<p>This paper discusses the finite-time synchronization (FTS) for a class of fractional-order fuzzy neural networks (FOFNNs). First, a general fractional differential inequality is developed, which provides new approaches for dealing with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3357_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>1</mn> </msub> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$l_{1}f^{-\beta}(t), (\beta >0)$</EquationSource> </InlineEquation>. Furthermore, a nonlinear fractional-order finite time inequality <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3357_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="244" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>D</mi> <mi>t</mi> <mi>α</mi> <mprescripts /> <msub> <mi>t</mi> <mn>0</mn> </msub> <mi>c</mi> </mmultiscripts> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo>−</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <msup> <mi>V</mi> <mrow> <mo>−</mo> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <msup> <mi>V</mi> <mrow> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$_{t_{0}}^{c}D_{t}^{\alpha}V(t)\leq -l_{1}V^{-\beta}(t)-l_{3}V^{- \gamma}(t)$</EquationSource> </InlineEquation> is also established. Next, three controllers are designed, namely two feedback nonlinear controllers and adaptive controller. Then, using mathematical analysis and the newly established inequality, several easily-validated algebraic conditions are derived to guarantee the FTS of FOFNNs. Moreover, the paper effectively estimates the upper bound of the settling time for FTS. Finally, a numerical example is provided to demonstrate the validity of the theoretical results.</p>

Finite-time synchronization of fuzzy neural networks with fractional order

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2025年9月25日

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Libo Wang,
Guigui Xu,
Pan Wang

摘要

This paper discusses the finite-time synchronization (FTS) for a class of fractional-order fuzzy neural networks (FOFNNs). First, a general fractional differential inequality is developed, which provides new approaches for dealing with $l_{1} f^{- β} (t), (β > 0)$ $l_{1}f^{-\beta}(t), (\beta >0)$ . Furthermore, a nonlinear fractional-order finite time inequality ${}_{t_{0}}^{c}D_{t}^{α} V (t) \leq - l_{1} V^{- β} (t) - l_{3} V^{- γ} (t)$ $_{t_{0}}^{c}D_{t}^{\alpha}V(t)\leq -l_{1}V^{-\beta}(t)-l_{3}V^{- \gamma}(t)$ is also established. Next, three controllers are designed, namely two feedback nonlinear controllers and adaptive controller. Then, using mathematical analysis and the newly established inequality, several easily-validated algebraic conditions are derived to guarantee the FTS of FOFNNs. Moreover, the paper effectively estimates the upper bound of the settling time for FTS. Finally, a numerical example is provided to demonstrate the validity of the theoretical results.