Search Theory is the field which deals with maximizing the detection of one or more targets, by optimizing the placement of detection resources. The optimization becomes particularly uneasy for moving targets, since it involves the full detection history. Combinatorial approaches are used for a small number of resources. For numerous resources, a continuous approach has been introduced by B.O. Koopman. And the resource optimization has been solved, for a target with a Markovian probabilistic prior. However, the evolution in time and the interactions of the resources remain difficult. Starting from the classical algorithm of Brown, we proposed an extension of the continuous approach involving the temporal behavior of the resources and their interaction. There is an algebraic framework associated with these new problems, and a description language has thus been defined. At last, a new modeling of the target prior is proposed, yielding a minimax optimization of the resources. This approach, involving a set of the available moving strategies of the target, avoids some limitations associated with the Markovian probabilistic prior.