doi: 10.1007/s00498-007-0014-8.  Google Scholar

[10]

R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques,, Birkhäuser, (1996).  doi: 10.1007/978-0-8176-4759-9.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[12]

B. Ingalls and E. D. Sontag, A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections,, Journal of the Franklin Institute, 339 (2002), 211.  doi: 10.1016/S0016-0032(02)00022-4.  Google Scholar

[13]

H. Ito, A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems,, IEEE Transactions on Automatic Control, 55 (2010), 702.  doi: 10.1109/TAC.2009.2037457.  Google Scholar

[14]

H. Ito, Z. P. Jiang, S. Dashkovskiy and B. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Transactions on Automatic Control, 58 (2013), 1192.  doi: 10.1109/TAC.2012.2231552.  Google Scholar

[15]

H. Ito and Z.-P. Jiang, Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective,, IEEE Transactions on Automatic Control, 54 (2009), 2389.  doi: 10.1109/TAC.2009.2028980.  Google Scholar

[16]

H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Transactions on Automatic Control, 51 (2006), 1626.  doi: 10.1109/TAC.2006.882930.  Google Scholar

[17]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces,, Springer, (2012).  doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[18]

B. Jayawardhana, H. Logemann and E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability,, Communications in Information and Systems, 8 (2008), 413.  doi: 10.4310/CIS.2008.v8.n4.a4.  Google Scholar

[19]

Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,, Automatica, 32 (1996), 1211.  doi: 10.1016/0005-1098(96)00051-9.  Google Scholar

[20]

Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications,, Mathematics of Control, 7 (1994), 95.  doi: 10.1007/BF01211469.  Google Scholar

[21]

Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems,, Automatica, 37 (2001), 857.  doi: 10.1016/S0005-1098(01)00028-0.  Google Scholar

[22]

Z.-P. Jiang and Y. Wang, A generalization of the nonlinear small-gain theorem for large-scale complex systems,, in 7th World Congress on Intelligent Control and Automation (WCICA), (2008), 1188.   Google Scholar

[23]

I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general non-linear control systems,, IMA Journal of Mathematical Control and Information, 28 (2011), 309.  doi: 10.1093/imamci/dnr001.  Google Scholar

[24]

I. Karafyllis and Z.-P. Jiang, A new small-gain theorem with an application to the stabilization of the chemostat,, Int. J. Robust. Nonlinear Control, 22 (2012), 1602.  doi: 10.1002/rnc.1773.  Google Scholar

[25]

P. Kokotović and M. Arcak, Constructive nonlinear control: a historical perspective,, Automatica, 37 (2001), 637.  doi: 10.1016/S0005-1098(01)00002-4.  Google Scholar

[26]

Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability,, SIAM Journal on Control and Optimization, 34 (1996), 124.  doi: 10.1137/S0363012993259981.  Google Scholar

[27]

H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback,, SIAM Journal on Control and Optimization, 51 (2013), 1203.  doi: 10.1137/110850396.  Google Scholar

[28]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations,, Mathematical Control and Related Fields, 1 (2011), 231.  doi: 10.3934/mcrf.2011.1.231.  Google Scholar

[29]

A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples,, Systems & Control Letters, 87 (2016), 23.  doi: 10.1016/j.sysconle.2015.10.014.  Google Scholar

[30]

A. Mironchenko and H. Ito, Integral input-to-state stability of bilinear infinite-dimensional systems,, in Proc. of the 53th IEEE Conference on Decision and Control, (2014), 3155.  doi: 10.1109/CDC.2014.7039876.  Google Scholar

[31]

A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach,, SIAM Journal on Control and Optimization, 53 (2015), 3364.  doi: 10.1137/14097269X.  Google Scholar

[32]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives,, Kluwer Academic Publishers Group, (1991).  doi: 10.1007/978-94-011-3562-7.  Google Scholar

[33]

P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems,, Systems & Control Letters, 55 (2006), 1006.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[34]

C. Prieur and F. Mazenc, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws,, Mathematics of Control, 24 (2012), 111.  doi: 10.1007/s00498-012-0074-2.  Google Scholar

[35]

H. Royden and P. Fitzpatrick, Real Analysis,, Prentice Hall, (2010).   Google Scholar

[36]

E. Sontag and A. Teel, Changing supply functions in input/state stable systems,, IEEE Transactions on Automatic Control, 40 (1995), 1476.  doi: 10.1109/9.402246.  Google Scholar

[37]

E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Transactions on Automatic Control, 34 (1989), 435.  doi: 10.1109/9.28018.  Google Scholar

[38]

E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property,, Systems & Control Letters, 24 (1995), 351.  doi: 10.1016/0167-6911(94)00050-6.  Google Scholar

[39]

E. D. Sontag, Comments on integral variants of ISS,, Systems & Control Letters, 34 (1998), 93.  doi: 10.1016/S0167-6911(98)00003-6.  Google Scholar

[40]

E. D. Sontag, Input to state stability: Basic concepts and results,, in Nonlinear and Optimal Control Theory, 1932 (2008), 163.  doi: 10.1007/978-3-540-77653-6_3.  Google Scholar

[41]

G. Teschl, Ordinary Differential Equations and Dynamical Systems,, American Mathematical Society, (2012).  doi: 10.1090/gsm/140.  Google Scholar

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control,, Consultants Bureau, (1987).  doi: 10.1007/978-1-4615-7551-1.  Google Scholar

[2]

D. Angeli and A. Astolfi, A tight small gain theorem for not necessarily ISS systems,, Syst. Control Lett., 56 (2007), 87.  doi: 10.1016/j.sysconle.2006.08.003.  Google Scholar

[3]

D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Transactions on Automatic Control, 45 (2000), 1082.  doi: 10.1109/9.863594.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction To Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[6]

S. Dashkovskiy and A. Mironchenko, On the uniform input-to-state stability of reaction-diffusion systems,, in Proc. of the 49th IEEE Conference on Decision and Control (CDC), (2010), 6547.  doi: 10.1109/CDC.2010.5717779.  Google Scholar

[7]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems,, Mathematics of Control, 25 (2013), 1.  doi: 10.1007/s00498-012-0090-2.  Google Scholar

[8]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems,, SIAM Journal on Control and Optimization, 51 (2013), 1962.  doi: 10.1137/120881993.  Google Scholar

[9]

S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks,, Mathematics of Control, 19 (2007), 93.  doi: 10.1007/s00498-007-0014-8.  Google Scholar

[10]

R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques,, Birkhäuser, (1996).  doi: 10.1007/978-0-8176-4759-9.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[12]

B. Ingalls and E. D. Sontag, A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections,, Journal of the Franklin Institute, 339 (2002), 211.  doi: 10.1016/S0016-0032(02)00022-4.  Google Scholar

[13]

H. Ito, A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems,, IEEE Transactions on Automatic Control, 55 (2010), 702.  doi: 10.1109/TAC.2009.2037457.  Google Scholar

[14]

H. Ito, Z. P. Jiang, S. Dashkovskiy and B. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Transactions on Automatic Control, 58 (2013), 1192.  doi: 10.1109/TAC.2012.2231552.  Google Scholar

[15]

H. Ito and Z.-P. Jiang, Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective,, IEEE Transactions on Automatic Control, 54 (2009), 2389.  doi: 10.1109/TAC.2009.2028980.  Google Scholar

[16]

H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Transactions on Automatic Control, 51 (2006), 1626.  doi: 10.1109/TAC.2006.882930.  Google Scholar

[17]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces,, Springer, (2012).  doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[18]

B. Jayawardhana, H. Logemann and E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability,, Communications in Information and Systems, 8 (2008), 413.  doi: 10.4310/CIS.2008.v8.n4.a4.  Google Scholar

[19]

Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,, Automatica, 32 (1996), 1211.  doi: 10.1016/0005-1098(96)00051-9.  Google Scholar

[20]

Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications,, Mathematics of Control, 7 (1994), 95.  doi: 10.1007/BF01211469.  Google Scholar

[21]

Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems,, Automatica, 37 (2001), 857.  doi: 10.1016/S0005-1098(01)00028-0.  Google Scholar

[22]

Z.-P. Jiang and Y. Wang, A generalization of the nonlinear small-gain theorem for large-scale complex systems,, in 7th World Congress on Intelligent Control and Automation (WCICA), (2008), 1188.   Google Scholar

[23]

I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general non-linear control systems,, IMA Journal of Mathematical Control and Information, 28 (2011), 309.  doi: 10.1093/imamci/dnr001.  Google Scholar

[24]

I. Karafyllis and Z.-P. Jiang, A new small-gain theorem with an application to the stabilization of the chemostat,, Int. J. Robust. Nonlinear Control, 22 (2012), 1602.  doi: 10.1002/rnc.1773.  Google Scholar

[25]

P. Kokotović and M. Arcak, Constructive nonlinear control: a historical perspective,, Automatica, 37 (2001), 637.  doi: 10.1016/S0005-1098(01)00002-4.  Google Scholar

[26]

Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability,, SIAM Journal on Control and Optimization, 34 (1996), 124.  doi: 10.1137/S0363012993259981.  Google Scholar

[27]

H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback,, SIAM Journal on Control and Optimization, 51 (2013), 1203.  doi: 10.1137/110850396.  Google Scholar

[28]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations,, Mathematical Control and Related Fields, 1 (2011), 231.  doi: 10.3934/mcrf.2011.1.231.  Google Scholar

[29]

A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples,, Systems & Control Letters, 87 (2016), 23.  doi: 10.1016/j.sysconle.2015.10.014.  Google Scholar

[30]

A. Mironchenko and H. Ito, Integral input-to-state stability of bilinear infinite-dimensional systems,, in Proc. of the 53th IEEE Conference on Decision and Control, (2014), 3155.  doi: 10.1109/CDC.2014.7039876.  Google Scholar

[31]

A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach,, SIAM Journal on Control and Optimization, 53 (2015), 3364.  doi: 10.1137/14097269X.  Google Scholar

[32]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives,, Kluwer Academic Publishers Group, (1991).  doi: 10.1007/978-94-011-3562-7.  Google Scholar

[33]

P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems,, Systems & Control Letters, 55 (2006), 1006.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[34]

C. Prieur and F. Mazenc, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws,, Mathematics of Control, 24 (2012), 111.  doi: 10.1007/s00498-012-0074-2.  Google Scholar

[35]

H. Royden and P. Fitzpatrick, Real Analysis,, Prentice Hall, (2010).   Google Scholar

[36]

E. Sontag and A. Teel, Changing supply functions in input/state stable systems,, IEEE Transactions on Automatic Control, 40 (1995), 1476.  doi: 10.1109/9.402246.  Google Scholar

[37]

E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Transactions on Automatic Control, 34 (1989), 435.  doi: 10.1109/9.28018.  Google Scholar

[38]

E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property,, Systems & Control Letters, 24 (1995), 351.  doi: 10.1016/0167-6911(94)00050-6.  Google Scholar

[39]

E. D. Sontag, Comments on integral variants of ISS,, Systems & Control Letters, 34 (1998), 93.  doi: 10.1016/S0167-6911(98)00003-6.  Google Scholar

[40]

E. D. Sontag, Input to state stability: Basic concepts and results,, in Nonlinear and Optimal Control Theory, 1932 (2008), 163.  doi: 10.1007/978-3-540-77653-6_3.  Google Scholar

[41]

G. Teschl, Ordinary Differential Equations and Dynamical Systems,, American Mathematical Society, (2012).  doi: 10.1090/gsm/140.  Google Scholar

[1]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[2]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[3]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[4]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[6]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[7]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[8]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[9]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[10]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[11]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[14]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[15]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[16]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[17]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[18]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[19]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[20]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]