A distributed algorithm is described for finding a common fixed point of a family of m > 1 nonlinear maps M i : R n → R n assuming that each map is a paracontraction and that at least one such common fixed point exists. The common fixed point is simultaneously computed by m agents assuming each agent i knows only M i , the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agent estimates to converge to the same common fixed point of the m nonlinear maps.

A hybrid observer is described for estimating the state of an m > 0 channel, n-dimensional, continuous-time, distributed linear system of the forṁ x = Ax, yi = Cix, i ∈ {1, 2, . . . , m}. The system's state x is simultaneously estimated by m agents assuming each agent i senses yi and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. Agent i updates its estimate xi of x at "event times" t1, t2, . . . using a local observer and a local parameter estimator. The local observer is a continuous time linear system whose input is yi and whose output wi is an asymptotically correct estimate of Lix where Li a matrix with kernel equaling the unobservable space of (Ci, A). The local parameter estimator is a recursive algorithm designed to estimate, prior to each event time tj, a constant parameter pj which satisfies the linear equations w k (tj −τ ) = L k pj +µ k (tj −τ ), k ∈ {1, 2, . . . , m}, where τ is a small positive constant and µ k is the state estimation error of local observer k. Agent i accomplishes this by iterating its parameter estimator state zi, q times within the interval [tj − τ, tj), and by making use of the state of each of its neighbors' parameter estimators at each iteration. The updated value of xi at event time tj is then xi(tj) = e Aτ zi(q). Subject to the assumptions that (i) none of the Ci are zero, (ii) the neighbor graph N(t) is strongly connected for all time, (iii) the system whose state is to be estimated is jointly observable, (iv) q is sufficiently large and nothing more, it is shown that each estimate xi converges to x exponentially fast as t → ∞ at a rate which can be controlled.

A distributed algorithm is proposed for solving a linear algebraic equation Ax = b over a multi-agent network, where A ∈ Rn ×n and the equation has a unique solution x * ∈ R n . Each agent knows only a subset of the rows of A b , controls a state vector x i (t) of size smaller than n and is able to receive information from its nearby neighbors. Neighbor relations are characterized by time-dependent directed graphs. It is shown that for a large class of time-varying networks, the proposed algorithm enables each agent to recursively update its own state by only using its neighbors' states such that all x i (t) converge exponentially fast to a specific part of x * of interest to agent i. Applications of the proposed algorithm include solving the least square solution problem and the network localization problem.

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This paper presents a stochastic model for block arrival times based on the difficulty retargeting rule used in Bitcoin, as well as other proof-of-work blockchains. Unlike some previous work, this paper explicitly models the difficulty target as a random variable which is a function of the previous block arrival times and affecting the block times in the next retargeting period. An explicit marginal distribution is derived for the time between successive blocks (the blocktime), while allowing for randomly changing difficulty. This paper also aims to serve as an introduction to Bitcoin and proof-ofwork blockchains for the controls community, focusing on the difficulty retargeting procedure used in Bitcoin.