# Understanding the terminal condition in the Taylor rule

I am currently reading this summary of various theories on how central banks control inflation. However, I got quite confused in section 3.2.4. Here is the context.

Suppose we have a log-linearized Fisher equation

$$i_t = r_t + \mathrm{E}_t\pi_{t + 1}$$

and Taylor rule

$$i_t = x_t^i + \phi \pi_t,$$ where $$i_t$$ is the policy rate set by a monetary authority, $$r_t$$ is the real rate, $$\pi_t$$ is inflation, $$x_t^i$$ is a variable summarizing exogenous factors determining the path of policy and $$\phi > 1$$. We can equivalently summarize the above with

$$\mathrm{E}_t[\pi_{t + 1} - \pi_{t + 1}^{*}] = \phi [\pi_t - \pi_t^*] - r_t + \pi_{t + 1}^* - \phi \pi_t^* + x_t^i,$$ where $$\pi_t^*$$ is the inflation target. Since the real interest rate and inflation target is exogenous, we can subsume it in $$x_t^i$$ to get

$$\mathrm{E}_t[\pi_{t + 1} - \pi_{t + 1}^{*}] = \phi [\pi_t - \pi_t^*] + x_t^i.$$

Solving this expectational difference equation forward, we have

$$\pi_t = \lim_{T \to \infty} \left\{\pi_t^* - \sum_{j = 0}^T \phi^{j - 1} E_t x_{t + j}^i + \phi^{-T} \mathrm{E}_t[\pi_{t + T} - \pi_{t + T}^*]\right\}.$$ To get a bounded solution, we have to assume that the latter term converges to zero. My question is: how does one interpret such an assumption economically? Specifically, suppose we take the example of Section 3.2.3 and assume that at time $$t$$, $$\pi_t - \pi_t^* = 1$$. Then, since $$\phi > 1$$, the monetary authority will increase $$i_t$$ by $$\phi$$. Then, by the Fisher equation, expected inflation increases by $$\phi$$, which in turn forces the monetary authority to increase $$i_{t + 1}$$ by $$\phi^2$$, and so on. But these paths are ruled out by the terminal condition, so what is happening in the economy instead? The Fisher equation still holds, so what are agents doing? I think I understand the math, but I am not getting the economic intuition here. Section 3.2.4 is not helping me much.