In this paper we address the problem of determining convex conditions for the convex hull of ellipsoids to be robustly invariant. The motivation for the study is from the vertex control. Based on the convex hull of a set of vertices, necessary and sufficient conditions were suggested in the literature for the stability analysis and controller design problems of a constrained uncertain time-varying linear discrete-time system. However it is not trivial to compute the vertices, especially for high dimensional systems. In the present paper, we propose to replace the convex hull of vertices by the convex hull of ellipsoids. The associated ellipsoids are not required to be robustly invariant. The conditions are given in terms of linear matrix inequalities (LMIs). Hence the main drawback of vertex control is removed. The conditions are losslessly extended to stabilization, providing then an LMI solution to the non-linear state feedback control design problem of constrained uncertain time-varying linear discretetime systems. Several examples are included with comparison to earlier solutions from the literature to illustrate the results.