Optimal control problems for certain linear fractional-order systems given by equations with Hilfer derivative

Two optimal control problems are investigated for linear time-invariant systems of fractional order with lumped parameters, which dynamics is described by equations with Hilfer derivative: control problem with minimal norm and time-optimal control problem with control norm constraint. Controls are considered that are the $p$-integrable or essentially bounded functions. Investigation is conducted by the method of moments. Correctness and solvability conditions of the problem of moments are obtained for the problem statement considered. The optimal control problems stated are solved analytically for several particular cases and the properties of the solutions are investigated depending on fractional differentiation indices and Hilfer fractional differential operator parameters. The comparison is conducted of the results obtained with the known results for integer-order systems and fractional-order systems described by equations with Riemann–Liouville or Caputo derivative.