As stated above, the sheer size of the network captured (implicitly) in our model (in excess of 10 reachable chemical species) has posed a barrier to simulation using conventional methods.
He said he expected the mayor's commissioners and deputy mayors to be reachable as well.
Define 0 < M ≤ 2 N as number of reachable system states, determined as soon as a state is repeated during the signal propagation.
First, a Hamilton Jacobi Isaas (HJI) partial differential equation (PDE) is constructed to describe the set of backward reachable states as a function of time starting from a target set of states.
If we consider a perturbation Δ = ( 0 0 0 0 0 δ 2 0 0 δ 1 0 0 0 ), the system ( A + B Δ, B ) has the same cone of positively reachable states as ( A, B ) and taking δ 1 < | I − A 1 | b 1 = 7 and δ 2 < | I − A 2 | b 2 = 4 we can ensure that the perturbed system ( A + B Δ, B ) is also asymptotically stable.
We first find matrices R and C. <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0002756.e058.PNG" class= inline-graphic"/> here comes, as before, from the states immediately reachable from green states.
An easy way of determining this is to make a new Markov chain Nabs with both e2 and e6 (more generally, states A−1 o)) set as absorbing states and see if one, both, or neither of these two absorbing states are reachable from each state in the chain.
An ergodic set of states C in a Markov chain is defined as a set of states where all states communicate and no state outside C is reachable from any state in Cd.
The hypothesis is that the final state μ G (D ) is a reachable state from the control state μ G (C ).
In total, there are 72 reachable states, with no end states.
This problem formulation becomes quickly intractable, as the number of reachable information states grows exponentially in the time horizon (Duff [2002]).
