Costate Variables and Adjoint Equations

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

Costate variables (also adjoint, conjugate variables, or the Lagrange multipliers paired with the state) are dual variables introduced in an optimal-control problem to enforce the dynamics. They have no directly measurable physical counterpart; geometrically they are the sensitivity of the optimal cost-to-go to the state: $\lambda_i(t)=\partial J^*/\partial x_i(t)$ (Bryson & Ho 1975; Betts 2010). Pontryagin's Maximum Principle couples the costate to the state through a Hamiltonian canonical system, turning the problem into a TPBVP in $(\mathbf{x},\boldsymbol{\lambda})$.

Hamiltonian form and adjoint equations

For $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u},t)$ and $J=\Phi(\mathbf{x}(t_f),t_f)+\int_{t_0}^{t_f}L\,dt$, define the control Hamiltonian $H=L+\boldsymbol{\lambda}^{\!\top}\mathbf{f}$. The two PMP conditions on the costate are (Conway 2010, Ch.1; Betts 2010):

1. Adjoint (costate) equations $\dot{\boldsymbol{\lambda}}=-\partial H/\partial\mathbf{x}$.
2. Transversality $\boldsymbol{\lambda}(t_f)=\partial \Phi/\partial\mathbf{x}|_{t_f}+(\partial\boldsymbol{\Psi}/\partial\mathbf{x})^{\!\top}\boldsymbol{\nu}$, where $\boldsymbol{\Psi}=0$ collects terminal equality constraints with multipliers $\boldsymbol{\nu}$. Free $t_f$ adds $H(t_f)=0$.

With $\dot{\mathbf{x}}=\partial H/\partial\boldsymbol{\lambda}$ and pointwise minimization of $H$ over $\mathbf{u}$, these define the optimal trajectory.

Costate components in trajectory optimization

For a continuous-thrust spacecraft with state $(\mathbf{r},\mathbf{v},m)$ and control $(\hat{\mathbf{T}},T)$,

• Position costate $\boldsymbol{\lambda}_r$: driven by the gravity gradient.
• Velocity costate $\boldsymbol{\lambda}_v$: $\dot{\boldsymbol{\lambda}}_v=-\boldsymbol{\lambda}_r$ when $H$ has no explicit $\mathbf{v}$. The optimal thrust direction is $-\boldsymbol{\lambda}_v/\|\boldsymbol{\lambda}_v\|$; hence $-\boldsymbol{\lambda}_v$ is the primer vector (Lawden 1963).
• Mass costate $\lambda_m$: monotone increasing, $\dot{\lambda}_m=\|\boldsymbol{\lambda}_v\|T/m^2\geq 0$. The switching function $\rho=1-\lambda_m I_{sp}g_0/m-\|\boldsymbol{\lambda}_v\|$ decides $T$ at its bounds, yielding bang-bang / bang-off-bang laws (see Bang-Bang Control, Primer Vector).

Costate normalization

For free-final-state problems, $(\boldsymbol{\lambda}(t_0),\lambda_0)$ can be rescaled by any positive constant without changing the trajectory (PMP homogeneity). Costate normalization fixes $\|\boldsymbol{\lambda}(t_0)\|=1$ (or one component to unity), reducing the search dimension by one and improving conditioning (Thorne 1996; Oshima et al. 2017). Minimum-time problems lose this homogeneity and replace it by $H\equiv -1$; minimum-fuel problems keep it.

Adjoint-control transformation

Initial costates have no physical meaning. The adjoint-control transformation (Kluever & Pierson 1995; Conway 2010, Ch.4) replaces $\boldsymbol{\lambda}(t_0)$ by intuitive variables: thrust angles $(\alpha,\beta)$, their rates $(\dot\alpha,\dot\beta)$, plus $\lambda_v$, $\dot{\lambda}_v$, $\lambda_m$. The implicit costate transformation (Pozzi et al. 2025) is the multi-arc analogue: a closed-form map carries the final costate of one arc into the initial costate of the next.

Initial-costate sensitivity

Indirect methods suffer because $\boldsymbol{\lambda}(t_0)$ is numerically unstable, small perturbations diverge by $t_f$. The optimal initial costate locus describes $\boldsymbol{\lambda}(t_0)$ as a curve over problem parameters with distinct behaviour in parabolic, elliptic, and spiral regimes (Thorne 1996). In practice, homotopy methods sweep from an easily solved energy-optimal problem to the fuel-optimal target.

Application notes

• The bottleneck in indirect methods is the initial-costate guess and homotopy design, an order of magnitude harder than guessing states.
• $\lambda_m$ spans many decades in large-scale transfers; log-scaling or independent normalization avoids ill-conditioning.
• In CR3BP-LT, $H_{lt}=H_{nat}-\boldsymbol{\lambda}^{\!\top}\mathbf{a}_{lt}$ becomes a new integral when $\mathbf{a}_{lt}$ is fixed in the rotating frame (Cox et al. 2021).

Related concepts

• Pontryagin's Maximum Principle
• Hamiltonian
• Two-Point Boundary-Value Problem (TPBVP)
• Primer Vector
• Bang-Bang Control
• Homotopy Method
• Indirect Methods

References

• Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control.
• Lawden, D. F. (1963). Optimal Trajectories for Space Navigation.
• Betts, J. T. (2010). Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed.
• Conway, B. A. (Ed.) (2010). Spacecraft Trajectory Optimization, Ch. 1–2, 4.
• Thorne, J. D. (1996). Optimal continuous-thrust orbit transfers. Acta Astronautica, 38(8), 565–578.
• Kluever, C. A., & Pierson, B. L. (1995). Optimal Earth-Moon trajectories using nuclear electric propulsion. JGCD.
• Oshima, K., et al. (2017). Earth-Moon transfer trajectories … JGCD.
• Cox, A. B., et al. (2021). CR3BP with low-thrust.
• Pozzi, E., et al. (2025). Implicit costate transformation for multi-arc optimal control.