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Abstract The Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.
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This content will become publicly available on February 1, 2026
Practical considerations for implementing robust-to-early termination model predictive control
Model Predictive Control (MPC) is widely used to achieve performance objectives, while enforcing operational and safety constraints. Despite its high performance, MPC often demands significant computational resources, making it challenging to implement in systems with limited computing capacity. A recent approach to address this challenge is to use the Robust-to-Early Termination (REAP) strategy. At any time instant, REAP converts the MPC problem into the evolution of a virtual dynamical system whose trajectory converges to the optimal solution, and provides guaranteed sub-optimal and feasible solution whenever its evolution is terminated due to limited computational power. REAP has been introduced as a continuous-time scheme and its theoretical properties have been derived under the assumption that it performs all the computations in continuous time. However, REAP should be practically implemented in discrete-time. This paper focuses on the discrete-time implementation of REAP, exploring conditions under which anytime feasibility and convergence properties are maintained when the computations are performed in discrete time. The proposed methodology is validated and evaluated through extensive simulation and experimental studies.
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- Award ID(s):
- 2244082
- PAR ID:
- 10601165
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Systems & Control Letters
- Volume:
- 196
- Issue:
- C
- ISSN:
- 0167-6911
- Page Range / eLocation ID:
- 106018
- Subject(s) / Keyword(s):
- Model predictive control Robust-to-early termination Discrete-time implementation Limited computing capacity
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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