Carnot problem of maximum work from a finite resource interacting with environment in a finite time

We treat an irreversible extension of the Carnot problem of maximum mechanical work delivered from a system of a finite-resource fluid and a bath fluid. For a derived work functional, Pontryaginʹs maximum principle is applied to determine optimal states and optimal controls. These are respectively represented by the temperature of the resource fluid, T, and the so-called driving temperature T′, an effective temperature which replaces T in the Carnot efficiency formula. Solution of canonical equations is given and free boundary conditions are discussed. Correspondence is proved between the canonical formalism and that governed by the Hamilton–Jacobi equation. A link is shown between the process duration and the optimal dissipation intensity. Hysteretic properties cause difference between the work supplied and delivered, for inverted end states of the process. The results prove that bounds of the classical availability should be replaced by stronger bounds obtained for finite time processes. Related statistical and stochastic approaches are outlined.