Abstract :
A discrete time invariant linear state/signal system Σ with a Hilbert state space X and a Kre˘ın signal
space W has trajectories (x(·),w(·)) that are solutions of the equation x(n + 1) = F x(n)
u(n) , where F is
a bounded linear operator from XW
into X with a closed domain whose projection onto X is all of X. This
system is passive if the graph of F is a maximal nonnegative subspace of the Kre˘ın space −X [ ]X [ ]W.
The future behavior Wfut of a passive system Σ is the set of all signal components w(·) of trajectories
(x(·),w(·)) of Σ on Z+ = {0, 1, 2, . . .} with x(0) = 0 and w(·) ∈ 2(Z+;W). This is always a maximal
nonnegative shift-invariant subspace of the Kre˘ın space k2(Z+;W), i.e., the space 2(Z+;W) endowed
with the indefinite inner product inherited from W. Subspaces of k2(Z+;W) with this property are called
passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future,
full, and past). In particular, we define and study the input and output maps of a passive state/signal system,
and the past/future map of a passive behavior.We then turn to the inverse problem, and construct two passive
state/signal realizations of a given passive future behavior W+, one of which is observable and backward
conservative, and the other controllable and forward conservative. Both of these are canonical in the sense
that they are uniquely determined by the given data W+, in contrast earlier realizations that depend not
only onW+, but also on some arbitrarily chosen fundamental decomposition of the signal space W. Fromour canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output
shift realizations of a given operator-valued Schur function in the unit disk.
© 2009 Elsevier Inc. All rights reserved
Keywords :
Output map , Past/future map , Kre??n space , Quotient space , Passive , Behavior , State/signal , Input map , De Branges–Rovnyak model , conservative
