Guesses, mesh, scaling

Direct collocation is a nonconvex NLP. A feasible hypothesis can fail if the starting point, the mesh, or the units are unsuitable.

Constant initial guess

initial_guess maps a symbolic component to a constant:

sol = ocp.solve(
    control_intervals=40,
    polynomial_degree=3,
    initial_guess={sys.x[0]: 1.0, sys.u[0]: 0.0},
    tf_guess=120.0,
)

Useful values already satisfy the boxes (non-negative states, a control in \([0,1]\)). The default fill is not a process trajectory. A zero guess on a quantity that appears in a denominator can make the first IPOPT iteration fail.

initial_guess is not a time series. ocp.simulate(sol) replays a solution; it does not load an arbitrary simulated \(u(t)\) into the collocation NLP. Forward integration of \(f\) with a prescribed open-loop \(u\) (a constant control, zero irrigation) is eco_opt.simulate(sys, x0, t, u), independent of solve.

Mesh refinement

A typical first mesh is 20–40 intervals and degree 3. After sol.success and a plausible \(u(t)\), control_intervals may be increased (and degree, if the state is poorly resolved). First-solve Goddard uses 59 intervals and degree 5 to match an archive; that mesh is not required for a new model.

"radau" is often more reliable when \(u^*\) is expected to switch; "legendre" matches the Goddard archive. See Transcription.

Refining the mesh rebuilds the NLP. The previous sol is not interpolated onto the new grid. The same constant initial_guess applies; tf_guess may be set near the coarse \(T^*\).

Sweeps on a frozen mesh

When only numeric bounds or pinned parameters change, reuse the collocation graph with build / set_constraints / optimize. That sequence is Build and optimize. A new path expression or a change of \(f\) requires st and build again.

Scaling

Automatic scaling is not implemented. If states differ by many orders of magnitude (a large inventory versus a fraction versus a thin-layer biomass), the model is rescaled: nondimensional groups, or each state divided by a typical size, so that NLP variables are O(1).

There is no ocp.scale method. Path residuals in the problem statement are likewise O(1) when a defect is divided by a typical scale.

Failed solves: Troubleshooting (return_status, print level, restoration).