CONSIDER the differential equation dx dt = A(t)x + B(t)u (1) where x = ( x 1,…, x n ), u = ( u 1,…, u r ) are elements of n- and r-dimensional Euclidean spaces respectively, and A( t), B( t) are n × n and n × r matrices respectively. Let u( t) ϵ M ⊂ E r when 0 ⩽ t ⩽ μ, where M is a set containing at least one point u 0 of spaces E r in its interior. We fix a point x 0, and select a control u( t) to which the solution x( t) of (1) corresponds when x(0) = x 0. If we now apply all the possible controls, the point x( μ) will describe a set which will be called the attainability set and written as A( μ, x 0). We shall be discussing the same topic for attainability sets as in [1], namely, the questions of when the set has an interior point and what radius, compared with μ, a sphere may have that is contained in the set (or the length of rib of an interior parallelepiped). We investigated in [1] the attainable set A( μ, x 0) in the case of a non-linear set of differential equations, when 1 ⩽ n r ⩽ 2 , n being the number of dimensions of the phase vector x, and r the number of dimensions of the vector u; in addition, we considered a linear set of differential equations with constant coefficients, with no restrictions on the ratio between n and r. We consider below a linear set of differential equations with variable coefficients, again with no restrictions on n r . It turns out that one of the most important conditions for the continuity of Bellman's function is that the set A( μ, x 0) contain a sphere, centre x 0. This set property is later used to prove the solvability of the elementary operation introduced in [2], when proving the convergence of the total sampling method. The conditions under which the set A( μ, x 0) has the properties discussed below, are the same as the conditions for complete controllability of the set (1). These latter conditions were obtained in [3]. They are obtained below as a consequence of the theorem concerning the properties of A( μ, x 0). In addition, it is shown that, for complete controllability, only piecewise-constant controls need be considered.