Fuel-optimal Control

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

Definition

Fuel-optimal control minimizes propellant consumption. Under the rocket equation it is equivalent to maximizing final mass or minimizing total delta-v. Together with energy-optimal control (minimize $\int\|\mathbf{u}\|^2 dt$ ) and time-optimal control (minimize $t_f-t_0$ ), it is one of the three canonical performance indices in spacecraft trajectory optimization; each leads to a different optimal-control structure under Pontryagin's Minimum Principle (Bryson & Ho 1975; Betts 1998; Conway 2010).

Index	Form	Control structure	Typical use
Fuel-optimal ( $L^1$ )	$J=\int_{t_0}^{t_f}\|\mathbf{u}\|\,dt=-m(t_f)/c$	Bang-off-Bang	Low-thrust deep-space, powered descent
Energy-optimal ( $L^2$ )	$J=\int_{t_0}^{t_f}\|\mathbf{u}\|^2\,dt$	Continuous smooth throttle	Homotopy start, power-limited
Time-optimal	$J=t_f-t_0$	Full thrust throughout	Time-critical transfers

Here $\mathbf{u}=T_{\max}u\boldsymbol{\alpha}/m$ is thrust acceleration and $u\in[0,1]$ is throttle.

Mathematical formulation

Dynamics and cost

In a central gravity field with variable mass:

\dot{\mathbf{r}}=\mathbf{v},\quad \dot{\mathbf{v}}=\mathbf{g}(\mathbf{r})+\frac{T_{\max}}{m}\,u\,\boldsymbol{\alpha},\quad \dot{m}=-\frac{T_{\max}}{c}\,u.

Fuel-optimal in Mayer form is $J=-m(t_f)$ , equivalent to $J=(T_{\max}/c)\int u\,dt$ in Lagrange form. The Hamiltonian is

H=\boldsymbol{\lambda}_r^{\mathrm{T}}\mathbf{v}+\boldsymbol{\lambda}_v^{\mathrm{T}}\mathbf{g}+u\,T_{\max}\!\left(\frac{\boldsymbol{\lambda}_v^{\mathrm{T}}\boldsymbol{\alpha}}{m}-\frac{\lambda_m}{c}\right).

Optimal control: primer vector and switching function

With the primer vector $\mathbf{p}\equiv-\boldsymbol{\lambda}_v$ and the dimensionless switching function

\rho(t)=1-\frac{c\,\|\mathbf{p}\|}{m}-\lambda_m,

minimizing $H$ yields the optimal direction $\boldsymbol{\alpha}^{*}=\mathbf{p}/\|\mathbf{p}\|$ and throttle (Zhu & Gao 2017; Caillau et al. 2012):

u^{*}=\begin{cases}0,&\rho>0\\1,&\rho<0\\\text{indeterminate},&\rho=0\end{cases}

$\rho=0$ corresponds to a singular arc, where the first-order condition is insufficient and the Legendre-Clebsch second-order condition is needed. Singular arcs are rare in standard fuel-optimal problems; most solutions exhibit Bang-off-Bang structure: alternating MT arcs ( $u=1$ ) and NT arcs ( $u=0$ ) with no intermediate thrust (Lawden 1963; see Bang-bang Control).

Energy-optimal as the homotopy start

The energy-optimal cost yields a smooth switching function $\rho_E=1-2\varepsilon c\|\mathbf{p}\|/m-\lambda_m$ (with $\varepsilon$ the homotopy parameter); the throttle is continuous and the convergence basin is wide. Bertrand and Epenoy (2002) introduce the regularized cost

J_\varepsilon=\int_{t_0}^{t_f}\!\bigl[u-\varepsilon\,u(1-u)\bigr]dt,

bridging $\varepsilon=1$ (energy-optimal, smooth) to $\varepsilon\to 0$ (fuel-optimal, Bang-off-Bang). Each subproblem along the path uses the previous solution as initial guess: the bridge that makes indirect methods engineering-feasible (see Homotopy Method).

The special case of time-optimal control

The time-optimal cost is independent of $u$ explicitly; combined with $H\equiv 0$ (free-final-time condition) it implies $u^{*}=1$ throughout, i.e., full thrust with direction $\boldsymbol{\alpha}^{*}=\mathbf{p}/\|\mathbf{p}\|$ . There are no switches, which makes time-optimal problems relatively tractable and a common starting point for thrust-amplitude homotopy (Caillau & Daoud 2012).

Fuel-time trade-off

Real missions usually bound the transfer time. Define the time ratio $c_{t_f}=t_f/t_f^{\min}$ ; final mass increases with $c_{t_f}$ but saturates for $c_{t_f}\gtrsim 2$ (Caillau et al. 2012, Fig. 4), diminishing returns. Designs pick a point on this Pareto front: $c_{t_f}\approx 1.5$ favors time for crewed missions; $c_{t_f}\approx 2$ --3 favors fuel for cargo.

Application notes

Powered descent

Lunar or planetary powered descent is a canonical fuel-optimal problem with terminal state constraint and cost $J=m(t_f)$ . The resulting thrust law typically shows always-braking or Bang-off-Bang behavior. For standard soft-landing formulations, non-trivial singular arcs can be ruled out (You & Dai 2022), justifying the Bang-off-Bang assumption.

Cislunar low-thrust transfers

LEO to $L_1$ / $L_2$ Halo: Zhang et al. (2025) solve $L_1$ -Halo to $L_2$ -Halo transfers with erf-smoothing homotopy, fuel consumption only 0.34% of initial mass.
Planar CR3BP minimum-fuel: Caillau et al. (2012) solve GEO-to- $L_1$ /Moon transfers at 0.3 N using $L^2$ -- $L^1$ and logarithmic-barrier homotopies, emphasizing the need for conjugate-point tests.
Multi-stage formulation: long transfers are split into thrust-coast-thrust legs, each a separate BVP stitched by match conditions (see Indirect Methods).

Station-keeping

Long-term maintenance of NRHO, Halo and other libration-point orbits is a sequence of small fuel-optimal problems: whenever deviations exceed a threshold, a minimum- $\Delta V$ correction over a fixed horizon is solved, with Bang-off-Bang characteristics (Zhang and Wang 2022).

Bridge to energy-fuel homotopy

Direct solution of Bang-off-Bang fuel-optimal control is infeasible (discontinuous control, unknown switch count). The mainstream route is the energy-fuel homotopy that continues a smooth energy-optimal solution toward the fuel-optimal one. Three smoothing families dominate the literature:

Polynomial smoothing ( $u(1-u)$ , Bertrand & Epenoy 2002): simplest, but degrades at low thrust;
Logarithmic barrier ( $-u\log u-(1-u)\log(1-u)$ , Caillau et al. 2012; Taheri et al. 2016): enforces $0<u<1$ , makes the Hamiltonian everywhere differentiable;
Sigmoid family ( $\tanh$ , algebraic, error function erf, Zhang et al. 2025): directly approximates $\mathrm{sign}(\rho)$ ; erf converges fastest.

Empirical rule: at $\varepsilon\sim 10^{-5}$ the throttle profile is visually indistinguishable from true Bang-off-Bang (Taheri et al. 2016; Zhang et al. 2025). See Homotopy Method.

Primer Vector: the adjoint quantity determining optimal thrust direction and impulse times
Bang-bang Control: the typical structure of fuel-optimal solutions
Homotopy Method: the numerical workhorse for Bang-off-Bang fuel-optimal solutions
Pontryagin's Minimum Principle: the theorem behind optimal control laws
Co-state Variables: the source of the switching function
Indirect Methods: the method framework for fuel-optimal problems
Electric Propulsion: the physical carrier of fuel-optimal low-thrust control

References

Lawden, D. F. 1963. Optimal Trajectories for Space Navigation. Butterworths, London.
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.
Bertrand, R., and Epenoy, R. 2002. "New Smoothing Techniques for Solving Bang–Bang Optimal Control Problems." Optim. Control Appl. Methods 23(4): 171–197.
Caillau, J.-B., Cerf, M., Dujols, A., et al. 2012. "Minimum Fuel Control of the Planar Circular Restricted Three-Body Problem." CEP.
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.
Zhang, et al. 2025. "Smoothing Technique for Indirect Low-Thrust Trajectory Optimization in Cislunar Space." Space Sci. Technol.
Zhu, Z., and Gao, Y. 2017. "Survey of Two Continuation Methods for Optimal Bang-Bang Control of Low-Thrust Trajectories." J. Deep Space Explor. 4(2): 101–110. (In Chinese.)
You, S., and Dai, R. 2022. "Fuel-Optimal Trajectory Generation via Down-To-The-Moon Approach." JGCD, doi:10.2514/1.G006815.