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A set of N independent Gaussian linear time-in variant systems is observed by M sensors whose task is to provide a steady-state causal estimate minimizing the mean-square error on the system states, subject to additional measurement costs. The sensors can switch between systems instantaneously, and there are additional resource constraints, for example on the number of sensors that can observe a given system simultaneously. We first derive a tractable relaxation of the problem, which provides a bound on the achievable performance. This bound can be computed by solving a convex program involving linear matrix inequalities, and moreover this program can be decomposed into coupled smaller dimensional problems. In the scalar case with identical sensors, we give an analytical expression of an index policy proposed in a more general context by Whittle. In the general case, we develop open-loop periodic switching policies whose performance matches the bound arbitrarily closely.