The optimal coding scheme for communicating a Gaussian message over an Additive White Gaussian noise (AWGN) channel with AWGN output feedback, with a limited number of transmissions is unknown. Even if we restrict the scope of the coding scheme to linear schemes, still, deriving the optimal coding scheme is a challenging task. The state-of-the-art linear scheme for channels with noisy feedback is by Chance and Love, where the coefficients of the linear scheme are numerically optimized based on unique observations [1]. In this paper, we introduce a new class of sequential linear schemes for this channel by introducing a novel linear state process at the transmitter and derive the optimal sequential scheme within this class of schemes in a closed-form by formulating a novel Dynamic Programming (DP). We empirically show that our scheme outperforms the state-of-the-art linear scheme in [1] for noisy feedback and coincides with the SK scheme for noiseless feedback. We also show that in communicating message bits as opposed to a Gaussian message, a learning-based approach further improves the reliability of sequential linear schemes. This problem is an instance of decentralized control without any common information and to the best of our knowledge the first such scenario where we can derive analytical solutions using a DP.