Partially Exact Controllability of Semilinear Heat Exchanger Systems

TL;DR

The paper investigates partial exact controllability of semilinear heat exchanger PDEs, showing that two representative models (monotubular and two-stream) are not exactly controllable but are partially exactly controllable on suitably restricted state spaces. It develops a general abstract framework giving a sufficient condition for L-partial exact controllability in semilinear systems and applies it by constructing subspaces $X_\varepsilon$ (and $X^2_\varepsilon$) where the restricted controllability Gramian $Q_{L_\varepsilon}(t)$ is coercive and satisfies a uniform bound on $t\|Q_{L_\varepsilon}^{-1}(t)\|$. For each model, the authors prove $L_\varepsilon$-partially exact controllability to the domain $D(A)$ (or its projection) for all $0<\varepsilon<1/2$, enabling constructive steering to terminal states on $[\varepsilon,1-\varepsilon]$. This provides a principled tool for designing controls in heat-exchanger settings and suggests extensions to multi-fluid configurations.

Abstract

In this paper, we study two semilinear systems describing a monotubular and a two-stream heat exchanger. Neither system is exactly controllable; however, for each we specify a subspace of the state space with respect to which the system is exactly controllable, thus establishing partial exact controllability.

Theorems & Definitions (7)

• Theorem 2.1
• Definition 3.1
• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Theorem 3.3
• proof