Coordinated missions with platoons of autonomous agents are rapidly becoming popular because of technological advances in computing, networking, miniaturization and combination of electromechanical systems. These multi-agents networks coordinate their actions to perform challenging spatially-distributed tasks such as search, survey, exploration, and mapping. Environmental monitoring and locational optimization are among the main applications of the emerging technology of wireless sensor networks where the optimality refers to the assignment of sub-regions to each agent, in such a way that a suitable coverage metric is maximized. Usually the coverage metric encodes a distribution of risk defined on the area, and a measure of the performance of individual robots with respect to points inside the region of interest. The risk density can be used to quantify spatial distributions of risk in the domain. The solution of the optimal control problem in which the risk measure is not time varying is well known in the literature, with the optimal con figuration of the robots given by the centroids of the Voronoi regions forming a centroidal Voronoi tessellation of the area. In other words, when the set of mobile robots converge to the corresponding centroids of the Voronoi tessellation dictated by the coverage metric, the coverage itself is maximized. In this work, it is considered a time-varying risk density evolving according to a diffusion equation with varying boundary conditions that quantify a time-varying risk on the border of the workspace. Boundary conditions model a time varying flux of external threats coming into the area, averaged over the boundary length, so that rather than considering individual kinematics of incoming threats it is considered an averaged, distributed effect. This approach is similar to the one commonly adopted in continuum physics, in which kinematic descriptors are averaged over spatial domain and suitable continuum fields are introduced to describe their evolution. By adopting a first gradient constitutive relation between the flux and the density, a simple diffusion equation is obtained. Asymptotic convergence and optimality of the non-autonomous system are studied by means of Barbalat's lemma and connections with varying boundary conditions are established. Some criteria on time-varying boundary conditions and evolution are established to guarantee the stabilities of agents' trajectories. A set of numerical simulations illustrate theoretical results.

Multi-agent systems are extensively used in several applications. An important class of applications involves the optimal spatial distribution of a group of mobile robots on a given area, where the optimality refers to the assignment of subregions to the robots, in such a way that a suitable coverage metric is maximized. Typically the coverage metric encodes a risk distribution defined on the area, and a measure of the performance of individual robots with respect to points inside the region of interest. The coverage metric will be maximized when the set of mobile robots configure themselves as the centroids of the Voronoi tessellation dictated by the risk density. In this work we advance on this result by considering a generalized area control problem in which the coverage metric is non-autonomous, that coverage metric is time varying independently of the states of the robots. This generalization is motivated by the study of coverage control problems in which the coordinated motion of a set of mobile robots accounts for the kinematics of objects penetrating from the outside. Asymptotic convergence and optimality of the non-autonmous system are studied by means of Barbalat's Lemma, and connections with the kinematics of the moving intruders is established. Several numerical simulation results are used to illustrate theoretical predictions.