Minimal Reachability Problems

TL;DR

This paper investigates minimal actuator placement problems in linear systems, proving NP-hardness for certain reachability tasks and providing polynomial algorithms with approximation guarantees, demonstrated on large networks.

Contribution

It extends previous work by addressing minimal reachability with efficient algorithms and approximation guarantees, including NP-hardness proofs.

Findings

NP-hardness of minimal reachability problems

Polynomial algorithms with approximation guarantees

Successful application on large random networks

Abstract

In this paper, we address a collection of state space reachability problems, for linear time-invariant systems, using a minimal number of actuators. In particular, we design a zero-one diagonal input matrix B, with a minimal number of non-zero entries, so that a specified state vector is reachable from a given initial state. Moreover, we design a B so that a system can be steered either into a given subspace, or sufficiently close to a desired state. This work extends the recent results of Olshevsky and Pequito, where a zero-one diagonal or column matrix B is constructed so that the involved system is controllable. Specifically, we prove that the first two of our aforementioned problems are NP-hard; these results hold for a zero-one column matrix B as well. Then, we provide efficient polynomial time algorithms for their general solution, along with their worst case approximation guarantees. Finally, we illustrate their performance over large random networks.