Let X be a compact Hausdorff topological space. The Banach space C(X) consists of all continuous complex value functions with the supnorm. An operator P on C(X) is called a generalized bicircular projection if P + £f(I − P) is an isometry for all |£f| = 1, £f in C and P2 = P. In this thesis, we study some projections which are the averages of two composition operators or two weighted composition operators on C(X). If a projection is the average of the identity and a composition operator, it is a generalized bicircular projection. And give an example of a projection which is the average of the identity and a weighted composition operator, but not a generalized bicircular projection. We also discuss some projections which are the average of two bounded linear operators on a Banach space. And the main result is that, let T1 and T2 are two bounded linear operators on a Banach space, and Q = T1+T2 2 . If T1 ¡CT2 = T2 ¡CT1 and T2 1 = T2 2 = Id then Q is a tripotent, i.e. Q3 = Q.

<div>After completion of a resupply mission to NASA’s proposed Lunar Orbital Platform - Gateway, safe disposal of the Logistics Module is required. One potential option is disposal to heliocentric space. This investigation includes an exploration of the trajectory escape dynamics from an Earth-Moon L2 Near Rectilinear Halo Orbit (NRHO). The effects of the solar gravitational perturbations are assessed in the Bicircular Restricted 4-Body Problem (BCR4BP), as defined in the Earth-Moon rotating frame and in the Sun-B1 rotating frame, where B1 is the Earth-Moon barycenter. Disposal trajectories candidates are classified in three outcomes: direct escape, in direct escapes and captures.</div><div>Characteristics of each outcome is defined in terms of three parameters: the location of the apoapses within to the Sun-B1 rotating frame, a characteristic Hamiltonian value, and the osculating eccentricity with respect to the Earth-Moon barycenter. Sample trajectories are presented for each outcome. Low-cost disposal options are introduced.</div>

<p>Many opportunities for frequent transit between the lunar vicinity and the heliocentric region will arise in the near future, including servicing missions to space telescopes and proposed missions to various asteroids and other destinations in the solar system. The overarching goal of this investigation is the development a framework for periodic and transit options in the Earth-Moon-Sun system. Rather than overlapping different dynamical models to capture the dynamics of the cislunar and heliocentric region, this analysis leverages a four-body dynamical model, the Bicircular Restricted Four-Body Problem (BCR4BP), that includes the dynamical structures that exist due to the combined influences of the Earth, the Moon, and the Sun. The BCR4BP is an intermediate step in fidelity between the CR3BP and the higher-fidelity ephemeris model. The results demonstrate that dynamical structures from the Earth-Moon-Sun BCR4BP provide valuable information on the flow between cislunar and heliocentric spaces. </p> <p><br></p> <p>Dynamical structures associated with periodic and bounded motion within the BCR4BP are successfully employed to construct transfers between the 9:2 NRHO and locations of interest in heliocentric space. The framework developed in this analysis is effective for transit between any cislunar orbit and the Sun-Earth libration point regions; a current important use case for this capability involves departures from the NRHOs, orbits that possess complex dynamics and near-stable properties. Leveraging this methodology, one-way trajectories from the lunar vicinity to a destination orbit in heliocentric space are constructed, as well as round-trip trajectories that returns to the NRHO after completion of the objectives in heliocentric space. The challenges of such trajectory design include the phasing of the trajectory with respect to the Earth, the Moon, the Sun, on both the outbound and inbound legs of the trajectory. Applications for this trajectory include servicing missions to a space telescope in heliocentric space, where the initial and final locations of the mission is the Gateway near the Moon. Lastly, the results of this analysis demonstrate that the properties and geometry of the periodic orbits, bounded motion, and transfers that are delivered from the BCR4BP are maintained when the trajectories are transitioned to the higher-fidelity ephemeris model. </p>

<p>An increasing interest in lunar exploration calls for low-cost techniques of reaching the Moon. Ballistic lunar transfers are long duration trajectories that leverage solar perturbations to reduce the multi-body energy of a spacecraft upon arrival into cislunar space. An investigation is conducted to explore methods of constructing ballistic lunar transfers. The techniques employ dynamical systems theory to leverage the underlying dynamical flow of the multi-body regime. Ballistic lunar transfers are governed by the gravitational influence of the Earth-Moon-Sun system; thus, multi-body gravity models are employed, i.e., the circular restricted three-body problem (CR3BP) and the bicircular restricted four-body problem (BCR4BP). The Sun-Earth CR3BP provides insight into the Sun’s effect on transfers near the Earth. The BCR4BP offers a coherent model for constructing end-to-end ballistic lunar transfers. Multiple techniques are employed to uncover ballistic transfers to conic and multi-body orbits in cislunar space. Initial conditions to deliver the spacecraft into various orbits emerge from Periapse Poincaré maps. From a chosen geometry, families of transfers from the Earth to conic orbits about the Moon are developed. Instantaneous equilibrium solutions in the BCR4BP provide an approximate for the theoretical minimum lunar orbit insertion costs, and are leveraged to create low-cost solutions. Trajectories to the <i>L</i>2 2:1 synodic resonant Lyapunov orbit, <i>L</i>2 2:1 synodic resonant Halo orbit, and the 3:1 synodic resonant Distant Retrograde Orbit (DRO) are investigated.</p>

<p>Secondary payloads have a rich and successful history of utilizing cheap rides to orbit to perform outstanding missions in Earth orbit, and more recently, in cislunar space and beyond. New launch vehicles, namely the Space Launch System (SLS), are increasing the science opportunity for rideshare class missions by providing regular service to the lunar vicinity. However, trajectory design in a multi-body regime brings a host of novel challenges, further exacerbated by constraints generated from the primary payload’s mission. Often, secondary payloads do not possess the fuel required to directly insert into lunar orbit and must instead perform a lunar flyby, traverse the Earth-Moon-Sun system, and later return to the lunar vicinity. This investigation develops a novel framework to construct low-cost, end-to-end lunar transfer trajectories for secondary payload missions. The proposed threephase approach provides unique insights into potential lunar transfer geometries. The phases consist of an arc from launch to initial perilune, an exterior transfer arc, and a lunar approach arc. The space of feasible transfers within each phase is determined through low-dimension grid searches and informed filtering techniques, while the problem of recombining the phases through differential corrections is kept tractable by reducing the dimensionality at each phase transition boundary. A sample mission demonstrates the trajectory design approach and example solutions are generated and discussed. Finally, alternate strategies are developed to both augment the analysis and for scenarios where the proposed three-phase technique does not deliver adequate solutions. The trajectory design methods described in this document are applicable to many upcoming secondary payload missions headed to lunar orbit, including spacecraft with only low-thrust, only high-thrust, or a combination of both. </p>

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<p dir="ltr"> With NASA's Artemis program and international collaborations focused on building a sustainable infrastructure for human exploration of the Moon, there is a growing demand for lunar exploration and complex spaceflight operations in cislunar space. However, designing efficient transfer trajectories between the Earth and the Moon remains complex and challenging. This investigation focuses on developing a dynamically informed framework for constructing low-energy transfers in the Earth-Moon-Sun Bicircular Restricted Four-body Problem (BCR4BP). Techniques within dynamical systems theory and numerical methods are exploited to construct transfers to various cislunar orbits. The analysis aims to contribute to a deeper understanding of the dynamical structures governing spacecraft motion. It addresses the characteristics of dynamical structures that facilitate the construction of propellant-efficient pathways between the Earth and the Moon, exploring periodic structures and energy properties from the Circular Restricted Three-body Problem (CR3BP) and BCR4BP. The investigation also focuses on constructing families of low-energy transfers by incorporating electric propulsion, i.e., low thrust, in an effort to reduce the time of flight and offer alternative transfer geometries. Additionally, the investigation introduces a process to transition solutions to the higher fidelity ephemeris force model to accurately model spacecraft motion through the Earth-Moon-Sun system. This research provides insights into constructing families of ballistic lunar transfers (BLTs) and cislunar low-energy flight paths (CLEFs), offering a foundation for future mission design and exploration of the Earth-Moon system.</p>