Single-Boundary Control of the Two-Phase Stefan System

Shumon Koga; Miroslav Krstic

Home/Archive/arXiv

arXiv · 1905.12735

Single-Boundary Control of the Two-Phase Stefan System

Shumon Koga·Miroslav Krstic

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents the control design of the two-phase Stefan problem. The two-phase Stefan problem is a representative model of liquid-solid phase transition by describing the time evolutions of the temperature profile which is divided by subdomains of liquid and solid phases as the liquid-solid moving interface position. The mathematical formulation is given by two diffusion partial differential equations (PDEs) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) driven by the Neumann boundary values of both PDE states, resulting in a nonlinear coupled PDE-ODE-PDE system. We design a state feedback control law by means of energy-shaping to stabilize the interface position to a desired setpoint by using single boundary heat input. We prove that the closed-loop system under the control law ensures some conditions for model validity and the global exponential stability estimate is shown in $L_2$ norm. Furthermore, the robustness of the closed-loop stability with respect to the uncertainties of the physical parameters is shown. Numerical simulation is provided to illustrate the good performance of the proposed control law in comparison to the control design for the one-phase Stefan problem.

Explore related subjects

math.OC

Keep this discovery

BibTeX RIS

Shumon Koga, Miroslav Krstic. 2019-05-29. Single-Boundary Control of the Two-Phase Stefan System. https://arxiv.org/abs/1905.12735

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

KEEP EXPLORING

Related papers

arXiv2026

Deterministic and Random Bipartite Matching on General Networks: Convex Flow Reformulation, Asymptotic Properties, and Fast Algorithms

Minimum-distance bipartite matching on general networks has numerous applications various fields. This paper first focuses on deterministic problems and presents an exact edgewise-separable convex-flow reformulation. By introducing a smooth monotone rearrangement approximation of the edge-wise imbalance profiles, the convex-flow reformulation's can be solved efficiently. If we further conduct a first-order resistance-based approximation of the convex program, a one-step Laplacian-based estimator can be analytically derived in closed forms. The paper also studies random problems where supply and demand points are randomly distributed. We show that the expected optimal matching distance scales with the square root of the number of points if the supply/demand point distributions are identical, or linearly otherwise. In the former case, the optimal flow is proven to be centered, symmetric, and sub-Gaussian. In the latter case, the limiting resistance network characterizes how supply-demand imbalance is redistributed and motivates a fast algorithm that approximate the optimal flow based on the limiting resistance. Numerical experiments show that the proposed estimators closely approximate the exact matching cost while substantially reducing computation time. The proven theoretical properties of the random matching solution are numerically verified by large-scale Monte Carlo simulations.

math.OC↗

arXiv2026

Conformal-DRO: Distributionally Robust Optimization with Conformalized Ambiguity Set

Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.

math.OC↗

arXiv2026

The best approximation tuple: an extension of the Cheney-Goldstein algorithm and results to the multiple sets case

In this paper we extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is the consideration of more than two sets. The other is the ability to handle each set as an intersections of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to this generalizations is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. This enable us to overcome a certain theoretical obstacle related to cycles and minimizers of general functionals. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fej\'er monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

math.OC↗