Fig. 7
From: Reducing M2 macrophage in lung fibrosis by controlling anti-M1 agent

Objective function. For the optimal regulator problem, a graph of the objective functional \({\hat{J}}(M2,\eta _{M1},t)\) in (33) versus time has been plotted, revealing a gradual decrease over time. It is evident that the objective function diminishes progressively. This implies that the objective function is being minimized or optimized as time advances, indicating that the optimal control problem is making progress toward achieving the desired goal.