Type of Lyapunov surfaces corresponding to a specific Lyapunov function | pressku.com

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Consider nan pursuing Lyapunov campaigner function:

$$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \tag1$$

where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary affirmative integer number. Suppose $x_{i}, y_{i}$ are known points successful nan 2D space. I would for illustration to understand what does nan usability $f(x,y)$ correspond for different values of nan arguments $x,y$ successful nan 2D space. If location wasn't nan summation term, past it would surely correspond a circle successful nan 2D space. I americium not judge astir its practice successful nan supra lawsuit wherever nan summation word exists. Does it still correspond a circle (or an ellipse) pinch different radius and halfway than nan lawsuit without nan summation word ?

What I americium looking for, is to understand nan type of nan Lyapunov surfaces (level surfaces) defined by equation (1). It should beryllium noted that (1) possesses nan characteristics of Lyapunov functions:

  1. $V(x,y,q_i) = 0$ if and only if $(x,y,q_i) = 0$.
  2. $V(\cdot)$ is simply a affirmative continuously differentiable usability satisfying $V(x,y,q_i) > 0 \ \forall (x,y,q_i)\neq (0,0,0)$.