In a factorial design, simple effects refer to the effect of one factor at a specific level of another factor. The way these are compared depends on whether a control group is included.

• Regular factorial design (without a control): When no explicit control is defined, comparisons of simple effects are usually made between all levels of the factors. For example, in a 2×2 factorial, you would test whether the effect of Factor A is different at each level of Factor B, or vice versa. All levels are treated symmetrically, and interpretation focuses on interaction patterns.
• Factorial design with a control group: If one level of a factor is designated as a "control," then comparisons often emphasize whether each treatment condition differs from that control, rather than comparing all levels against each other. Statistically, this is often done with planned contrasts or dummy coding where the control serves as a reference. The analysis of simple effects still follows the same principles, but the interpretation changes: the focus is no longer only on the interaction but also on whether treatments deviate significantly from the control condition

Are you talking about a factorial design plus a control ?

If so, then, yes, this is usually analyzed with orthogonal contrasts to compare, e.g., the control vs. all the treatments; the simple effects in the treatments; and the interaction effect within the factorial.

I didn't read this paper, but it looks like it covers a few different approaches to analyze this kind of design: https://pure.psu.edu/en/publications/approaches-to-analyzing-experiments-with-factorial-arrangements-o