Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz's variational problem and when in presence of time delays are also considered.

Automated summary

In this work, we investigate variational problems involving the use of generalized proportional fractional derivatives in place of ordinary derivatives. These fractional derivatives depend on a fixed parameter, which acts as a weight on the state function and its first-order derivative. We examine the problems with and without boundary conditions, and explore additional constraints such as isoperimetric and holonomic conditions. We also consider Herglotz's variational problem and the presence of time delays.