Nondifferentiable pulse (non_diff)

Question. Can collocation handle a rectangular pulse in \(f\) written with if_else?

This is a numerical pulse, not phenology or ecological regime switching. The pulse is written with CasADi if_else because a Python if cannot branch on symbolic time; see Transcription.

Formulation

Dynamics.

\[ \dot x_1=x_2,\qquad \dot x_2=-x_1-x_2+u+d(t),\qquad \dot x_3=5x_1^2+2.5 x_2^2+0.5 u^2, \]

with \(d(t)\) a pulse of height 100 on \([0.5,0.6]\) via nested if_else.

Objective. \(\min x_3(T)\). Horizon. Fixed, \(T=2\). \(\mathbf{x}(0)=\mathbf{0}\).

What was done

Radau, degree 3, 100 intervals; control guess \(-5\).

What we found

Solver succeeds. Regression \(J^*\approx 58.07\). The pulse is visible in \(x_2\).

Interpretation

A jump in forcing is not a jump in the control. Bang–bang allocation (Cohen, Lindh) is a different object. Do not read this example as a phenology switch.

Public API

Bare dx (examples/non_diff/non_diff.py):

import eco_opt


def nondiff_model(t, x, u):
    ...
    return dx


sys = eco_opt.System(states=3, controls=1, model=nondiff_model)
ocp = eco_opt.Ocp()
ocp.min(eco_opt.t_f(sys.x[2]))
ocp.st(system=sys, constraints=[...])

Run and tests

cd examples/non_diff
python non_diff.py
python plot_non_diff.py
python -m pytest ../../tests/test_non_diff.py -q

References

Yop example Nondifferentiable system. The text and code here are ours.