Article Contents
Article Contents

# Construction of Lyapunov functions using Helmholtz–Hodge decomposition

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.

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

 Citation:

• 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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