Introduction

Production–regeneration systems describe processes that must alternate between a productive phase and a recovery phase. During production, the system generates an output (for example filtered water), but this phase also progressively degrades an internal state (such as fouling or resistance). The regeneration phase does the opposite: it restores the system’s internal condition, but does not directly produce useful output.

Such a system can naturally be formulated as an optimal control problem. The main objective of this package is to provide tools to analyze and solve problems of this type.

Problem statement

We consider the following three-dimensional dynamical system defined for both production and regeneration modes:

\[u = +1 : \left\{ \begin{array}{rl} \dot c(t) &= l_p(x(t)) \\[0.5em] \dot x(t) &= f_p(x(t)) \\[0.5em] \dot y(t) &= g_p(x(t)) \end{array} \right. \quad \text{and} \quad u = -1 : \left\{ \begin{array}{rl} \dot c(t) &= l_r(x(t)), \\[0.5em] \dot x(t) &= f_r(x(t)), \\[0.5em] \dot y(t) &= g_r(x(t)), \end{array} \right.\]

where :

• $c$ denotes the cost,
• $x$ represents the internal state of the system,
• $y$ is an auxiliary state associated with a target value (or budget constraint), denoted by $T$.

Let $u \in [-1, 1]$ be the control variable, where $u = +1$ denotes production mode and $u = -1$ denotes regeneration mode, we consider the following optimal control problem

\[\text{(OCP)} \quad \left\{ \begin {array}{ll} \displaystyle \min_{x,y,t_f} \int_{t_0}^{t_f} \frac{1 + u(t)}{2} l_p(x(t)) + \frac{1-u(t)}{2} l_r(x(t)) \, \mathrm dt, & t \in [t_0, t_f] \ \mathrm{a.e.}, \\[1em] \displaystyle \mathrm{s.t.} \ \dot x(t) = \frac{1 + u(t)}{2} f_p(x(t)) + \frac{1-u(t)}{2} f_r(x(t)), & t \in [t_0, t_f] \ \mathrm{a.e.}, \\[1em] \displaystyle \phantom{\mathrm{s.t.} \ } \dot y(t) = \frac{1 + u(t)}{2} g_p(x(t)) + \frac{1-u(t)}{2} g_r(x(t)), \, & t \in [t_0, t_f] \ \mathrm{a.e.}, \\[1em] \phantom{\mathrm{s.t.} \ } u(t) \in [-1, 1], & t \in [t_0, t_f], \\[1em] \phantom{\mathrm{s.t.} \ } x(t_0) = x_0, \quad y(t_0) = y_0, \quad y(t_f) = T, \end{array} \right.\]

where $t_0 \in \mathbb R$, $x_0 > 0$, $y_0 > 0$ and $T > 0$ are provided.

Main theorical results

Based on the theoretical developments presented in Dutto et al., 2026, all possible optimal solution structures are characterized by the following result:

Here, $u_s$ denotes a singular control such that $\dot x = 0$. See Dutto et al., 2026 for a detailed presentation and proof.