This paper is concerned with γ-optimal control for linear discrete time varying systems on bounded time intervals driven by statistically uncertain random disturbances with nonzero mean values. The uncertainty is described in information theoretic terms using a previously introduced anisotropy functional. It is shown that, under additional constraints on the mean value and the covariance matrix of the input disturbance, the γ-optimal anisotropy-based controller design problem can be reduced to a multiobjective control problem. In comparison with the original anisotropy-based control setting, which uses the anisotropic norm of a system as a performance criterion, the suboptimal control approach, considered in the present paper, employs a modified version of the norm. The modified anisotropic norm is related to the original norm at a reduced anisotropy level and the H∞-norm of the system. Using this connection, we obtain sufficient conditions for γ-optimal anisotropy-based control. The resulting controller involves the solution of a certain convex optimization problem. A numerical example is presented in order to demonstrate the method proposed.