<p>This paper addresses the regional boundary observability of linear time-fractional systems characterized by the Hilfer fractional derivative of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha \in ]0,1[,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> with type <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0 \le \beta \le 1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>β</mi> <mo>≤</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The primary objective is to reconstruct the initial state of the system within a subregion <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> of the boundary <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> of the evolution domain <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. To achieve this, we establish the connection between regional boundary observability on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and regional observability in a suitable subregion <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ \omega _\texttt{r} \subset \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi mathvariant="monospace">r</mi> </msub> <mo>⊂</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, defined such that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ \Gamma \subset \partial \omega _\texttt{r} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>⊂</mo> <mi>∂</mi> <msub> <mi>ω</mi> <mi mathvariant="monospace">r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. This method enables the recovery of the initial state within the subregion <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$ \omega _\texttt{r} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mi mathvariant="monospace">r</mi> </msub> </math></EquationSource> </InlineEquation> by utilizing an extended version of the Hilbert Uniqueness Method (HUM) tailored for fractional systems. Following this, the trace on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$ \partial \omega _\texttt{r} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <msub> <mi>ω</mi> <mi mathvariant="monospace">r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is restricted to <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$ \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, allowing the determination of the initial state on <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$ \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. We also propose an algorithm to recover the initial state in <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$ \omega _\texttt{r} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mi mathvariant="monospace">r</mi> </msub> </math></EquationSource> </InlineEquation> and ultimately on <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$ \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, achieving a satisfactory reconstruction error. Notably, the reconstruction error associated with the initial state is remarkably low, affirming the efficacy of the approach employed in this investigation.</p>

Regional boundary observability of time-fractional systems with Hilfer derivatives of order $\alpha $ and type $\beta $

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发表日期：

2026年5月25日

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

摘要

This paper addresses the regional boundary observability of linear time-fractional systems characterized by the Hilfer fractional derivative of order $\alpha \in ]0,1[,$ $α \in] 0, 1 [,$ with type $0 \le \beta \le 1.$ $0 \leq β \leq 1 .$ The primary objective is to reconstruct the initial state of the system within a subregion $ \Gamma $ $Γ$ of the boundary $ \partial \Omega $ $\partial Ω$ of the evolution domain $ \Omega $ $Ω$ . To achieve this, we establish the connection between regional boundary observability on $ \Gamma $ $Γ$ and regional observability in a suitable subregion $ \omega _\texttt{r} \subset \Omega $ $ω_{r} \subset Ω$ , defined such that $ \Gamma \subset \partial \omega _\texttt{r} $ $Γ \subset \partial ω_{r}$ . This method enables the recovery of the initial state within the subregion $ \omega _\texttt{r} $ $ω_{r}$ by utilizing an extended version of the Hilbert Uniqueness Method (HUM) tailored for fractional systems. Following this, the trace on $ \partial \omega _\texttt{r} $ $\partial ω_{r}$ is restricted to $ \Gamma $ $Γ$ , allowing the determination of the initial state on $ \Gamma $ $Γ$ . We also propose an algorithm to recover the initial state in $ \omega _\texttt{r} $ $ω_{r}$ and ultimately on $ \Gamma $ $Γ$ , achieving a satisfactory reconstruction error. Notably, the reconstruction error associated with the initial state is remarkably low, affirming the efficacy of the approach employed in this investigation.

Regional boundary observability of time-fractional systems with Hilfer derivatives of order \(\alpha \) and type \(\beta \)

摘要