Cusp exponent for the amplitude-augmented functional

Determine how adding the amplitude term \(\lambda_3\int u^p/p\) to the functional \(\int_0^{e_0}(\lambda_1|u'|^2/2+\lambda_2/u)\,e\) affects the cusp exponent of the minimizing dissipation schedule.

Background

The paper completely solves the regularized least-time problem with Dirichlet roughness penalty and reciprocal-time cost, obtaining a unique cycloidal minimizer whose behavior near zero risk is $u^*(e)\sim (9/(2\lambda))^{1/3}e^{2/3}$. It also discusses a more general functional from prior work that includes an additional nonnegative amplitude term $\lambda_3\int u^p/p$.

Although the existence argument is stated to extend to this amplitude-augmented functional, the authors do not obtain a closed-form identification of the minimizer or determine whether the additional term changes the near-zero cusp exponent. Establishing that effect is therefore explicitly left unresolved.