A Conjecture on Fields of Extremals with Slopes Diverging (Open)

Summary. In the study of the extremals of a functional of the type $F(z)=\int_{0}^{T}L(t,z(t),z^{\prime }(t))dt$, the following situation often arises: the extremals $f_{\lambda }$ that satisfy $f_{\lambda } (0)=a$ and $f_{\lambda }^{\prime }(0)=\lambda$ constitute a central field, and the slope of the field at each point diverges, i.e., $\lim_{\lambda \rightarrow \pm \infty } f_{\lambda}^{\prime }(t)=\pm \infty $. Under these conditions, we conjecture that $\lim\limits_{\lambda \rightarrow \pm \infty} f_{\lambda}(T)=\pm \infty $ and, hence, that an extremal will exist for any condition at the end point.

Classification: Primary, Optimization; Secondary, Calculus of Variations

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