Figure 8
The graph is titled “Analytical properties of the sustainability state”. The horizontal axis is labeled “Operational time”, ranging from 0 to 50 with an interval of 10, and the vertical axis is labeled “Sustainability state, tau (t)”, ranging from 0.0 to 1.0 with an interval of 0.2. Four curves are plotted and identified in a legend at the upper left: a blue line labeled “tau (0) equals 0.05”, an orange line labeled “tau (0) equals 0.15”, a green line labeled “tau (0) equals 0.3”, and a red line labeled “tau (0) equals 0.5”. All curves increase monotonically over time and approach a saturation value near 1.0. The red curve (tau (0) equals 0.5) starts highest at around (0, 0.5) and remains above the others throughout. The green, orange, and blue curves start at progressively lower initial values and follow similar rising trends, with the blue curve (tau (0) equals 0.05) starting lowest at (0, 0.05). Over time, all curves gradually converge toward similar values close to 0.95, showing asymptotic behavior. Note: All numerical values are approximated.Analytical properties of the physics-informed sustainability state. Evolution of the sustainability state governed by irreversible first-order kinetics for different initial conditions. The formulation guarantees monotonic evolution, boundedness within , and asymptotic saturation independent of the initial state, demonstrating physically admissible and stable long-term behaviour