Controllability in Dimensions One and Two of Sobolev-Type Equations in Banach Spaces

We establish conditions necessary for $\varepsilon$-controllability in dimension one of first-order singular linear differential equation in Banach spaces. This result generalizes similar results for regular equations. For this class of equations, we show that the notion of $\varepsilon$-controllability in dimension two is more natural, and moreover, analogous necessary conditions are sufficient in the case of dimension two. Using an abstract approach, we derive sufficient conditions for the $\varepsilon$-controllability in dimension two of the Cauchy–Dirichlet problem for the Barenblatt–Zheltov–Kochina equation.