## Appendix A Proofs  
**Theorem A1**. : _We assume that each agent follows the LCPSO Equation (11), with \(r_{\text{com}}\to+\infty\). Then the center of gravity of the swarm is heading towards the attractor \(\mathbf{p}^{\text{g}}(t)\), and its velocity vector is equal to \(\bar{\mathbf{v}}(t+\Delta t)=c_{0}\bar{\mathbf{v}}(t)+c_{2}\big{(}\mathbf{p}^{ \text{g}}(t)-\bar{\mathbf{p}}(t)\big{)}\). Furthermore, if the swarm is stable in the sense of Definition 2, all agents follow the speed vector of the center of gravity._Proof. : We calculate the speed vector of the center of gravity as follows:  
\[\bar{\mathbf{v}}(t+\Delta t)= \frac{c_{0}}{N}\sum_{i=1}^{N}\mathbf{v}_{i}(t)+\frac{1}{N}\sum_{i=1 }^{N}\mathbf{v}_{i}(t+\Delta t)\] \[\Rightarrow = c_{0}\bar{\mathbf{v}}(t)+\frac{1}{N}\sum_{i=1}^{N}\left(c_{2} \Big{(}\mathbf{p}^{g}(t)-\mathbf{p}_{i}(t)\Big{)}+c_{3}\sum_{j=1,j\
eq i}^{N} \
abla_{\mathbf{p}_{i}}P^{r}\Big{(}d_{i,j}(t)\Big{)}\right)\] \[\Rightarrow = c_{0}\bar{\mathbf{v}}(t)+\frac{c_{2}}{N}\left(\left(N\mathbf{p }^{g}(t)-\sum_{i=1}^{N}\mathbf{p}_{i}(t)\right)+c_{3}\sum_{i=1}^{N-1}\sum_{j=i +1}^{N}\left(\
abla_{\mathbf{p}_{j}}P^{r}\big{(}d_{i,j}(t)\big{)}+\
abla_{ \mathbf{p}_{i}}P^{r}\big{(}d_{i,j}(t)\big{)}\right)\right)\]  
In the last line of development, the second part contains all the gradients of the repulsion potentials. Since this potential meets Definition 1, their gradients are odd. Consequently, we have \(\
abla_{\mathbf{p}_{i}(t)}P^{r}\big{(}d_{i,j}(t)\big{)}=-\
abla_{\mathbf{p}_{ i}(t)}P^{r}\big{(}d_{i,j}(t)\big{)}\), so the sum is null. If we also remember the definition of the center of gravity (15), we have:  
\[\bar{\mathbf{v}}(t+\Delta t)=c_{0}\bar{\mathbf{v}}(t)+c_{2}\big{(}\mathbf{p}^ {g}(t)-\bar{\mathbf{p}}(t)\big{)}\]  
Equation (17) is equal to the gradient of the model of attractive potential defined in relation (7). Thus, we have \(\bar{\mathbf{v}}(t+\Delta t)=-\
abla_{\mathbf{p}}P^{a}\big{(}d_{att,\mathbf{p }}(t)\big{)}\), and we can see that it is indeed the center of gravity that is attracted by the attractor \(\mathbf{p}^{g}(t)\). Moreover, if we are stable in the sense of Definition 2, we have :  
\[\bar{\mathbf{v}}(t+\Delta t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{v}_{i}(t+\Delta t )=\mathbf{v}_{i}(t+\Delta t)\forall i\in[1,\cdots,N]\]  
**Theorem A2**. : _A swarm of \(N=2\) agents following the potential field \(P(t)=P_{1}(t)+P_{2}(t)\) in a gradient descent strategy will converge to an inter-agent distance at equilibrium \(r_{eq}=\sqrt{\frac{2c_{3}}{c_{2}}}\)._  
Proof. : In order not to burden the analysis, we set \(c_{0}\) to \(0\), hence there are only interaction forces between the agents. The velocity vectors will only fit on the \(\big{(}\mathbf{p}_{1}(0)\mathbf{p}_{2}(0)\big{)}\) line, which depends on the initial position of the agents \(1\) and \(2\). We can therefore perform our analysis in dimension \(1\) without loss of generality, and the position and speed of the agents are respectively the scalar values \(p_{i}(t)\) and \(v_{i}(t)\). We suppose that there is no collision, so \(d_{1,2}(t)\in]0;+\infty[\). In this interval, \(P(t)\in\mathcal{C}^{\infty}\). So we can use the property of a convex function: if the second derivative of \(P(t)\) is null or positive, then this function is convex. \[P(t) = \frac{c_{2}}{2}d_{1,2}^{2}(t)-2c_{3}\log\big{(}d_{1,2}(t)\big{)}\] \[\Rightarrow \frac{\partial P(t)}{\partial d_{1,2}(t)} = c_{2}d_{1,2}(t)-\text{sign}\big{(}p_{1}(t)-p_{2}(t)\big{)}\frac{ 2c_{3}}{d_{1,2}(t)}\] \[\Rightarrow \frac{\partial^{2}P(t)}{\partial d_{1,2}^{2}(t)} = c_{2}+\frac{2c_{3}}{d_{1,2}^{2}(t)}\]As \(d_{1,2}(t)\in]0;+\infty[\), \(\frac{\partial^{2}P(d_{1,2}(t))}{\partial d_{1,2}^{2}(t)}\) is always positive, so the potential field \(P(t)\) is convex.