Consider the following Lyapunov candidate 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 positive integer number. The symbols $x,y\in \mathbb{R} $ are scalars representing the position of a rigid body in 2D space. The positive integer $n$ corresponds to the number of other units in the 2D space, the position of which is known. So, the pairs $(x_i, y_i), i=1,\dots,n$ represent the positions of $n$ bodies in the 2D space. Consequently, the summation term denotes the potential energy of the system with $(x_i,y_i)$ being known.
I would like to understand what does the function $V(x,y,q_i)$ represent for different values of the arguments $x,y$ in the 2D space. If there wasn't the summation term, then it would certainly represent a circle in the 2D space. I am not sure about its representation in the above case where the summation term exists. Does it still represent a circle (or an ellipse) with different radius and center than the case without the summation term ?
What I am looking for, is to understand the type of the Lyapunov surfaces (level surfaces) defined by equation (1). It should be noted that (1) possesses the characteristics of Lyapunov functions:
- $V(x,y,q_i) = 0$ if and only if $(x,y,q_i) = 0$.
- $V(\cdot)$ is a positive continuously differentiable function satisfying $V(x,y,q_i) > 0 \ \forall (x,y,q_i)\neq (0,0,0)$.