Multi-arc Optimal Control

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Multi-arc optimal control is the extension of the continuous-time optimal control problem to a trajectory composed of $N$ arcs, where each arc $j$ can use a different state representation and the arcs are joined at $N-1$ interfaces by matching (corner) conditions. The single-arc counterpart treats the transfer as one continuous IVP-to-TPBVP with one state representation throughout (Pozzi et al. 2025; Betts 1998).

Splitting is forced whenever a single state representation cannot describe the entire path efficiently: a cislunar low-thrust transfer from the Gateway NRHO to a Low Earth Orbit crosses the Earth-Moon boundary, where modified equinoctial elements (MEE) about the Moon degenerate near $e = 1$ and terrestrial MEE about Earth are ill-conditioned near the Lagrange points. The multi-arc formulation sidesteps the singularity by switching coordinates at the interface, at the cost of additional junction conditions.

Mathematical Structure

For each arc $j \in \{1,\dots,N\}$ , the state equations

\dot{\boldsymbol{x}}^{(j)} = \boldsymbol{f}^{(j)}\!\left(\boldsymbol{x}^{(j)}, \boldsymbol{u}^{(j)}, \boldsymbol{p}, \tau\right),

share the auxiliary independent variable $\tau \in [0, t_f - t_0]$ (forward or backward), but each $\boldsymbol{x}^{(j)}$ can be a different coordinate set. At each interface time $\tau_j$ between arc $j$ and arc $j+1$ , three conditions are imposed (Pozzi et al. 2025):

a scalar transition function $\zeta_j(\boldsymbol{x}^{(j+1)}_{\text{ini}}, \boldsymbol{x}^{(j)}_{\text{fin}}, \tau_j) = 0$ that flags when the switch happens (e.g., geocentric distance crosses a threshold);
a vector matching function $\boldsymbol{\chi}_j$ that maps the outgoing arc's terminal state to the incoming arc's initial state - generally an implicit nonlinear function, not a bare equality;
a time-consistency equality tying the per-arc durations to the overall epoch parameters $\boldsymbol{p} = [t_0, t_f]^T$ .

Together with the dynamics, the matching conditions form an augmented TPBVP. Its dimension is $\sum_j n_j$ state variables plus interface multipliers.

Corner Conditions and Costate Matching

Variational necessary conditions at a smooth interior corner $\tau_j$ reduce to the Weierstrass-Erdmann conditions:

\boldsymbol{\lambda}^{(j)}(\tau_j^{-}) = \boldsymbol{\lambda}^{(j+1)}(\tau_j^{+}), \qquad H^{(j)}(\tau_j^{-}) = H^{(j+1)}(\tau_j^{+}),

i.e., the costate (transformed to a common representation) and the Hamiltonian are continuous across the interface. When the matching function $\boldsymbol{\chi}_j$ is implicit, the costate continuity generalizes to $\boldsymbol{\lambda}^{(j)} = (\partial \boldsymbol{\chi}_j / \partial \boldsymbol{x}^{(j)}_{\text{fin}})^{-T} (\partial \boldsymbol{\chi}_j / \partial \boldsymbol{x}^{(j+1)}_{\text{ini}})^{T} \boldsymbol{\lambda}^{(j+1)}$ plus interface multipliers.

Implicit Costate Transformation

When the matching functions define a bijective map between adjacent arcs' states, Pozzi et al. (2025) show the corner conditions can be solved sequentially - the costate on one arc is recovered from the costate on the next by a closed-form transformation (the implicit costate transformation). The multi-arc TPBVP then has the same number of unknowns as the single-arc problem: the initial costate $\boldsymbol{\lambda}^{(1)}(t_0)$ , the final time $t_f$ , and any arc parameters. This avoids the exponential growth of unknowns that a naive $N$ -arc shooting implementation would suffer, and is the main reason the indirect heuristic method remains tractable for Earth-Gateway-Moon transfers.

Relation to Patch Points and Defect Constraints

Multi-arc optimal control is the theoretical side of trajectory splitting; the algorithmic side is multiple shooting with patch points:

Patch points are the discrete nodes at which inter-arc continuity is enforced by Newton-type iteration; they belong to the numerical method.
Corner conditions are the necessary conditions for optimality at those nodes when the trajectory is treated as an extremal of a multi-arc variational problem; they belong to optimal control theory.
In direct transcription, continuity at interior nodes appears as defect constraints in the NLP, with no explicit costate.

In short: the same physical trajectory can be split either way, but multi-arc OC carries analytical structure (costate matching, Hamiltonian continuity) that pure multiple shooting ignores.

Comparison: Single-arc vs Multi-arc

Aspect	Single-arc OC	Multi-arc OC
State representation	one set throughout	one set per arc, switch at interfaces
Interface conditions	none	$\zeta_j, \boldsymbol{\chi}_j, \tau_j$ matching
Costate treatment	standard TPBVP	corner conditions; can be sequentialized
Singularity handling	ill-conditioned near $e = 1$ , $h = k = 0$	each arc uses its non-singular coordinates
Typical use	two-body transfers, single-resonance arcs	multi-body low-thrust, Earth-Moon crossings

Application: Cislunar Low-Thrust Transfers

Pozzi et al. (2025) formulate two-way minimum-time low-thrust transfers between Gateway (NRHO) and both LEO and a Low Lunar Orbit in a full ephemeris model with Sun/Earth/Moon gravity. The Gateway-to-LLO leg stays in a perturbed two-body framework and is solved as a single-arc OC; the Gateway-to-LEO leg, dominated by two attracting bodies, is cast as a five-arc multi-arc problem - three of the arcs have zero length (pure coordinate changes), and the remaining two use terrestrial and lunar MEE respectively, joined at the Earth-Moon boundary. Combined with an indirect heuristic algorithm (particle swarm + costate integration), the multi-arc framework yields complete two-way transfer solutions that respect high-fidelity ephemeris dynamics.

Practical Notes

The number of arcs is dictated by where a single state representation fails, not by accuracy considerations alone; adding arcs for their own sake multiplies interface constraints without benefit.
Sequential solvability via the implicit costate transformation is special to bijective matching maps; non-bijective interfaces (e.g., collision/impact) require the full augmented TPBVP.
For direct-method implementations, multi-arc structure maps naturally onto multiphase pseudospectral collocation, with phase-linkage constraints replacing corner conditions.

References

Pozzi, E., Morselli, A., Masdemont, J., & Gomez, G. (2025). Optimal low-thrust orbit transfers connecting Gateway with Earth and Moon. Celestial Mechanics and Dynamical Astronomy.
Pozzi, E., Morselli, A., Masdemont, J., & Gomez, G. (2024). Optimization, guidance, and control of low-thrust transfers from the lunar Gateway to low lunar orbit.
Beolchi, A. C., Morselli, A., & Topputo, F. (2023). A multi-arc formulation for low-thrust orbit transfers in a high-fidelity multibody ephemeris model.
Betts, J. T. (1998). Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21(2), 193-207.
Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control. (Weierstrass-Erdmann corner conditions.)
Conway, B. A. (Ed.). (2010). Spacecraft Trajectory Optimization. Cambridge University Press.
Howell, K. C., & Pernicka, H. J. (1987). Numerical determination of Lissajous trajectories in the restricted three-body problem. Celestial Mechanics, 41(1-4), 107-124. (multiple shooting at patch points)