Pontryagin's Maximum Principle

Author: Tianjiang Says

Contributing institutions: School of Astronautics, Harbin Institute of Technology; National Key Laboratory of Rapid Design and Intelligent Swarm for Micro/Nano Spacecraft

References: Guan Yutong et al. Hyperparameter Auto-Tuning and Homotopy Methods for Spacecraft Long-Range Cooperative Rendezvous, Spacecraft Environment Engineering, 2026.

Definition

Pontryagin's Maximum Principle is a core theorem of optimal control theory proposed by the Soviet mathematician Pontryagin and colleagues in 1958. It provides the first-order necessary conditions for solutions to continuous optimal control problems and serves as the theoretical foundation for indirect methods in solving optimal control problems.

Core Formulas

Hamiltonian

Construct the Hamiltonian:

H=λvTf(x,u,t)+λxTx+λmm˙H = \lambda_v^T f(x, u, t) + \lambda_x^T x + \lambda_m \dot{m}

where λ=[λv;λx;λm]\lambda = [\lambda_v; \lambda_x; \lambda_m] are the costate variables.

Costate Equations

Costate variables satisfy:

λ˙=−∂H∂x\dot{\lambda} = -\frac{\partial H}{\partial x}

Extremality Condition

For fuel-optimal control, the optimal thrust ratio satisfies:

uj∗={0,ρj>01,ρj<0∈(0,1),ρj=0u_j^* = \begin{cases} 0, & \rho_j > 0 \\ 1, & \rho_j < 0 \\ \in (0,1), & \rho_j = 0 \end{cases}

where ρj=∂H∂uj\rho_j = \frac{\partial H}{\partial u_j} is the switching function.

Application in Trajectory Optimization

Application by Zhao Han et al. (2026)

In spacecraft cooperative rendezvous fuel-optimal problems:

  1. Constructing the performance index:

J=∑j=12FjIspg0∫t0tfujdtJ = \sum_{j=1}^{2} \frac{F_j}{I_{sp}g_0} \int_{t_0}^{t_f} u_j dt

  1. Establishing costate equations: Costate differential equations are obtained by taking partial derivatives of the Hamiltonian

  2. Determining optimal control: Thrust direction and thrust ratio are determined from the extremality conditions

  3. Shooting solution: The two-point boundary value problem is transformed into a shooting problem for solution

Physical Meaning of Costate Variables

Costate variables have profound physical significance:

  • Position costate λr\lambda_r: Related to position gradient, affects orbit shape
  • Velocity costate λv\lambda_v: Related to velocity gradient, determines thrust direction
  • Mass costate λm\lambda_m: Related to mass gradient, determines fuel consumption

Relationship with Bang-Bang Control

Pontryagin's Maximum Principle directly leads to the bang-bang characteristic of fuel-optimal control:

  • When the switching function ρj>0\rho_j > 0, thrust is zero (coasting)
  • When the switching function ρj<0\rho_j < 0, thrust is at maximum
  • Switching occurs when ρj=0\rho_j = 0

References

  • Pontryagin L S, et al. The Mathematical Theory of Optimal Processes[M]. Wiley, 1962.
  • Bryson A E, Ho Y C. Applied Optimal Control[M]. Hemisphere, 1975.
  • Guan Yutong, Gao Changsheng, Hu Yudong, Zhao Han. Hyperparameter Auto-Tuning and Homotopy Methods for Spacecraft Long-Range Cooperative Rendezvous[J]. Spacecraft Environment Engineering, 2026. [in Chinese]