former analytically and the latter numerically.

In Sec. VII, we conclude this paper.

In Appendix A, we discuss the physical meaning of entropy production to elucidate the physical contents of our results in two typical situations.

Review of Stochastic thermodynamics
===================================

In this section, we briefly review thermodynamics of classical stochastic systems and introduce notations that will be used later.

Dynamics
--------

We consider a classical stochastic system $\textbf{S}$ that is in contact with heat baths $\textbf{B}_1$, $\textbf{B}_2$, $\cdots$, $\textbf{B}_n$ at respective temperatures $T_1 = (k_{\rm B}\beta_1)^{-1}$, $T_2 = (k_{\rm B}\beta_2)^{-1}$, $\cdots$, $T_n = (k_{\rm B}\beta_n)^{-1}$. Let $x$ be the phase-space point of system $\textbf{S}$ and $\lambda$ be a set of external parameters such as the volume of a gas or the frequency of an optical tweezers. We control the system from time $0$ to $\tau$ with control protocol $\lambda (t)$. Let $x(t)$ be a trajectory of the system.

To formulate the stochastic dynamics, we discretize the time interval $[0, \tau]$ by dividing it into $N$ small intervals with width $\Delta t := \tau / N$. The original continuous-time dynamics is recovered by taking the limit of $N \to \infty$ or equivalently $\Delta t \to 0$. Let $t = n 