Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion

TL;DR

This paper develops new dimension reduction techniques for large-scale controlled stochastic differential equations with non-zero initial states, providing error bounds and strategies for separate control and initial state reduction.

Contribution

It introduces two novel strategies for reducing the dimension of controlled stochastic systems with non-zero initial states, including error bounds and the use of Hankel singular values.

Findings

Transformation to zero initial states simplifies analysis.

Error bounds are derived using Hankel singular values.

Separate reduction of control and initial states is effective.

Abstract

In this paper, we establish new strategies to reduce the dimension of large-scale controlled stochastic differential equations with non-zero initial states. The first approach transforms the original setting into a stochastic system with zero initial states. This transformation naturally leads to equations with controlled diffusion. A detailed analysis of dominant subspaces and bounds for the reduction error is provided in this controlled diffusion framework. Subsequently, we introduce a reduced system for the original framework and prove an a-priori error bound for the first ansatz. This bound involves so-called Hankel singular values that are linked to a new pair of Gramians. A second strategy is presented that is based on the idea of reducing control and initial state dynamics separately. Here, different Gramians are used in order to derive a reduced model and their relation to dominant subspaces are pointed out. We also show an a posteriori error bound for the second approach involving two types of Hankel singular values.