Optimal low-thrust GTO–GSO transfers using differential evolution


Transfers from geosynchronous transfer orbits (GTO) to geosynchronous orbits (GSO) using electric propulsion have been optimized in the current study. Both time-optimal and fuel-optimal trajectories are generated. Three-dimensional equations of motion are considered for the system dynamics. The indirect approach based on optimal control theory is followed and the resulting two-point boundary value problem is solved using differential evolution, a search-based global optimization technique. Optimal trajectories for various mission scenarios are obtained using differential evolution and compared to transfers presented in literature.

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Correspondence to Padmanabha Prasanna Simha.

List of symbols

List of symbols


Differential evolution


Two-point boundary value problem


Geosynchronous transfer orbit


Geosynchronous orbit


Crossover ratio


Mutation factor


Population size

\(\mu _p\) :

Gravitational parameter of the] primary body (Earth)

\(\mu _s\) :

Gravitational parameter of the secondary body

r :

Radial distance from the centre of the Earth

\(r_s\) :

Radial distance from the centre of the secondary body

\({\vec {\alpha }}\) :

Acceleration vector due to thrust

m :

Spacecraft mass

\(I_{sp}\) :

Specific impulse

\({\vec{\mathbf {x}}}\) :

State vector

\({\vec {\lambda} }\) :

Costate vector

\(\Phi _f\) :

Terminal cost function

\(J_{time}\) :

Time-optimal cost function

\(J_{fuel}\) :

Fuel-optimal cost function

\({\hat{h}}\) :

Angular momentum unit vector

\({\vec{\mathbf {e}}}\) :

Eccentricity vector

\(T_{max}\) :

Maximum available thrust

\(\varvec{H_{time}}\) :

Time-optimal Hamiltonian

\(\varvec{H_{fuel}}\) :

Fuel-optimal Hamiltonian

\({\vec {\alpha }}^*\) :

Optimal acceleration vector

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Prasanna Simha, P., Ramanan, R.V. Optimal low-thrust GTO–GSO transfers using differential evolution. Sādhanā 46, 1 (2021). https://doi.org/10.1007/s12046-020-01523-x

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  • Low thrust
  • trajectory optimization
  • differential evolution