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<p>In this study, we present the concept of global observability for a specific class of linear time-fractional systems defined by the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-Caputo derivatives and characterized by a differentiation order between 1 and 2. We begin by defining the concept of global observability and examining its fundamental properties. Next, we introduce a method for determining the system’s state at <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( t=0 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> through an extension of the principles of the Hilbert uniqueness method (HUM). Notably, this approach operates without any prior knowledge of the initial state vector, and its main idea is to transform the reconstruction problem into a more manageable task, facilitating the development of an algorithm for state estimation. Furthermore, we provide practical demonstrations by varying the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\Psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation> function to substantiate our theoretical findings. The effectiveness of the proposed algorithm is validated through comprehensive numerical simulations presented in the concluding section.</p>

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Observability for a class of \(\Psi \)-Caputo time-fractional distributed-parameter linear systems

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Hamza Ben Brahim,
  • Khalid Zguaid,
  • Fatima-Zahrae El Alaoui

摘要

In this study, we present the concept of global observability for a specific class of linear time-fractional systems defined by the \(\Psi \) Ψ -Caputo derivatives and characterized by a differentiation order between 1 and 2. We begin by defining the concept of global observability and examining its fundamental properties. Next, we introduce a method for determining the system’s state at \( t=0 \) t = 0 through an extension of the principles of the Hilbert uniqueness method (HUM). Notably, this approach operates without any prior knowledge of the initial state vector, and its main idea is to transform the reconstruction problem into a more manageable task, facilitating the development of an algorithm for state estimation. Furthermore, we provide practical demonstrations by varying the \(\Psi \) Ψ function to substantiate our theoretical findings. The effectiveness of the proposed algorithm is validated through comprehensive numerical simulations presented in the concluding section.