On the Controllability of Linear Sobolev Type Equations with Relatively Sectorial Operator

$ \varepsilon $-controllability of linear first order differential equations not resolved with respect to the time derivative $L \dot{x} (t) = Mx (t) + Bu (t), \quad 0<t<T$ are studied. It is assumed that $\ker L \ne \{0 \}$ and the operator $M$ is strongly $(L, p)$-sectorial. These conditions guarantee the existence of an analytic semigroup in the sector of the resolution of the homogeneous equation $ L \dot{x} (t) = Mx (t) $. Using the theory of degenerate semigroups of operators with kernels the original equation is reduced to a system of two equations: regular, i.e. solved for the derivative (on the image of the semigroup of the homogeneous equation) and the singular (on the kernel of the semigroup) with a nilpotent operator at the derivative. Using the results of $\varepsilon$-controllability of the regular and singular equations, necessary and sufficient conditions of $\varepsilon $-controllability of the original equation of Sobolev type with respect to $p$-sectorial operator in terms of the operators are obtained. Abstract results are applied to the study of $\varepsilon$-controllability of a particular boundary-value problem, which is the linearization at zero phase–field equations describing the theory in the framework of mesoscopic phase transition.