Abstract — The trajectories of spacecrafts eith the higu limited thrust rocket engine with the thrust vector control brtween the circular orbit of Earth artificial satellite and the Moon are defined on the basis of numerical solution of corresponding maximum principle boundary problems by "shooting" method within the scope of the circular restricted three bodies problem. Flights in the Moon orbit plane with minimum mass expenditure and limited flight time and fastest flights with limited mass expenditure are begin considered. The optimal trajectories of hit to the Moon, soft landing on its surfase and launching on fixed circular orbit of the Moon satellite as well as multi-turning optimal trajectories are defined here.

Abstract — Optimal trajectories fo the return of a spacecraft with jet engine of high limited thrust, controlled by the thrust vector, from the Moon to Earth are determined. These trajectories correspond either to spacecraft flights in the lunar orbit plane with the maximum terminal mass ina limited flight time, or to the fastest flights with a limited terminal mass. Optimal flight trajectories, in particular, multiple-loop trajectories from an orbit of an artifical Moon satellite or from the lunar surface to an artificial Earth satellite and to the boundary of Earth's atmosphere uhder specified conditions of the atmosphere entry are considered. The investigations are carried out on the basis of the maximum principle within the framework og the restricted circular three-body problem with solutions to bouhdary value problems by means of the shooting method.

Abstract — Optimum spatial trajectories for the flight of a spacecraft with a jet engine of a large limited thrust, controlled by the thrust vector, from a circular orbit of the Earth artifical satellite to the orbit of the Moon artificall satellite are investigated. Minimized are consumptions of mass under the condition of boundedness for the time of flight, or the time of flight under the condition of boundedness for the consumption of mass. The studies are conducted in the context of a restricted circular three-body problem, on the basis of the maximum principle. Maximum principle boundary value problems are solved by the shooting method.

Within the circular restricted problem of three bodies (points) optimal flights of a spacecraft with rocket high limited thrust engine between artificial satellites orbits around the Earth (ASE) and around the Moon (ASM), and also between orbits ASM and ASE (returning trajectories) are considered. Trajectories corresponding to flights in the lunar orbit plane are investigated. Minimization of the flight time with limited mass expenditure, or the mass expenditure with limited flight time are obtained. Corresponding optimal control problems are solved on the basis of the maximum principle. Boundary-value problems of the maximum principle are solved numerically using shooting method. Flight trajectories between circular orbits ASE and ASM with single or multiple (in case of multiturning trajectories) swicthing engine near the Earth and near the Moon (Cosmic Research No.6 1994, No.5 1995) have been determined. Trajectories which have up to 3 active sections in the Earth-orbital space and up to 2 in the Moon-orbital space have been obtained. Active sections in these trajectories are located near minimum distances of the spacecraft from the Earth and the Moon. Also flight trajectories between circular and elliptic orbits ASM and ASE have been constructed. Flight trajectories between positive (the Earth-Moon system circulation) and negative (on the reverse of the Earth-Moon system circulation) circulation directions as in orbits ASE and ASM have been obtained. The analysis of calculation results showed, that the shortest flight time is the flight time between orbits of the positive direction of the spacecraft circulation by ASE and ASM orbits (circular as well as elliptical). The time expenditure on the flight around an attractive centre of the transitional elliptical orbit is compensated by the decrease of the flight time in the central passive section of the trajectory. It was found, that there are more than one solution to the boundary-value problem of the maximum principle. Trajectories which have two active sections in the Earth-orbital have been obtained for the distinct flight time with the same mass expenditure. In comparison with the one-turn trajectory the time gain may be up to 5 hours, and the time loss may be about the flight time around the Earth along the circular orbit. The elliptical orbital injection makes possible to decrease the flight time with the limited mass (or, which is the same, to decrease the mass expenditure with the limited flight time) in comparison with the circular orbital injection.

Problems on trajectories optimization of flights between the Earth and the Moon of a spacecraft equippet with the high limited thrust rocket engine with the thrust vector control are discussed. Mass expenditure under constraint on the flight time is minimized. Optimal control problems are solved numerically on the base of the maximum principle with boundary problems solutions by shootihg method. Optimal trajectories of flights between the Earth artificial satellite orbit and the Moon are received - trajectories of hit to the Moon, the soft landing on its surface and the orbital injection of the lunar artificial satellite , and optimal trajectories of flights between the Moon and the Earth artificial satellite orbit when the spacecraft starts from the lunar surface and from the orbit of its artificial satellite. The flights in the lunar orbit plane are considered. Investigations within the scope of the three bodies circular limited problem are fulfilled.

Abstract — The fastest maneuvers controlled by a thrust vector are considered for a spacecraft with a jet engine of a large limited thrust, at a final mass restricted or unrestricted beforehand. A method of numerical solution based on the maximum principle is discussed for the problems of optimal control over these maneuvers. As examples, we consider the problems of spacecraft flights between points of a circular orbit, including the problem of the fastest flight at a final mass restricted and unrestricted beforehand, and, incidentally, the problem of a flight with minimal mass expenditure at a restricted flight time.

Abstract — A mathematically well-posed technique is suggested to obtain first-order necessary conditions of local optimality for the problems of optimization to be solved in a pulse formulation for flight trajectories of a spacecraft with a high-thrust jet engine (HTJE) in an arbitrary gravitational field in vacuum. The technique is based on the Lagrange principle of derestriction for conditional extremum problems in a function space. It allows one to formalize an algorithm of change from the problems of optimization to a boundary-value problem for a system of ordinary differential equations in the case of any optimization problem for which the pulse formulation makes sense. In this work, such a change is made for the case of optimizing the flight trajectories of a spacecraft with a HTJE when terminal and intermediate conditions (like equalities, inequalities, and the terminal functional of minimization) are taken in a general form. As an example of the application of the suggested technique, we consider in this work, within the framework of a bounded circular three-point problem in pulse formulation, the problem of constructing the flight trajectories of a spacecraft with a HTJE through one or several libration points (including the case of going through all libration points) of the Earth–Moon system. The spacecraft is launched from a circular orbit of an Earth’s artificial satellite and, upon passing through a point (or points) of libration, returns to the initial orbit. The expenditure of mass (characteristic velocity) is minimized at a restricted time of transfer.

Abstract — The necessary first-order conditions of strong local optimality (conditions of maximum principle) are considered for the problems of optimal control over a set of dynamic systems. To derive them a method is suggested based on the Lagrange principle of removing constraints in the problems on a conditional extremum in a functional space. An algorithm of conversion from the problem of optimal control of an aggregate of dynamic systems to a multipoint boundary value problem is suggested for a set of systems of ordinary differential equations with the complete set of conditions necessary for its solution. An example of application of the methods and algorithm proposed is considered: the solution of the problem of constructing the trajectories of a spacecraft flight at a constant altitude above a preset area (or above a preset point) of a planets surface in a vacuum (for a planet with atmosphere beyond the atmosphere). The spacecraft is launched from a certain circular orbit of a planets satellite. This orbit is to be determined (optimized). Then the satellite is injected to the desired trajectory segment (or desired point) of a flyby above the planets surface at a specified altitude. After the flyby the satellite is returned to the initial circular orbit. A method is proposed of correct accounting for constraints imposed on overload (mixed restrictions of inequality type) and on the distance from the planet center: extended (nonpointlike) intermediate (phase) restrictions of the equality type.

Abstract — The problem of optimal control over many-revolution spacecraft orbit transfers between circular coplanar orbits of satellites is considered. The spacecraft flight is controlled by a thrust vector of a jet engine with restricted thrust (JERT). The mass expenditure is minimized at a limited time of flight. The optimal control problem is solved based on the maximum principle. The boundary value problem of the maximum principle is solved numerically using the shooting method. A modified computation scheme of the shooting method is suggested (multi-point shooting), as well as a method (correlated with the scheme) of choosing the initial approximation with the use of a solution to the optimization problem in the impulse formulation. The scheme and method allow one to construct many-revolution spacecraft orbit transfers.

Abstract — The paper presents the solution of the first ACT global trajectory optimisation competition problem on the transfer of a spacecraft from Earth to asteroid 2001 TW-229. The initial optimal control problem is reduced to an auxiliary mathematical programming problem by means of simplifying assumptions. The auxiliary problem is solved numerically with the use of a decomposition method. The solution is then employed to construct the Pontrjagin extremals of the initial problem.

Abstract — In this first part of our paper, it is suggested to use solutions to boundary value problems in the optimization problems (in impulse formulation) for spacecraft trajectories in order to obtain the initial approximation, when boundary value problems of the maximum principle are solved numerically by the shooting method. The technique suggested is applied to the problems of optimal control over motion of the center of mass of a spacecraft controlled by the thrust vector of jet engine with limited thrust in an arbitrary gravitational field in a vacuum. The method is based on a modified (in comparison to the classic scheme) shooting method computation together with the method of continuation along a parameter (maximum reactive acceleration, initial thrust-to-weight ratio, or any other parameter equivalent to them). This technique allows one to obtain the initial approximation with a high precision, and it is applicable to a wide range of optimal control problems solved using the maximum principle, if the impulse formulation makes sense for these problems.

Abstract — As examples of application of the technique suggested in the first part of this work, the problems of optimizing the trajectories of spacecraft transfers between circular coplanar orbits are considered in this second part. During the transfer the spacecraft is controlled by the vector of thrust of a limited-thrust jet engine. The mass consumption is minimized for a limited time of transfer. Extreme trajectories with two and three powered sections (Homan-type and bi-elliptic transfer trajectories) are numerically determined. The solution of these well-studied problems allows one to compare the results of applying the suggested technique with the results of application of other previously used techniques.

Abstract — The optimization problem for trajectories of spacecraft flight from the Earth to an asteroid is considered in this paper. The flight is realized in the central Newtonian gravitational field of the Sun with a possibility of gravitational maneuvers near planets. Perturbation maneuvers are taken into account using the method of point area of action with a limitation on the flyby altitude. The spacecraft is controlled by changing the value and direction of the engine thrust. The problem is solved taking into account constraints on the launch time, flight duration, and minimum distance to the Sun.