Optimal Control and Signaling Strategies of Control-Coding Capacity of General Decision Models: Applications to Gaussian Models and Decentralized Strategies

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 600-629, February 2024.
Abstract. We investigate the control-coding (CC) capacity of general dynamical decision models (DMs) that involve nonlinear filtering, which is absent in the specific DMs investigated in [C. K. Kourtellaris and C. D. Charalambous, IEEE Trans. Inform. Theory, 64 (2018), pp. 4962–4992]. We derive characterizations of CC capacity and we show their equivalence to extremum problems of maximizing the information theoretic measure of directed information from the input process to the output process of the DM over randomized strategies. Due to the generality of the DMs, the CC capacity is shown to be equivalent to partially observable Markov decision problems, contrary to the DMs in the above mentioned paper, which give rise to fully observable Markov decision problems. Subsequently, the CC capacity is transformed, using nonlinear filtering theory, to fully observable Markov decision problems. For the application example of a Gaussian DM with past dependence on inputs and outputs, we prove a decentralized separation principle that states optimal inputs are Gaussian and consist of (i) a control, (ii) an estimation, and (iii) an information transmission part, which interact in a specific order. The optimal control and estimation parts are related to linear-quadratic Gaussian stochastic optimal control problems with partial information. Various degenerated cases are discussed, including examples from the above mentioned paper, which do not involve estimation.