\beta(t) is a function that depends on time (not \beta^{t}), but I don’t know how to included in the model using Dynare’s syntax. Suggestions are welcome.

Could you explain the intuition of what you are trying to do? If beta is really depending on the time point t, you model is not stationary and Dynare won’t solve it.

Hello professor,
I have the same question, if i need to put any time-dependent variable, for example: r(t)=alpha1+beta1*time, how can i generate time variable?
I tried to do it with equation t=t(-1)+1 but this makes the existence of steady state impossible.
In advance, thanks for the attention.
Greetings from Colombia.

It depends on how you plan to solve the model. If you choose a local approximation of the model (ie use stoch_simul command) you cannot have this variable since (i) t does not admit a steady state (ii) more generally your model is not stationary because of the time trend. If you consider instead a deterministic model simulated with a perfect foresight solver (ie the simul command) then you can have the trend variable since the existence of the steady state is not mandatory.