Lenhart and Workman (lenhart_workman)

Question. Can the same API recover three textbook OCPs: bang–bang, smooth tracking, and minimum time?

Formulation

17.2 — linear bang–bang.

\[ \max\int_0^2(2x-3u)\,\mathrm{d}t,\qquad \dot x=x+u,\quad 0\le u\le 2,\quad x(0)=5. \]

Analytic switch \(t^*=2-\ln 2.5\approx 1.084\).

17.3 — tracking.

\[ \min\int_0^2(x-t^2)^2\,\mathrm{d}t,\qquad \dot x=u,\quad 0\le u\le 4,\quad x(0)=1. \]

Fixed horizon; free terminal state.

20.4 — min-time trolley. Double integrator, free \(T\), drive to rest. Analytic \(T\) from the textbook.

What was done

Three small System/Ocp scripts in one folder. Same constraint tuples as the ecological examples.

What we found

17.2 recovers the switch. 17.3 is smooth (not bang–bang); objective regression-tracked. 20.4 matches the analytic \(T\).

Interpretation

Textbook biology OCPs are small enough that a wrong transcription is obvious. They are not substitutes for the process-model examples; they gate bang–bang, tracking, and free time separately.

Public API

Example 17.2 returns bare dx; 20.4 uses Namespace (examples/lenhart_workman/):

import eco_opt
from eco_opt import Namespace

sys = eco_opt.System(states=1, controls=1, model=linear_bangbang_model)
ocp = eco_opt.Ocp()
ocp.max([2 * sys.x[0] - 3 * sys.u[0]])
ocp.st(system=sys, constraints=[...])

Run and tests

cd examples/lenhart_workman
python linear_bangbang_ex17_2.py
python lq_tracking_ex17_3.py
python min_time_trolley_ex20_4.py
python plot_lenhart_workman.py
python -m pytest ../../tests/test_lenhart_workman.py -q

References

Lenhart and Workman, Optimal Control Applied to Biological Models, examples 17.2, 17.3, and 20.4.

Not validated against the book’s figures

Where the book gives a closed form (switch time, \(T\)), tests check that. Trajectories are not compared to plotted solutions in Lenhart and Workman.