May 2019, 39(5): 2437-2454. doi: 10.3934/dcds.2019103

Construction of Lyapunov functions using Helmholtz–Hodge decomposition

Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, Japan

Received  November 2017 Revised  May 2018 Published  January 2019

Fund Project: The first author is supported by Grant-in-Aid for JSPS Fellows (17J03931)

The Helmholtz–Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.

Citation: Tomoharu Suda. Construction of Lyapunov functions using Helmholtz–Hodge decomposition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2437-2454. doi: 10.3934/dcds.2019103
References:
[1]

H. BhatiaG. Norgard and V. Pascucci, The Helmholtz-Hodge decomposition-a survey, IEEE T. Vis. Comput. Gr., 19 (2013), 1386-1404.

[2]

H. BhatiaV. Pascucci and P. Bremer, The natural Helmholtz-Hodge decomposition for Open-Boundary flow analysis, IEEE T. Vis. Comput. Gr., 20 (2014), 1566-1578. doi: 10.1109/TVCG.2014.2312012.

[3]

J. DemongeotN. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition Ⅰ -methodology, C. R. Math., 344 (2007), 121-126. doi: 10.1016/j.crma.2006.10.016.

[4]

J. DemongeotN. Glade and L. Forest, Liénard systems and {potential-Hamiltonian} decomposition Ⅱ -algorithm, C. R. Math., 344 (2007), 191-194. doi: 10.1016/j.crma.2006.10.013.

[5]

M. Denaro, On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions, Int. J. Numer. Methods Fluids, 43 (2003), 43-69. doi: 10.1002/fld.598.

[6]

T. Duarte and R. Mendes, Deformation of Hamiltonian dynamics and constants of motion in dissipative systems, J. Math. Phys., 24 (1983), 1772-1778. doi: 10.1063/1.525894.

[7]

E. Fuselier and G. Wright, A radial basis function method for computing Helmholtz-Hodge decompositions, IMA J. Numer. Anal., 37 (2017), 774-797. doi: 10.1093/imanum/drw027.

[8]

J. Geng and Z. Shen, The Neumann problem and Helmholtz decomposition in convex domains, J. Funct. Anal., 259 (2010), 2147-2164. doi: 10.1016/j.jfa.2010.07.005.

[9]

P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291.

[10]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Springer–Verlag Berlin Heidelberg, 2007.

[11]

J. LaSalle, Some extensions of Liapunov's second method, IRE Trans. Circuit Theory, 7 (1960), 520-527.

[12]

A. Polanski, Lyapunov function construction by linear programming, IEEE Trans. Autom. Control., 42 (1997), 1013-1016. doi: 10.1109/9.599986.

[13]

K. Polthier and E. Preuß, Identifying vector field singularities using a discrete Hodge decomposition, Visualization and Mathematics III, 2003, 113–134.

[14]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics, CRC-Press, 1999.

[15]

G. Strang, Linear Algebra and Its Applications, Thomson, Brooks/Cole, 2006.

[16]

A. Wiebel, Feature detection in vector fields using the Helmholtz-Hodge decomposition Diploma-Thesis.,

show all references

References:
[1]

H. BhatiaG. Norgard and V. Pascucci, The Helmholtz-Hodge decomposition-a survey, IEEE T. Vis. Comput. Gr., 19 (2013), 1386-1404.

[2]

H. BhatiaV. Pascucci and P. Bremer, The natural Helmholtz-Hodge decomposition for Open-Boundary flow analysis, IEEE T. Vis. Comput. Gr., 20 (2014), 1566-1578. doi: 10.1109/TVCG.2014.2312012.

[3]

J. DemongeotN. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition Ⅰ -methodology, C. R. Math., 344 (2007), 121-126. doi: 10.1016/j.crma.2006.10.016.

[4]

J. DemongeotN. Glade and L. Forest, Liénard systems and {potential-Hamiltonian} decomposition Ⅱ -algorithm, C. R. Math., 344 (2007), 191-194. doi: 10.1016/j.crma.2006.10.013.

[5]

M. Denaro, On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions, Int. J. Numer. Methods Fluids, 43 (2003), 43-69. doi: 10.1002/fld.598.

[6]

T. Duarte and R. Mendes, Deformation of Hamiltonian dynamics and constants of motion in dissipative systems, J. Math. Phys., 24 (1983), 1772-1778. doi: 10.1063/1.525894.

[7]

E. Fuselier and G. Wright, A radial basis function method for computing Helmholtz-Hodge decompositions, IMA J. Numer. Anal., 37 (2017), 774-797. doi: 10.1093/imanum/drw027.

[8]

J. Geng and Z. Shen, The Neumann problem and Helmholtz decomposition in convex domains, J. Funct. Anal., 259 (2010), 2147-2164. doi: 10.1016/j.jfa.2010.07.005.

[9]

P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291.

[10]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Springer–Verlag Berlin Heidelberg, 2007.

[11]

J. LaSalle, Some extensions of Liapunov's second method, IRE Trans. Circuit Theory, 7 (1960), 520-527.

[12]

A. Polanski, Lyapunov function construction by linear programming, IEEE Trans. Autom. Control., 42 (1997), 1013-1016. doi: 10.1109/9.599986.

[13]

K. Polthier and E. Preuß, Identifying vector field singularities using a discrete Hodge decomposition, Visualization and Mathematics III, 2003, 113–134.

[14]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics, CRC-Press, 1999.

[15]

G. Strang, Linear Algebra and Its Applications, Thomson, Brooks/Cole, 2006.

[16]

A. Wiebel, Feature detection in vector fields using the Helmholtz-Hodge decomposition Diploma-Thesis.,

Figure 1.  Left: Contours of $ V_1 $ and the sign of $ \dot{V_1} $. In the shaded domain, $ \dot{V_1} $ is positive. Right: Solution curves of Equation (2). A contour of $ V_1 $ is given for comparison with the left panel
Figure 2.  Contours of $ V_2 $ and the sign of $ \dot{V_2} $. In the shaded domain, $ \dot{V_2} $ is positive
Figure 3.  Solution curves of the vector field (4)
Figure 4.  Strictly orthogonal HHD of the vector field (4). Left: solution curves of $ -\nabla V $. Right: solution curves of $ {\bf u} $
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