People working in adaptive systems desire to have a fractional derivative having a characteristic similar to the classic: being null at an extremum. However and as it is well-known, such goal cannot be achieved with the most used derivatives, namely, Riemann-Liouville and Caputo. As an attempt to solving the problem, a conjecture is formulated: let f(t) be a piecewise continuous bounded function defined on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> having Fourier transform. Assume it has an extremum at <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$t=t_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Then, there are orders of derivative, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in \mathbb {R}^+,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and one asymmetry parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta \in [-2,2],$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> such that the unified fractional derivative, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$D^\alpha _{\theta } f(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>D</mi> <mi>θ</mi> <mi>α</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is null at the point <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_438_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$t_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. With this conjecture, it is shown that such objective is possible using two-sided derivatives. Therefore, we lose causality, but, in practice, such shortcoming will not impede its use, since the two-sided memory is not long.

Finding extrema using the unified fractional derivative: a conjecture

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发表日期：

2025年8月25日

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Manuel D. Ortigueira

摘要

People working in adaptive systems desire to have a fractional derivative having a characteristic similar to the classic: being null at an extremum. However and as it is well-known, such goal cannot be achieved with the most used derivatives, namely, Riemann-Liouville and Caputo. As an attempt to solving the problem, a conjecture is formulated: let f(t) be a piecewise continuous bounded function defined on $\mathbb {R},$ $R,$ having Fourier transform. Assume it has an extremum at $t=t_0$ $t = t_{0}$ . Then, there are orders of derivative, $\alpha \in \mathbb {R}^+,$ $α \in R^{+},$ and one asymmetry parameter $\theta \in [-2,2],$ $θ \in [- 2, 2],$ such that the unified fractional derivative, $D^\alpha _{\theta } f(t)$ $D_{θ}^{α} f (t)$ , is null at the point $t_0$ $t_{0}$ . With this conjecture, it is shown that such objective is possible using two-sided derivatives. Therefore, we lose causality, but, in practice, such shortcoming will not impede its use, since the two-sided memory is not long.