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Construction of Lyapunov functions using Helmholtz–Hodge decomposition

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

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  • 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.

    Mathematics Subject Classification: Primary: 37B25; Secondary: 37C10, 37B35.

    Citation:

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  • 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} $

  • [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.,
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