How to set steady state values of endogenous variables when solving for deterministic model

I am working on a very simple deterministic RBC model with government to see how the economy transits under a permanent improvement of technology A_t to 1.1 from 1.
In the model, government spending Gt follows a simple rule, i.e. it is assumed that Gt reacts to the economic conditions:log(Gt )=ρ_G *log(G(t-1) )+(1-ρ_G )(log(Gs)+φY_G *log(Yt/Ys)). So the steady state values of government spending(Gt) and output(Yt) appears in the model and these steady state values are viewed as parameters. When technology improve permanently, the initial steady state values of government spending and output are different from the end values.
My question is how to set these parameters(steady state values of endogenous variables) to gurantee residuals of the static equations equal to 0? the initial steady state values or the end steady state values ?

You need to be careful here. As you are building a model with permanent shocks, there will be a permanent change in the steady state. For that reason, Gs and Ys are not parameters as they are going to change. So one question is at which point does the value and therefore the rule change. Is that immediately when the TFP shock hits? If yes, define the two steady states as exogenous variables varexo and then define the respective steady state values.

Dear Professor jpfeifer,
Thank you for you reply, and I have solved the problem according to your suggestion.
Best regards.