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Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/off independently and "optimal" operating power of each engine depends on the available solar power, which depends on the distance from the Sun. Application of the Composite Smooth Control (CSC) framework to a heliocentric fuel-optimal trajectory optimization from the Earth to the comet 67P/Churyumov-Gerasimenko is demonstrated, which presents a new approach to deal with multiple-engine problems. Operation of engine clusters with 4, 6, 10 and even 20 engines of the same type can be optimized. Moreover, engine clusters with different/mixed electric engines are considered with either 2, 3 or 4 different types of engines. Remarkably, the CSC framework allows us 1) to reduce the original multi-point boundary-value problem to a two-point boundary-value problem (TPBVP), and 2) to solve the resulting TPBVPs using a single-shooting solution scheme and with a random initialization of the missing costates. While the approach we present is a continuous neighbor of the discontinuous extremals, we show that the discontinuous necessary conditions are satisfied in the asymptotic limit. We believe this is the first indirect method to accommodate a multi-mode control of this level of complexity with realistic engine performance curves. The results are interesting and promising for dealing with a large family of such challenging multi-mode optimal control problems. • Optimization of low-thrust interplanetary trajectories with multiple engines. • Engine clusters with the same or different/mixed engine types are considered. • Same-type engine cluster with 4, 6, 10 and 20 engines are investigated. • Different-type engine clusters with 2, 3 and 4 engines are investigated. • Fuel-optimal trajectories from Earth to comet 67P/Churyumov-Gerasimenko. [ABSTRACT FROM AUTHOR]

Efficient performance of a number of engineering systems is achieved through different modes of operation - yielding systems described as "hybrid", containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable- I sp , variable-thrust engine or 2) multiple engines each capable of switching on/off independently. To alleviate the challenges that arise when an indirect optimization method is used, a new framework — Composite Smooth Control (CSC) — is proposed that seeks smoothness over the entire spectrum of distinct control inputs. A salient aftermath of the application of the CSC framework is that the original multi-point boundary-value problem can be treated as a two-point boundary-value problem with smooth, differentiable control inputs; the latter is notably easier to solve, yet can be made to accurately approximate the former hybrid problem. The utility of the CSC framework is demonstrated through a multi-year, multi-revolution heliocentric fuel-optimal trajectory for a spacecraft equipped with a variable- I sp , variable-thrust engine. • Optimization of low-thrust interplanetary trajectories with variable specific impulse, variable thrust engines. • A new framework, called Composite Smooth Control (CSC), is developed to handle various types of state-triggered constraints. • Indirect optimization method is used to formulate fuel-optimal problems. • Complex-based derivative approach is used to construct co-state dynamics numerically. • Multi-revolution, multi-year trajectory optimization is analyzed. [ABSTRACT FROM AUTHOR]

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework. • Co-optimization of trajectory and solar electric propulsion system parameters. • Comparison of constant and variable- l s p thruster for low-thrust trajectory design. • Solve multi-point boundary value problem using Composite Smoothing Control (CSC). • Optimization over bounds of the l s p , variable thruster efficiency, and power is achieved. • Linear relation between thruster efficiency and l s p is near-accurate for early studies. [ABSTRACT FROM AUTHOR]