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arXiv · 2609.38728

On the NP-Hardness of Unconstrained Static Output Feedback Stabilization

Gal Barkai·Iman Shames
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Abstract

In this paper we give a polynomial-time reduction from the Subset Sum Problem to unconstrained static output feedback stabilization, establishing its NP-hardness. The construction uses two scalar building blocks to impose approximate discrete choices and a weighted-sum constraint. Once the problem is encoded, we relate the stability of the $N+1$ decoupled loops encoding the problem to that of a coupled plant. This is accomplished by leveraging frequency separation via a band-pass transformation, and comparing the root counts of the true characteristic polynomial with that of the different blocks in different regions of the right half-plane. The construction then naturally bounds every stabilizing gain and gives explicit modulus-margin estimates, ensuring that the plant data have polynomial binary encoding length.

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Gal Barkai, Iman Shames. 2026-09-30. On the NP-Hardness of Unconstrained Static Output Feedback Stabilization. https://arxiv.org/abs/2609.38728

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