44 research outputs found
Weighted Hardy spaces: shift invariant and coinvariant subspaces, linear systems and operator model theory
The Sz.-Nagy--Foias model theory for contraction operators
combined with the Beurling-Lax theorem establishes a correspondence between any
two of four kinds of objects: shift-invariant subspaces, operator-valued inner
functions, conservative discrete-time input/state/output linear systems, and
Hilbert-space contraction operators. We discuss an analogue of
all these ideas in the context of weighted Hardy spaces over the unit disk and
an associated class of hypercontraction operators
Discrete analytic Schur functions
We introduce the Schur class of functions, discrete analytic on the integer
lattice in the complex plane. As a special case, we derive the explicit form of
discrete analytic Blaschke factors and solve the related basic interpolation
problem
Schur analysis over the unit spectral ball
We define the corresponding Hardy space, Schur multipliers and their
realizations, and interpolation. Possible applications of the present work
include matrices of quaternions, matrices of split quaternions, and other
algebras of hypercomplex numbers
Optimal transport over a linear dynamical system
We consider the problem of steering an initial probability density for the state vector of a linear system
to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator () this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schroedinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases