Controllability is investigated for a class of degenerate linear distributed control systems with the Caputo derivative such that the degeneracy subspace coincides with the kernel of the operator at the derivative. For such a system with a bounded independent of the time operator acting on the control function, criteria of e-controllability in time T and of e-controllability in free time are derived in terms of the operators from the considered differential equation. General results are used for the study of e-controllability of this type of systems with a finite-dimensional input. The obtained criteria are illustrated by examples of control systems described by partial differential equations and systems of equations not solvable with respect to the time-fractional derivative.
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