电推进比化学推进更有助于减少燃料消耗、增加有效载荷质量，因此近几年越来越广泛地应用于深空探测任务中。比冲是评估燃料消耗和衡量推进效果的重要的参量，无论是霍尔效应推进器还是离子电推进发动机，它们的比冲都是变化的。所以在轨迹设计中，变比冲电推进模型是更符合工程实际的，变比冲连续小推力轨迹优化问题的研究是具有工程应用前景的。本论文关注三个变比冲电推进简化数学模型下的轨迹优化问题，因此面临着应用间接法求解的常见的困难以及挑战，除此之外，还考虑了变比冲小推力多次引力辅助燃料最优轨迹的求解，新增的时变的多内点约束也会增加问题的求解难度和规模。为了克服上述困难，提高间接法求解变比冲连续小推力轨迹优化问题的效率，本文在同伦方法、开关函数检测、多内点约束问题的处理方法等方面进行了深入研究。首先，在比冲区间变化的数学模型下，采用间接法构造两点边值问题，并辅助以同伦方法求解燃料最优轨迹。给出了改进的同伦函数来解决应用常用的二次型和对数同伦函数造成的控制量耦合的问题。通过地球到火星转移的单圈和两圈算例，以及与双比冲和定比冲模型下的剩余燃料的对比，验证了可行性和有效性。其次，考虑比冲随距离变化的发动机模型，比冲和推力是发动机输入功率的函数。应用变分法和庞得里亚金极小值原理推导了时间最优和燃料最优问题的一阶必要条件。并且推导了解析的雅各比矩阵估计打靶方程的梯度信息以提高收敛效率。再采用牛顿法和二分法结合的开关函数检测技术提高积分精度，可以高效求解真实发动机参数拟合模型下的轨迹优化问题。再次，在功率分档的简化模型下，推导了时间最优和燃料最优问题的最优控制律。并且针对该多离散点模型，提出了功率档位检测方法，解决微分方程右端项不连续造成的问题，提高了积分精度和收敛效率。该方法不仅适用于单个发动机具有多个离散工作档位的情况，还可用于求解多个定比冲定推力推进系统的轨迹优化问题。最后，对于区间变比冲模型的多次引力辅助的轨迹优化问题，给出了两套完整的系统的求解燃料最优轨迹的方案，均包含等效脉冲模型下的初始轨道设计以及变比冲小推力多内点约束问题的求解。方案一是应用间接法常规地推导了变比冲多次引力辅助的一阶必要条件，而方案二借用协态变量转换的思想重新推导了多点边值问题的一阶必要条件。通过主带小行星探测的单次和三次引力辅助算例对比，验证了方案二相较于方案一的高效性，可以对未来深空探测的轨迹设计提供参考。

Electric propulsion has more advantages in propellant saving compared to chemical propulsion. Thus, in recent years, it was widely applied in deep-space exploration missions. Specific impulse is an important parameter to estimate the fuel consumption and evaluate the propulsive effect. The specific impulse is varied either in hall effect thrusters or ion electric engines. Therefore, the variable-specific-impulse low-thrust mathematical model is considered more practical in mission design. The research on optimization of variable-specific-impulse continuous low-thrust trajectories is of broad prospect in engineering practice. This thesis focuses on trajectory optimization of three simplified variable-specific-impulse mathematical models. The difficulties and challenges arise in using indirect methods. Moreover, the optimization of variable-specific-impulse multi-gravity-assist trajectories is also studied in this thesis. The added multiple interior-point constraints result in the increment of the solving scale and the problem becomes more difficult to solve.In order to overcome the difficulties above and improve the solving efficiency of the variable-specific-impulse optimal trajectories in using indirect methods, homotopy methods, switching detection techniques, methods to deal with the multiple interior-point constraints and other methods are applied in this work.Firstly, when considering the variable-specific-impulse model in which the specific impulse varies within a certain range, two-point boundary value problems are established by indirect methods. Additionally, the homotopy methods are used to optimize the fuel-optimal trajectories. Modified homotopy functions are given to overcome the difficulties that the optimal control variables are coupled caused by using the two commonly-used homotopy functions. Numerical simulations, which are based on the one- and two-revolution Earth-to-Mars transfers and the comparison of the fuel consumption with dual- and constant-specific-impulse model, validate the feasibility and efficiency of the method.Secondly, the thruster model that the specific impulse varies with the Sun-spacecraft distance is considered in this work. The specific impulse and the maximum thrust are served as a function of thruster input power. The first-order necessary conditions of minimum-time and minimum-fuel problems are formulated by calculus of variations and Pontryagin's minimum principle. Moreover, in order to increase the convergence rate of the shooting function, analytic Jacobians are derived. Besides, a switching detection technique combining Newton's and bisection methods is used to improve the integration accuracy. It can be used to efficiently optimize the trajectories considering real gridded ion thruster model with point-fitting lines.Thirdly, the thruster model is described using a finite number of operation points, which are characterized by different pairs of thruster input power. The optimal control law is derived for time-optimal and fuel-optimal problems. In order to guarantee the integral accuracy and improve the convergence rate for the discrete power-limited problem, a power operation detection method is proposed to overcome the difficulties caused by the discontinuous right-hand sides of the differential equations. The proposed method is proved to be feasible for the thruster model with multiple operation points. In addition, it can be also applied to the trajectory optimization problems with more than one constant-thrust engines.Finally, two systematic schemes are given to optimize the variable-specific-impulse multi-gravity-assist fuel-optimal trajectories. These schemes all include two steps: the preliminary design by patched-conic approximation and the optimization of the variable-specific-impulse low-thrust multi-gravity-assist trajectories. In the first scheme, the first-order necessary conditions are derived for variable-specific-impulse multiple interior-point problems. While in the second scheme, the optimality-preserving transformation is applied to reformulate the first-order necessary conditions for the multi-point boundary-value problem. Main-belt asteroids exploration missions with one and three gravity assists are given to substantiate that the second scheme has more advantages in solving efficiency compared to the first one. They are expected to be taken as references for the future deep-space mission design.