An investigation of control on the interaction of Airy-Gaussian beams (AiGBs) in cubic focusing and quintic defocusing competing nonlinear medium is performed by the split-step Fourier transform method. When the initial launch angle v is zero, two in-phase AiGBs attract each other and the breathing soliton with decaying period or soliton with invariant intensity can form in the case of weaker quintic defocusing nonlinearity. However, the interaction between two in-phase AiGBs under stronger quintic defocusing causes the average width of the breathing soliton to increase and even the beam bifurcation to occur, leading to the generation of soliton pairs. For the out-of-phase case, they repel each other, and the repulsive force increases monotonically with the increase of the quintic defocusing nonlinearity. When the initial launch angle for each of AiGBs is not zero, mutual attraction and repulsion can be exhibited during their interactions by adjusting the sign of v and the interval d. For the in-phase case, if v < 0 and d < 0 or v > 0 and d > 0, there are strong repulsive force and weak attraction between the two AiGBs, resulting in the formation of soliton pairs, and with the decrease of the interval, the attraction becomes greater. When the interval is small enough, the overlapping of the light field can make the nonlinear effect identical to the diffraction effect, the attraction between the two AiGBs increases, while the repulsion force is almost zero, and then a single breathing soliton can be generated in the center of the two AiGBs. If v < 0 and d > 0 (big enough) or v > 0 and d < 0, the constructive interference between two AiGBs causes the autofocusing beams first to be generated, then to repel each other, and the soliton pairs can form. For the out-of-phase case, if v < 0 and d < 0 or v > 0 and d > 0, the repulsion between the two AiGBs becomes bigger, and the repulsion increases with | v| monotonically. If v < 0 and d > 0 or v > 0 and d < 0, the elastic collision between the two AiGBs shows the phenomenon: first attracting and then repelling mutually. When both v and d are small enough, soliton pairs cannot form due to the unbalance between the strong diffraction effect and weaker nonlinear effect induced by the destructive interference.