Start control problems for fractional order evolution equations

Unique solvability conditions are found for the Cauchy initial value problem to linear and nonlinear evolution equations with the Gerasimov — Caputo fractional derivatives in Banach spaces. For start control problems with various quality functionals to systems that described by such equations, solution existence theorems are proved, and in some linear cases the uniqueness of the problem solution is proved also. Abstract results are demonstrated on problems for the linearized Oskolkov — Benjamin — Bona — Mahony — Burgers equation and for the nonlinear equation of semiconductors metastable states.