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Cooperative systems are systems in which the forces among agents arenon-repulsive. The free evolution of such systems can tend to the formation ofpatterns, such as consensus or clustering, depending on the properties andintensity of the interaction forces between agents. The kinetic cooperativesystems are obtained as the mean field limits of these systems when the numberof agents goes to infinity. These limit dynamics are described by transportpartial differential equations involving non-local terms. In this article, wedesign a simple and robust control strategy steering any kinetic cooperativesystem to approximate alignment. The computation of the control at each instantwill only require knowledge of the size of the support of the crowd in thephase space and of the Lipschitz constant of the interaction forces. Besides,the control we apply to our system is sparse, in the sense that it acts only ona small portion of the total population at each time. It also presents thefeatures of being obtained through a constructive procedure and to beindependent on the number of agents, making it convenient for applications.

Sparse Control of Kinetic Cooperative Systems to Approximate Alignment