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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.


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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} $

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