Optimal Control and Trajectory Optimization in Aerospace Propulsion Systems

Optimal control and trajectory optimization constitute the theoretical and computational framework by which aerospace vehicles—ranging from chemical rockets to electric-propulsion spacecraft and hypersonic vehicles—achieve mission objectives with minimal fuel, time or energy expenditure. Rooted in the calculus of variations and Pontryagin’s minimum principle, this discipline addresses the selection of control inputs (such as thrust magnitude and direction) and state trajectories (orbital elements or flight-path angles) that steer a vehicle from an initial to a final condition while satisfying physical constraints. Two principal solution paradigms prevail: indirect methods, which derive necessary conditions for optimality and solve resultant two-point boundary-value problems; and direct methods, which transcribe the control problem into a finite-dimensional nonlinear programme via collocation or pseudospectral schemes. Recent advances have focused on enhancing numerical robustness, reducing sensitivity to initial guesses and enabling onboard real-time computation through surrogate models or machine-learning surrogates. Electric propulsion missions, with their low-thrust continuous trajectories, have driven the development of multi-revolution optimisation techniques and homotopy-based approaches to identify multiple local minima. In parallel, hypersonic and reusable launch systems have motivated rapid optimisation algorithms that cope with atmospheric dynamics and stringent thermal or structural constraints. The integration of neural networks and deep-learning frameworks now offers promising routes to approximate Hamilton–Jacobi–Bellman solutions for guidance, navigation and control tasks, thereby bridging the gap between offline optimal trajectories and real-time execution.

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Optimal Control and Trajectory Optimization in Aerospace Propulsion Systems publication trend

The graph below shows the total number of articles in optimal control and trajectory optimization in aerospace propulsion systems across all publications each year (not limited to Nature Index journals).

Technical terms

Optimal control: Mathematical formulation to determine control inputs that optimise a performance index subject to dynamic constraints.

Trajectory optimization: Numerical process of finding state and control histories that minimise cost (fuel, time) while satisfying boundary-value and path constraints.

Direct transcription: Method that discretises control and state variables into finite nodes and solves the resulting nonlinear programme.

Pseudospectral method: Collocation technique using global polynomial approximations at Gauss-type nodes for high-accuracy trajectory solutions.

Homotopy method: Continuation approach that tracks solution paths from a simple problem to the target problem to overcome nonlinearity and multiple minima.

Bang-bang control: Control law characterised by instantaneous switches between extreme limits of the available control authority.

References

1. Survey of Direct Transcription for Low‐Thrust Space Trajectory Optimization with Applications. Abstract and Applied Analysis (2014).
2. A Survey on Low-Thrust Trajectory Optimization Approaches. Aerospace (2021).
3. Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem. IEEE Access (2020).
4. A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network. Aerospace (2022).
5. Deep Learning and Artificial Neural Networks for Spacecraft Dynamics, Navigation and Control. Drones (2022).