Multi-phase Cohen (`multiphase_cohen`)¶

Question. Can Cohen’s vegetative-then-reproductive switch be posed as two sequential phases with a free switch, instead of one jumping control?

Here plant is the same annual-plant model as cohen_king_roughgarden. Two phases because \(f\) and the role of the state change at a free switch; see Constraints and time.

Formulation¶

Two phases, no control. The bangs are built into each \(f\).

Phase 1 (vegetative).

\[ \dot x_1=x_1,\qquad \dot x_2=0. \]

Phase 2 (reproductive).

\[ \dot x_1=0,\qquad \dot x_2=x_1. \]

Objective. \(\max x_2(T)\). Horizon. Season \(T\) fixed; switch time \(t_s\) free. Continuity of time and state at \(t_s\). \(x_1(0)=X_{10}\), \(x_2(0)=0\).

Analytic: \(t_s^*=T-1\), \(x_2(T)=X_{10}e^{T-1}\).

What was done¶

Phase objects and ocp.st(phases=[p1, p2], continuity=True). Each phase has its own System and collocation grid.

What we found¶

Switch time and yield match the single-phase Cohen closed form (defaults \(T=5\), \(X_{10}=1\): \(t_s=4\), \(x_2(T)=e^4\)).

Two-phase Cohen

Interpretation¶

If the biology is “first grow, then reproduce,” that structure can sit in the phase graph rather than in a relaxed \(u(t)\). The optimum is the same; the decision variable is the switch instant.

Public API¶

Full script: examples/multiphase_cohen/cohen_two_phase.py.

import eco_opt
from eco_opt import Namespace, Phase

sys_v = eco_opt.System(states=2, controls=0, model=vegetative_model)
sys_r = eco_opt.System(states=2, controls=0, model=reproductive_model)
p1 = Phase(sys_v, constraints=[...], name="vegetative")
p2 = Phase(sys_r, constraints=[...], name="reproductive")
ocp = eco_opt.Ocp()
ocp.max(eco_opt.t_f(sys_r.y.repro))
ocp.st(phases=[p1, p2], continuity=True)

Run and tests¶

cd examples/multiphase_cohen
python cohen_two_phase.py
python plot_cohen_two_phase.py
python -m pytest ../../tests/test_multiphase.py -q

References¶

Cohen (1971). Same biology as the single-phase Cohen example; this folder tests the Phase API.

Not validated against published figures

Numerics are checked against Cohen’s closed form, not against figures from that paper.

Multi-phase Cohen (multiphase_cohen)¶