Indirect Methods

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

Definition

Indirect methods solve trajectory optimization problems by first applying Pontryagin's Minimum Principle to derive a set of necessary conditions — the state-costate Hamilton canonical equations plus transversality conditions — that together form a two-point boundary-value problem (TPBVP), which is then solved numerically by shooting, collocation, or other root-finding techniques (Betts 1998; Conway 2010; Bryson & Ho 1975).

In contrast to direct methods (which discretize the state-control history directly into NLP parameters), indirect methods first resolve the optimality conditions analytically and only then seek a numerical solution. Their strengths are strict satisfaction of first-order necessary conditions, low parameter dimension, and high accuracy (routinely to machine precision). Their weaknesses are a narrow convergence basin, extreme sensitivity to the initial costate guess, and Jacobian difficulties at discontinuous (Bang-bang) controls.

Mathematical framework

Optimal control problem

General Bolza form:

\min_{u}\;J=\Phi(\mathbf{x}(t_f),t_f)+\int_{t_0}^{t_f}L(\mathbf{x},\mathbf{u},t)\,dt,

subject to $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u},t)$ , endpoint constraints, and path constraints.

Hamilton canonical equations

Introduce the costate $\boldsymbol{\lambda}(t)$ and Hamiltonian $H=L+\boldsymbol{\lambda}^{\mathrm{T}}\mathbf{f}$ . The Minimum Principle gives:

\dot{\mathbf{x}}=\frac{\partial H}{\partial\boldsymbol{\lambda}},\qquad \dot{\boldsymbol{\lambda}}=-\frac{\partial H}{\partial\mathbf{x}},\qquad \mathbf{u}^{*}=\arg\min_{\mathbf{u}\in\mathcal{U}}H.

The first equation is the state equation, the second the costate equation, the third the extremal condition (which, after analytical elimination, expresses control as a function of state and costate). Transversality at the terminal time reads

\boldsymbol{\lambda}(t_f)=\frac{\partial\Phi}{\partial\mathbf{x}}\bigg|_{t_f}+\boldsymbol{\nu}^{\mathrm{T}}\frac{\partial\boldsymbol{\psi}}{\partial\mathbf{x}}\bigg|_{t_f},

with terminal constraint $\boldsymbol{\psi}(\mathbf{x}(t_f),t_f)=\mathbf{0}$ and multipliers $\boldsymbol{\nu}$ . If the final time is free, $H(t_f)+\partial\Phi/\partial t+\boldsymbol{\nu}^{\mathrm{T}}\partial\boldsymbol{\psi}/\partial t=0$ .

TPBVP and the shooting function

Stacking state and costate into $\mathbf{y}=[\mathbf{x};\boldsymbol{\lambda}]$ with $\dot{\mathbf{y}}=\mathbf{F}(\mathbf{y})$ , the initial state $\mathbf{x}(t_0)$ is known and the initial costate $\boldsymbol{\lambda}(t_0)=\mathbf{z}$ is unknown. Numerically integrating to $t_f$ yields a residual $\mathbf{s}(\mathbf{z})=\boldsymbol{\psi}(\mathbf{x}(t_f;\mathbf{z}))$ . The shooting function $\mathbf{s}(\mathbf{z})=\mathbf{0}$ is the nonlinear system to solve.

Numerical strategies

Single vs. multiple shooting

Single shooting: integrate the whole trajectory as one IVP, apply Newton's method to $\mathbf{z}$ . Simple but ill-conditioned for long transfers.
Multiple shooting: divide $[t_0,t_f]$ into segments, integrate each independently, and stitch with continuity match conditions. The free-variable/constraint method is the engineering form of multiple shooting — all nodal states, costates, and parameters are stacked into a free-variable vector $\mathbf{V}$ , all continuity and endpoint conditions into a constraint vector $\mathbf{C}(\mathbf{V})=\mathbf{0}$ , solved by Newton iteration (Spreen 2021; Differential Correction).

Indirect collocation

Indirect collocation discretizes both state and costate equations (e.g., Dickmanns-Wells Hermite-Simpson) and solves the resulting algebraic system, eliminating the forward-integration ill-conditioning of shooting. The trade-off is much larger parameter dimension.

Note: Hargraves and Paris (1987) observed that eliminating the costate and collocating state-control directly (i.e., direct collocation) is more robust — the starting point of the direct-method era. Modern indirect collocation is mostly used when strict PMP satisfaction is mandatory (e.g., inside differentiable NLP solvers).

Indirect heuristic methods

Indirect heuristic methods combine PMP with metaheuristics: genetic algorithms, particle swarm, or differential evolution search the initial costate $\mathbf{z}$ (and possibly discrete variables like impulse count or staging structure); each candidate evaluation applies the PMP control law analytically and integrates the trajectory (Pontani & Conway 2009; Conway 2010). This bypasses Newton's sensitivity at the cost of many evaluations and lower precision than pure shooting. Useful for problems with discrete decisions or many local optima.

Hybrid direct-indirect methods

Hybrid direct-indirect strategies use indirect optimality conditions to reduce the control parameter dimension, then a direct-method NLP solver for the remaining parameters (Kluever & Pierson 1997). For example, the per-arc thrust direction is given in closed form by $\boldsymbol{\alpha}^{*}=\mathbf{p}/\|\mathbf{p}\|$ , and only impulse times, throttle switches, and arc-boundary times are NLP variables. This combines indirect "control reduction" with direct "costate insensitivity" — a common engineering compromise.

Indirect Multi-Stage Formulation (IMF)

The Indirect Multi-Stage Formulation (IMF) partitions the trajectory into stages, each with its own state-costate equations and control law, stitched by equality constraints on position, velocity, mass, and costate continuity (Bowerfind & Taheri 2024). Especially suited for missions with mode switches (thrust-aerodynamic-coast combinations, multi-impulse cruise legs). The stage count may be fixed or itself an integer variable.

Key numerical tools

Costate normalization

Because the Hamiltonian system is invariant under a positive scaling of $\boldsymbol{\lambda}$ ( $H$ and $\boldsymbol{\lambda}$ scaled by the same constant still satisfy the NC), one can fix $\|\boldsymbol{\lambda}(t_0)\|=1$ , restricting the search from $\mathbb{R}^n$ to the unit sphere $S^{n-1}$ — one dimension fewer, with improved conditioning (Taheri et al. 2016; Co-state Variables).

Switching-point detection

In Bang-bang control, switching instants are roots of $\rho(t)=0$ that must be located precisely to avoid Jacobian discontinuities. The Newton-bisection hybrid uses Newton first (typically 4–5 iterations to machine precision), falling back to bisection if Newton fails or leaves the bracket (Zhang et al. 2015; Martinon & Gergaud 2010). Standard component of indirect methods for Bang-bang control.

Jump conditions

With state constraints (e.g., maximum load factor, minimum perilune altitude), costate and Hamiltonian can jump at the entry/exit of active intervals. The jump magnitude is set by the constraint gradient and a non-negative multiplier $\eta$ :

\boldsymbol{\lambda}(t_c^{+})=\boldsymbol{\lambda}(t_c^{-})-\eta\,\nabla_{\mathbf{x}}g(\mathbf{x}(t_c),t_c).

When the constraint is time-independent, $H$ stays continuous across the jump. The switching surface is the codimension-one submanifold $\rho=0$ in costate-state space, where the optimal trajectory may switch control or enter a singular arc (Hartl et al. 1995; Caillau & Daoud 2012).

Smoothing and homotopy

To bypass Bang-bang discontinuity and costate sensitivity, the mainstream approach is the homotopy method that gradually deforms a smooth "easy sister problem" (e.g., energy-optimal) into the target (e.g., fuel-optimal Bang-off-Bang). Cost-function homotopy, thrust-amplitude homotopy, and sigmoid smoothing are standardized in the literature (Bertrand & Epenoy 2002; Taheri et al. 2016; Zhang et al. 2025). See Homotopy Method.

Inner-loop / outer-loop structure

Parametric optimal control (e.g., Theory-of-Functional-Connections shape approximations) often uses an inner-loop outer-loop structure: the inner loop minimizes residuals for state and cost vectors; the outer loop optimizes switching times and final time (Johnston et al. 2020). Conceptually akin to multiple shooting — separating "fast" and "slow" variables into different solvers.

Comparison with direct methods

Axis	Indirect	Direct
Decision variables	Initial costate $\boldsymbol{\lambda}(t_0)$ , parameters	Discrete state-control history
Necessary conditions	Strictly satisfies PMP	Numerical approximation, post-hoc check
Initial guess	Costate guess hard, narrow basin	Any feasible guess, wide basin
Accuracy	High (routinely $10^{-12}$ )	Grid-limited
Control structure	Bang-bang emerges naturally	Impulse count needs post-hoc check
Parameter dimension	Small (tens)	Large (hundreds to thousands)

A common engineering practice is to obtain a coarse solution via direct or heuristic methods, then refine it to machine precision with an indirect method, using primer vector and switching function for post-hoc optimality verification (Conway 2010; Betts 1998).

Application notes

Cislunar low-thrust transfers: $L_1$ / $L_2$ Halo-to-Halo and LEO-to-NRHO fuel-optimal transfers are the main theater for indirect + homotopy methods.
Multi-impulse optimization: primer vector gradients decide impulse count and times; indirect methods then refine the impulsive model to finite thrust.
Station-keeping and collision avoidance: small fixed-horizon TPBVPs solved in real time for NRHO continuous-thrust station-keeping and collision avoidance.
Relative motion optimal control: controlling a deputy relative to a chief in three-body dynamics — via variational equations and pre-computed state transition matrices, any boundary condition can be approximated in $O(mn^3)$ operations (Kulik et al. 2023).

Pontryagin's Minimum Principle — the mathematical starting point
Co-state Variables — the central unknown of the TPBVP
Primer Vector — the physical embodiment of the velocity costate, fixing optimal thrust direction
Homotopy Method — the numerical workhorse for indirect Bang-bang control
Bang-bang Control — the typical control structure derived indirectly
Fuel-optimal Control — the dominant problem class for indirect methods
Adjoint-Control Transformation — reducing shooting dimension via the primer vector
Differential Correction — the Newton iteration behind the free-variable/constraint method
Circular Restricted Three-Body Problem (CR3BP) — the dynamical setting for cislunar indirect optimization

References

Bryson, A. E., and Ho, Y.-C. 1975. Applied Optimal Control. Hemisphere.
Betts, J. T. 1998. "Survey of Numerical Methods for Trajectory Optimization." JGCD 21(2): 193–207.
Conway, B. A. (ed.) 2010. Spacecraft Trajectory Optimization. Cambridge Univ. Press. Ch. 1, 2, 3, 7.
Dickmanns, E. D., and Wells, K. H. 1974. "Approximate Solution of Optimal Control Problems Using Hermite-Simpson Collocation."
Hargraves, C. R., and Paris, S. W. 1987. "Direct Trajectory Optimization Using Nonlinear Programming and Collocation." JGCD 10(4): 338–342.
Pontani, M., and Conway, B. A. 2009. "Numerical Solution of the Three-Dimensional Orbital Pursuit-Evasion Game." JGCD.
Kluever, C. A., and Pierson, B. L. 1997. "Optimal Earth-Moon Trajectories Using Nuclear Electric Propulsion." JGCD.
Bowerfind, W. M., and Taheri, E. 2024. "Rapid Approximation of Low-Thrust Spacecraft Reachable Sets."
Spreen, J. S. 2021. Robust Spacecraft Trajectory Optimization via Convex and Least-Squares Approaches. PhD Thesis, Univ. of Colorado.
Zhang, B., et al. 2015. "Switching Detection for Bang-Bang Control in Low-Thrust Trajectory Optimization." JGCD, doi:10.2514/1.G001080.
Martinon, P., and Gergaud, J. 2010. "Switching Time Detection for Optimal Control Problems." INRIA TR-7380.
Hartl, R. F., Sethi, S. P., and Vickson, R. G. 1995. "A Survey of the Maximum Principles for Optimal Control Problems with State Constraints." SIAM Review 37(2): 181–218.
Caillau, J.-B., and Daoud, B. 2012. "Minimum Time Control of the Restricted Three-Body Problem." SIAM J. Control Optim. 50(6).
Taheri, E., Kolmanovsky, I., and Atkins, E. 2016. "Enhanced Smoothing Technique for Indirect Optimization of Minimum-Fuel Low-Thrust Trajectories." JGCD 39(11): 2500–2511.
Kulik, S., et al. 2023. "Relative Motion Optimal Control via Variational Equations." JGCD, doi:10.2514/1.G007311.
Johnston, B., et al. 2020. Theory of Functional Connections Applied to Optimal Control.