Infinite-dimensional and finite-dimensional $\varepsilon$-controllability for a class of fractional order degenerate evolution equations

Issues of $\varepsilon$-controllability is researched for linear weakly degenerate fractional order evolution control systems with distributed parameters. The case of 0-bounded pair of operators in the system is considered. Using the generalized Showalter — Sidorov conditions instead of the Cauchy conditions significantly simplified the technical part of the study. Criteria and convenient in applications sufficient conditions of the $\varepsilon$-controllability in time $T$ and of the $\varepsilon$-controllability in free time are derived for this type systems in the cases of infinite-dimensional and finite-dimensional input. It is shown that for the finite-dimensional $\varepsilon$-controllability of the system finite dimensionality of its degeneracy subspace is necesarry. The obtained results are illustrated by examples of control systems described by differential equations and systems of equations not solvable with respect to the time-fractional derivative.