Conjugate Point, Extremal, and Second-Order Optimality Conditions

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

Definition

Pontryagin's Maximum Principle gives first-order necessary conditions; whether a candidate extremal is truly a local optimum is decided by second-order conditions anchored on the conjugate point concept. Along a reference extremal curve, a Jacobi field $\mathbf{z}(t)$ satisfies the variational equation

\dot{\mathbf{z}}=A(t)\mathbf{z},\quad A(t)=\frac{\partial \mathbf{F}}{\partial (\mathbf{x},\boldsymbol{\lambda})}\bigg|_{\text{extremal}},

where $\mathbf{F}=(\partial H/\partial\boldsymbol{\lambda},-\partial H/\partial\mathbf{x})$ is the Hamiltonian vector field. A time $t_c>t_0$ at which a non-trivial Jacobi field with $\mathbf{z}(t_0)=0$ returns to $\mathbf{z}(t_c)=0$ is a conjugate time, and the corresponding state point is the conjugate point. If no conjugate point exists before $t_f$ , the extremal is a weak local optimum in the $C^0$ topology (Caillau & Daoud 2012; Bryson & Ho 1975).

Extremal classification

A pair $(\mathbf{x}(t),\boldsymbol{\lambda}(t),\mathbf{u}^*(t))$ satisfying PMP is an extremal. By the value of the cost multiplier $\lambda_0$ :

Normal extremal ( $\lambda_0\neq 0$ ): can normalize $\lambda_0=-1$ ; the cost contributes to $H$ with non-zero weight. Generic case.
Abnormal extremal ( $\lambda_0=0$ ): the cost drops out of $H$ ; the optimal control is determined solely by the dynamics. Caillau et al. (2012) show abnormal extremals are absent from minimum-fuel problems when $t_f$ exceeds the minimum time: only the normal case needs consideration.

The family of extremals through a fixed initial point forms the extremal flow, classified by the switching function into bang extremals (switching function non-zero, control on the boundary) and singular extremals (switching function identically zero on a finite arc, control determined by higher-order conditions).

Legendre–Clebsch conditions

The Legendre–Clebsch condition is the second-order necessary condition: along the extremal,

\frac{\partial^2 H}{\partial \mathbf{u}^2}\succeq 0

(for minimization). The strengthened Legendre–Clebsch (or strong Legendre) condition requires strict positivity $\partial^2 H/\partial\mathbf{u}^2\succ 0$ , which guarantees the extremal can be embedded in an extremal field and yields local optimality when combined with no conjugate point (Kluever & Pierson 1995; Caillau et al. 2012). It also fixes the sign of switching-function-related quantities such as the thrust-direction cosine. The original $L^1$ minimum-fuel problem does not satisfy the strengthened condition; the logarithmic-barrier homotopy restores it, which is one reason smoothing techniques work for indirect low-thrust optimization.

Geometric optimal control

Geometric optimal control treats the optimal-control problem as a geometric object on the state manifold: extremal curves, the extremal flow, conjugate loci, and the cut locus. Tools from differential geometry (distributions, Lie brackets of vector fields, sub-Riemannian structures) provide global structural results that complement the pointwise PMP. For CR3BP-type problems with thrust-direction constraints, the control distribution has rank smaller than the state dimension; together with a Riemannian metric it defines a sub-Riemannian structure, and the abnormal vs normal dichotomy becomes a question of Chow's theorem on the reachability of the distribution (Caillau & Daoud 2012).

Conjugate-point mapping and applications

In libration-point mission design, conjugate-point mapping uses a Poincaré section (e.g. $x=K$ or $y=0$ in the synodic frame) to identify connecting arcs between transfer phases (Vaquero & Howell 2014). Conjugate-point tests are also used to verify the local optimality of continuous-thrust arcs (Prussing & Sandrik 2005).

Application notes

A PMP solution without a conjugate-point and Legendre–Clebsch check is only a candidate optimum, not a proven one.
Numerical computation of conjugate points integrates the variational equation alongside the state/costate system and detects sign changes of the determinant of the Jacobi-field matrix.
In smoothed minimum-fuel formulations, the strengthened Legendre–Clebsch condition holds by construction, eliminating abnormal extremals.

References

Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control.
Caillau, J.-B., Cots, O., & Gergaud, J. (2012). Minimum fuel control of the planar circular restricted three-body problem. CEAS Space Journal.
Caillau, J.-B., & Daoud, B. (2012). Minimum time for the restricted three-body problem. SIAM J. Control Optim., 50(6).
Kluever, C. A., & Pierson, B. L. (1995). Optimal Earth-Moon trajectories using nuclear electric propulsion. JGCD.
Prussing, J. E., & Sandrik, S. L. (2005). Second-order necessary conditions and sufficient conditions applied to continuous-thrust trajectories. JGCD.
Vaquero, M., & Howell, K. C. (2014). Conjugate-point mapping in the restricted problem.
Agrachev, A. A., & Sachkov, Y. L. (2004). Control Theory from the Geometric Viewpoint.