Zero-order-hold dynamics

Phase 12 provides a common ControlSchedule<C> and four evaluated models:

• ZohKeplerDynamics: [x,y,z,vx,vy,vz,mass], normalized mu = 1;
• ZohCr3bpDynamics: the same seven-state layout in the CR3BP synodic frame;
• ZohEquinoctialDynamics: [p,f,g,h,k,L,mass], normalized mu = 1;
• ZohSolarSailDynamics: six Cartesian states with cone/clock attitude.

The first three control rows are [thrust, i1, i2, i3]. The direction is Cartesian for Kepler/CR3BP and radial-transverse-normal for equinoctial dynamics. Kepler/equinoctial constants contain mass-flow coefficient c; CR3BP constants are [c, mu]. The preserved upstream mass equation is dm/dt = -c thrust exp(-1 / (mass 1e16)).

Solar-sail rows are [alpha, beta] in radians. Its constant c scales ideal sail acceleration as c cos(alpha)^2 / r².

Schedule and switch contract

A schedule has S + 1 strictly increasing finite boundaries and exactly S finite control rows. Construction performs all dimension, finiteness, and monotonicity checks.

For lookup, segment i owns [t_i, t_(i+1)); the final boundary belongs to the final segment. Forward propagation ends each integration exactly at a switch and starts the next segment with its new control. Backward propagation uses the interval to the left of each encountered switch. This directional rule avoids evaluating a segment across a discontinuity.

Each segment is integrated exactly once. The active control is copied into a fixed-size parameter array before the solve, so no allocation or schedule scan occurs in an RHS call. control_at uses binary search for occasional external lookup.

Sensitivities

ZohSensitivitySeeds carries an arbitrary compile-time seed width through the schedule in one augmented solve per segment. It contains:

• initial state seeds;
• one control seed matrix per segment;
• constant-parameter seeds.

State sensitivities are continuous at switches. A control column for a future segment remains exactly zero until that segment becomes active. Runtime is linear in the segment count for a fixed seed width; the implementation does not repropagate every prior segment for every control.

Model Jacobians use fixed-size, allocation-free central differentiation of the evaluated source equations. They are checked against C++ Taylor variations and end-to-end schedule finite differences. The longest CR3BP reference uses a 2e-5 relative/absolute variation comparison; nominal state parity remains 3e-10. This looser derivative tolerance is explicit and will be revisited if later leg gradients require analytic Jacobians.

Python API

Python exposes all four RHS functions and propagate_zoh_* functions. Boundaries and controls are ordinary nested sequences. Propagation releases the GIL and accepts backward, scalar tolerances, and maximum_step. Malformed row widths and grids are rejected before native integration.