Communications for Control in Cyber Physical Systems

Nonequilibrium equalities for feedback control

The following theorem provides an equality for feedback control:

Theorem 29

(Generalized Detailed Fluctuation Theorem With Feedback Control)

The following equality holds when feedback control exists:

$\begin{array}{l}\frac{P^{†}(X_N^{†},Y_N)}{P(X_N,Y_N)}={\mathrm{e}}^{-\sigma (X_N,{\Lambda }_N(Y_N)-I_c(X_N;Y_N))},\hfill \end{array}$

  (5.307)

where the “backward probability distribution” is defined as

$\begin{array}{l}{P}^{†}(X_N^{†},Y_N)=P(X_N^{†}{\Lambda }_N(Y_{N-1}))P(Y_N).\hfill \end{array}$

  (5.308)

The proof is similar to that of Theorem 26. Note that the physical meaning of $P^{†}(X_N^{†},Y_N)$ is the probability obtained from the following two-step ...