  In this paper, we investigate the use of relaxed logarithmic barrier
functions in the context of linear model predictive control. We present results
that allow to guarantee asymptotic stability of the corresponding closed-loop
system, and discuss further properties like performance and constraint
satisfaction in dependence of the underlying relaxation. The proposed
stabilizing MPC schemes are not necessarily based on an explicit terminal set
or state constraint and allow to characterize the stabilizing control input
sequence as the minimizer of a globally defined, continuously differentiable,
and strongly convex function. The results are illustrated by means of a
numerical example.
