The conventional friction bearings have limitations in controlling the structures’ vibration. To overcome their limitations and upgrade their seismic performances, the inertial amplifiers and inerters are applied to the core material of the conventional friction bearings. Accordingly, this paper introduces two kinds of upgraded friction bearings: inertial amplifier friction bearings and inerter-based friction bearings. These upgraded friction bearings are installed at the base of the single and multi-story buildings with an adjacent retaining wall. The impact between the building and the retaining wall is considered. Following Newton’s second law, the governing equations of motion for the isolated structures, including the impact, are derived. The impact is formulated by the signum function to derive analytical optimal closed-form solutions for the design parameters of these upgraded base isolators. H2 and H∞ optimization methods are applied to derive the exact closed-form expression for the optimal design parameters. To employ the H2 optimization method, the statistical linearization method is applied to linearize each nonlinear element of the governing equations of motion. Parametric studies show that optimum frequency and damping ratios decrease with increasing isolator mass ratio, increase with increasing inertial angle, and decrease with increasing isolator mass ratio. Transfer function development is the first step in obtaining dynamic reactions of isolated structures. Furthermore, Newmark-beta method is employed to validate the results of the frequency domain analysis and obtained dynamic response histories for the isolated single-degree-of-freedom systems. According to the results, the dynamic response reduction capacities of the inerter-based friction bearing and inertial amplifier friction bearing are significantly 97.16% and 96.62% superior to the conventional base isolators. All results are mathematically derived and accurate; hence, applicable to practical implementations.