Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.29512

Dissipation Bounds and Hardness in Globally Asymptotically Stable Quadratic Systems

Subhransu S. Bhattacharjee
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study worst-case finite-time energy dissipation in quadratic systems with proved global asymptotic stability (GAS). Estimating the maximum retained energy fraction to any fixed positive additive error below $1/2$ is NP-hard, even when the input includes a strict quartic Lyapunov function and proof of strictness, both constructed in polynomial time. NP denotes nondeterministic polynomial time. For a partially damped family, the unit-sphere minimum norm of the coupling map equals the structured coefficient distance to loss of GAS. Positive distances at most $1/2$ give two-sided inverse-square halving-time bounds with explicit coefficient factors. Polynomial-factor approximation of the least dissipated fraction, halving time or coefficient distance is NP-hard. Additive hardness persists throughout an explicit coefficient neighborhood and under direct damping with exponential stability, a strict quadratic Lyapunov function and a Hurwitz Jacobian. Unless $\mathrm{P}=\mathrm{NP}$, where P denotes deterministic polynomial time, no polynomial-time algorithm attains these guarantees on all inputs.

Explore related subjects

eess.SYcs.CCcs.SYmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Subhransu S. Bhattacharjee. 2026-10-21. Dissipation Bounds and Hardness in Globally Asymptotically Stable Quadratic Systems. https://arxiv.org/abs/2610.29512

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimal Control of Behavioral-Feedback SIR Epidemic Model

We consider a behavioral-feedback SIR epidemic model, in which the infection rate depends in feedback on the fractions of susceptible and infected agents. Contrary to the classic SIR dynamics, the considered model allows one to account for endogenous adaptation mechanisms of the agents in response to the epidemics, such as voluntary social distancing, or the adoption of face masks. For this model, we formulate an optimal control problem for a social planner that has the ability to reduce the infection rate to keep the infection curve below a certain threshold within an infinite time horizon, while minimizing the intervention cost. Based on the dynamic properties of the model, we prove that, under quite general conditions on the infection rate, the filling-the-box strategy is the optimal control. This strategy consists in letting the epidemics spread without intervention until the threshold is reached, then applying the minimum control that leaves the fraction of infected individuals constantly at the threshold until the reproduction number becomes less than one and the infection naturally fades out. Our contribution enhances the understanding of epidemic management while accounting for adaptive human behavior, offering novel insights for robust containment strategies.

eess.SY↗
arXiv2026

The Cheap Hours Don't Stay Cheap: Price-Chasing Data Centers Reshape Grids and Markets

Data centers unsettle the paradigm of location-bound electricity demand: computing workloads can migrate across sites and hours, whereas facilities and grid connections remain fixed. This mobility reduces electricity costs but reshapes the spatiotemporal structure of nodal prices. A model-driven framework couples US wholesale market signals with optimal fleet scheduling, showing how their interaction alters prices and other participants' responses. Greater flexibility enlarges savings yet can erode network benefits: in a Texas study, temporal shifting and cross-site allocation each relieve congestion, but their combination leaves total congestion nearly unchanged. At continental scale, computing demand gravitating toward cheaper electricity narrows price differences and compresses the low-price hours storage arbitrage relies on, threatening storage investment recovery. Cost-minimizing scheduling may therefore undermine grid-supporting resources. Comparing job-level and aggregate fleet representations reveals a trade-off between operational visibility and information disclosure, informing requirements and incentives that could turn workload mobility into a lever for system-wide efficiency.

eess.SY↗
arXiv2026

Traffic State Prediction and Control Using a Koopman Operator Framework with Polynomial Observables

In this paper, we develop a Koopman operator framework for modeling and control of traffic flow governed by the Lighthill--Whitham--Richards (LWR) conservation law. We show that the nonlinear dynamics induced by the Greenshields flow--density relation naturally motivate a dictionary of polynomial observables, while the controlled dynamics become bilinear in the lifted Koopman coordinates. A reduced-order Koopman model is identified from spatio-temporal traffic-density snapshots and used as the prediction model in a model predictive control (MPC) framework for ramp metering. The resulting Koopman-MPC is compared with two predictive control baselines that use, respectively, a fixed linearized LWR model and the nonlinear finite-volume LWR model. In a one-hour freeway experiment, the Koopman-based controller maintains the traffic density below the critical Greenshields value and closely reproduces the behavior of the nonlinear PDE-based MPC, with a maximum-density difference below 1%. The Koopman predictor also reduces the mean online model-evaluation cost by approximately one order of magnitude relative to the nonlinear PDE predictor. In contrast, the linearized LWR predictor is computationally cheaper but exhibits substantially larger prediction errors and yields a more conservative metering policy. These results demonstrate that the analytical structure of the nonlinear traffic conservation law can be exploited to construct an interpretable data-driven surrogate for efficient traffic prediction and control

eess.SY↗