On control of degenerate distributed systems

We study the control of linear distributed control systems described by differential equations in Banach spaces with a degenerate operator at the derivative. A homogeneous part of equations has a degenerate strongly continuous resolving semigroup. For such system with generally speaking time-dependent bounded operator at the control function we find the criteria of the $\varepsilon$-control for time $T$ and of the $\varepsilon$-control in for a free time in terms of the operators involved in the equation. General results are used for studying of the $\varepsilon$-control of the considered systems with a finite-dimensional input. The obtained conditions are demonstrated by examples of control systems described by partial differential equations and systems of equations unsolved with respect to the time derivative.