Ensuring stability, feasibility, robustness, and online tractability when integrating learned components into nonlinear control

Determine conditions and develop methods that guarantee stability, recursive feasibility, robustness, and online computational tractability for nonlinear control systems when incorporating learned components (such as learned dynamics models, cost functions, or constraints) into Model Predictive Control and related learning-based controllers, in order to reduce conservatism without sacrificing formal guarantees.

Background

The paper highlights that robust and stochastic MPC formulations can be overly conservative, limiting performance, and that learning-based approaches (e.g., differentiable MPC) can adapt constraints and costs from data to reduce conservatism. However, integrating learned elements into nonlinear control raises questions about maintaining feasibility, stability, robustness, and real-time solvability.

This open problem targets the theoretical and algorithmic foundations needed to endow learning-enhanced MPC formulations for nonlinear systems with the same rigorous guarantees traditionally available for purely model-based controllers, while ensuring practical online tractability.

References

However, ensuring stability, feasibility, robustness, and online computational tractability while incorporating learned components for nonlinear systems remains an open question.

— Safe Physics-Informed Machine Learning for Dynamics and Control  (2504.12952 - Drgona et al., 17 Apr 2025) in Section 6: Challenges and Opportunities

Characterizing $X(\ell)$ without knowing $0,$ is still an open challenge, and it is out of the scope of the current paper.

— Asynchronous Model Predictive Control Under Model Mismatch: Stability and Performance Guarantees  (2609.08836 - Liu et al., 8 Sep 2026) in Section 2.2, immediately following Definition 1 (Cost-induced ROA)

Learning-based control is developing rapidly, but raises questions regarding stability guarantees and certification that are still open.

— Dynamic Inversion: An Incrementally Evolving Methodology for Flight Control Design  (2608.23229 - Milz et al., 24 Aug 2026) in Conclusions

It is still unclear how much of the recent success in combining Koopman operator theory and MPC is due to the approximation capabilities of linear operators, as compared to the robustness of MPC to model-plant mismatch .

— Model predictive traction control system based on the Koopman operator  (2608.25753 - Hromatko et al., 26 Aug 2026) in Section “Results and discussion,” subsection “Comments”

Finally, developing a stronger theoretical understanding of how solver guidance interacts with stochastic policy-gradient optimization remains an important open research direction.

— Accelerating Reinforcement Learning via MPC Solver-Gradient Guidance for Weights-varying MPC  (2609.01061 - Zarrouki et al., 1 Sep 2026) in Future work, Conclusions

Sensitivity to reward design, hyperparameters and scenario shift is documented but unresolved , degradation under non-stationary conditions has been observed in the field , and simplified occupancy or load assumptions are known to inflate simulated performance .

— Artificial Intelligence for Energy Optimization in Data Centers  (2609.03716 - Ullah et al., 3 Sep 2026) in Section II, subsection “Deep Reinforcement Learning for Cooling,” Gap Identification

Reducing this collision rate, whether through the cost tuning above or by feeding the model's predictive uncertainty back into the planner as a runtime safety signal, remains open work.

— GPU-Accelerated Astrodynamics World Models for Spacecraft Rendezvous and Proximity Operations  (2609.03067 - Eddy et al., 2 Sep 2026) in Section Results, subsection 4.2, “Docking Performance”

A full nonlinear theory combining the present collective calibration idea with such nonlinear tube constructions is therefore left as an open problem.

— Collective Tube Model Predictive Control With Distribution-Free Joint Safety Certificates  (2609.26339 - Calafiore, 22 Sep 2026) in Introduction, paragraph beginning “We adopt linear dynamics in the main development”

Formal transfer of these certificates to the learning phase, in which model, cost, constraints and terminal cost evolve online, remains an open theoretical question; in this work it is addressed pragmatically through a two-timescale update strategy (fast ENMPC updates per step, slow RTO updates per episode) and empirically verified on the rig.

— Two-Timescale Reinforcement Learning for Real-Time Optimization and Economic NMPC: Experimental Validation  (2609.35141 - Adhau et al., 28 Sep 2026) in Section 5, Conclusion

The proposed framework is evaluated entirely through numerical experiments, and no theoretical guarantees of recursive feasibility, constraint satisfaction, or closed-loop stability are established. The numerical studies are used to investigate whether the online-identified DMAC dynamics can provide a useful prediction model for reference governance across systems of differing complexity. A theoretical treatment of the resulting time-varying prediction model and associated feasibility and stability properties is left for future work.

— Constraint Enforcement via Reference Governor in Dynamic Mode Adaptive Control  (2610.12342 - Oveissi et al., 8 Oct 2026) in Introduction, final paragraphs before Section 2; Section 1; Conclusion

Future work will develop conditions under which feasibility can be guaranteed for the reference governor under general indirect adaptive control, as the identified model evolves online.

— Constraint Enforcement via Reference Governor in Dynamic Mode Adaptive Control  (2610.12342 - Oveissi et al., 8 Oct 2026) in Conclusion