In this paper, we consider the optimal disturbance rejection problem for
possibly infinite dimensional linear time-varying (LTV) systems using a
framework based on operator algebras of classes of bounded linear operators.
This approach does not assume any state space representation and views LTV
systems as causal operators. After reducing the problem to a shortest distance
minimization in a space of bounded linear operators, duality theory is applied
to show existence of optimal solutions, which satisfy a "time-varying" allpass
or flatness condition. Under mild assumptions the optimal TV controller is
shown to be essentially unique. Next, the concept of M-ideals of operators is
used to show that the computation of time-varying (TV) controllers reduces to a
search over compact TV Youla parameters. This involves the norm of a TV compact
Hankel operator defined on the space of causal trace-class 2 operators and its
maximal vectors. Moreover, an operator identity to compute the optimal TV Youla
parameter is provided. These results are generalized to the mixed sensitivity
problem for TV systems as well, where it is shown that the optimum is equal to
the operator induced of a TV mixed Hankel-Toeplitz. The final outcome of the
approach developed here is that it leads to two tractable finite dimensional
convex optimizations producing estimates to the optimum within desired
tolerances, and a method to compute optimal time-varying controllers.Comment: 30 pages, 1 figur