Fixed-time impulsive control of a class of chaotic systems via the state observer

In this paper, an impulsive fixed-time control (IFTC) technique is developed to address the control problem of a class of chaotic systems, particularly relevant in secure communications and networked control systems, under the assumption that only the first state variable is measurable through potentially bandwidth-limited communication channels. A novel fixed-time stability theory with optimal fixed-time convergence characteristics is first established. Based on this theory, a fixed-time observer utilizing merely the first state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> is constructed, making it highly suitable for resource-constrained embedded computing platforms. Subsequently, a novel fixed-time impulsive control scheme is proposed, which effectively reduces both the communication burden (by minimizing required state feedback) and computational control costs (through sparse actuation). Finally, numerical simulation results validate the feasibility and rationality of the proposed theoretical findings, demonstrating its potential for efficient implementation in computer-based control systems and communication networks requiring precise timing and low overhead.