## 6 Conclusions  
In this paper, we carried out an analysis of LCPSO algorithm that merges the spirit of PSO and flocking algorithms. This analysis is supported by mathematical theorems, that apply regardless of the dimension, or when necessary, by Monte-Carlo results, especially concerning communication constraints. Using only one attractor in a limited area permits to follow a target accurately. We summarise the contributions of this paper by the following points. First, the formation at equilibrium is resilient to communication limits and the brutal moves of the target because Equation (18) is only based on measurements at time \(t\). Moreover, we have proven analytically that the formation will stay stable in the sense of Definition 2 whatever the dimension and the place of the attractor(s) in the swarm. The strength of this formation avoids collisions between agents and losing agents with communication constraints. Finally, the speed is intrinsically limited and predictable thanks to Equation (17). The stability of the swarm formation whatever the conditions (communication limits and the behavior of the target) makes our algorithm applicable in very constrained environments, like underwater scenario for example. LCPSO algorithm is resilient to the  
Figure 8: Tracking fail of agents following the LCPSO algorithm (18) as a function of \(N\). \(c_{0}=0.5\), \(c_{2}=0.5\), \(r_{eq}=7\) m, \(r_{com}=20\) m.  
Figure 9: \(D_{100}\) (m) evolution as a function of \(r_{eq}\) and \(r_{com}\) in dimension 2. \(N=10\), \(c_{0}=0.8\), \(c_{2}=2\), \(\beta(\mathbf{p},t)\sim\mathcal{N}(0,0.25)\), \(\tau=1\). breakdown of some agents because the attractor depends on measurements and can be exchanged easily with another agent. Communication limits do not degrade our swarm formation, and the simplicity of LCPSO allows the robot to embed only few computing power. Our work still has many limitations, which we acknowledge here. First, the plume model should be modified to be more realistic [31; 43] to include the problem of measurement noise with a low SNR and the problem of noise correlation (in our case the measurement noise was uncorrelated in time and space). Second, we left out some constraints in our algorithm. For instance, we do not consider localization problems of agents with real sensors: while an exact absolute position is not important for our algorithm, a correct _relative_ position is still necessary. This is an issue in the underwater environment, for instance. However, better positioning in challenging environments can often be enhanced using techniques such as Simultaneous Localization And Mapping (SLAM) [49; 50] using variants of Kalman filters [48] or Interval Analysis [51] to take position uncertainty into account; our work can integrate these methods. Third, robotic constraints on the motion of agents could be applied on our model, in particular for heading and speed or with a linearisation of the agents' trajectory [12]. Finally, we only used information in a scalar form. If we considered, when feasible, non-scalar information, resolving our problem could be easier. For example, we could measure a local gradient for function \(f\) to indicate the direction of its maximum. All of these limitations and possible solutions are left out for future work. Conceptualization, C.C. and A.A.; methodology, A.A. and P.-J.B.; software, C.C. ; validation, A.A. and P.-J.B.; formal analysis, C.C. ; investigation, C.C. ; writing--original draft preparation, C.C. ; writing--review and editing, A.A. and P.-J.B.; visualization, C.C. All authors have read and agreed to the published version of the manuscript. This research received external funding from ANRT and Brittany region. Not applicable. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Samples of the compounds are available from the authors. The following abbreviations are used in this manuscript: