Adaptive output tracking for 1-D unstable heat equations with unknown multi-frequency exosystem

• Published in: Communications in nonlinear science & numerical simulation Vol. 152; p. 109142
• Main Authors: Chen, Jun, Wang, Junmin, Yu, Xiufang
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2026
• Subjects: Adaptive control; Exponential stability; Heat equation; Output tracking; Unknown harmonic signals; Unknown harmonic signals; Output tracking; Heat equation; Exponential stability; Adaptive control
• ISSN: 1007-5704
• DOI: 10.1016/j.cnsns.2025.109142

In this paper, we address the adaptive output tracking problem for a 1-D unstable heat equation with non-collocated configurations. The system is subject to disturbances and reference signals generated by a finite-dimensional exosystem, which possess multiple unknown frequencies and amplitudes. By constructing an auxiliary system and internal model dynamics, we restructure the measurable tracking error composed of available states. An adaptive method is proposed to estimate the uncertain parameters. We design an error-based adaptive dynamic compensator to achieve exponential output tracking using the backstepping method. A numerical example is provided to illustrate the effectiveness of the proposed strategy. •Independent multi-channel handling of unknown multi-frequency signals.•Control of non-conservative systems with boundary instability.•Full-channel multi-frequency composite disturbance rejection.•Exponential convergence under unbounded operator framework.