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We develop a probabilistic control algorithm, $\texttt{GTLProCo}$, for swarmsof agents with heterogeneous dynamics and objectives, subject to high-leveltask specifications. The resulting algorithm not only achieves decentralizedcontrol of the swarm but also significantly improves scalability overstate-of-the-art existing algorithms. Specifically, we study a setting in whichthe agents move along the nodes of a graph, and the high-level taskspecifications for the swarm are expressed in a recently-proposed languagecalled graph temporal logic (GTL). By constraining the distribution of theswarm over the nodes of the graph, GTL can specify a wide range of properties,including safety, progress, and response. $\texttt{GTLProCo}$, agnostic to thenumber of agents comprising the swarm, controls the density distribution of theswarm in a decentralized and probabilistic manner. To this end, it synthesizesa time-varying Markov chain modeling the time evolution of the densitydistribution under the GTL constraints. We first identify a subset of GTL,namely reach-avoid specifications, for which we can reduce the synthesis ofsuch a Markov chain to either linear or semi-definite programs. Then, in thegeneral case, we formulate the synthesis of the Markov chain as a mixed-integernonlinear program (MINLP). We exploit the structure of the problem to providean efficient sequential mixed-integer linear programming scheme with trustregions to solve the MINLP. We empirically demonstrate that our sequentialscheme is at least three orders of magnitude faster than off-the-shelf MINLPsolvers and illustrate the effectiveness of $\texttt{GTLProCo}$ in severalswarm scenarios.

Probabilistic Control of Heterogeneous Swarms Subject to Graph Temporal Logic Specifications: A Decentralized and Scalable Approach