Control Problems for One-Dimensional Fluids and Reactive Fluids with Moving Interfaces

Abstract

Abstract. The purpose of this paper is to expose several recent challenging control problems for mono-dimensional fluids or reactive fluids. These problems have in common the existence of a moving interface separating two spatial zones where the dynamics are rather different. All these problems are grounded on topics of engineering interest. The aim of the author is to expose the main control issues, possible solutions and to spur an interest for other future contributors. As will appear, mobile interfaces play key roles in various problems, and truly capture main phenomena at stake in the dynamics of the considered systems.

Delay compensated control of the Stefan problem and robustness to delay mismatch

“…Under Assumptions 1 and 3, and assuming that Δ ≥ 0 and Δ ≥ 0, there exists a positive constant R > 0 such that if Δ 2 + Δ 2 < R then the closed-loop system consisting of the plant (155) to (158) and the control law (15) maintains the model validity (6) and is exponentially stable in the sense of the norm (16).…”

“…Motion planning boundary control has been adopted in Reference to ensure asymptotic stability of a one‐dimensional (1D) one‐phase nonlinear Stefan problem assuming a prior known moving boundary and deriving the manipulated input from the solutions of the inverse problem. However, the series representation introduced in Reference leads to highly complex solutions that reduce controller design possibilities.…”

“…While the numerical analysis of the one‐phase Stefan problem is broadly covered in the literature, their control‐related problems have been addressed relatively fewer. In addition to it, most of the proposed control approaches are based on finite‐dimensional approximations with the assumption of an explicitly given moving boundary dynamics . Diffusion‐reaction processes with an explicitly known moving boundary dynamics are investigated in based on the concept of inertial manifold and the partitioning of the infinite dimensional dynamics into slow and fast finite dimensional modes.…”

“…In addition to it, most of the proposed control approaches are based on finite-dimensional approximations with the assumption of an explicitly given moving boundary dynamics. [14][15][16] Diffusion-reaction processes with an explicitly known moving boundary dynamics are investigated in 15 based on the concept of inertial manifold 17 and the partitioning of the infinite dimensional dynamics into slow and fast finite dimensional modes. Motion planning boundary control has been adopted in Reference 16 to ensure asymptotic stability of a one-dimensional (1D) one-phase nonlinear Stefan problem assuming a prior known moving boundary and deriving the manipulated input from the solutions of the inverse problem.…”

Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

“…While the numerical analysis of the one-phase Stefan problem is broadly covered in the literature, their control related problems have been addressed relatively fewer. In addition to it, most of the proposed control approaches are based on finite dimensional approximations with the assumption of an explicitly given moving boundary dynamics [10], [1], [26]. Diffusion-reaction processes with an explicitly known moving boundary dynamics are investigated in [1] based on the concept of inertial manifold [8] and the partitioning of the infinite dimensional dynamics into slow and fast finite dimensional modes.…”