This Faculty Early Career Development Program (CAREER) grant funds research that enables general purpose robotic systems that can change their forms to optimally achieve a range of functions. This research introduces a systematic framework for synthesizing form and function within a dynamical system, replacing manual motion design with a scalable, tractable, and data-efficient approach. Unlike traditional fixed-form systems, whose shapes are determined at design time and tailored to specific tasks, this research enables next-generation platforms that can continuously morph to select shapes for solving complex multi-stage tasks in an optimal manner, thereby promoting the progress of science, advancing national prosperity and welfare, and securing the national defense. Tightly integrated with the research activities, this grant also funds a comprehensive outreach strategy to engage participants across various educational levels, including K-12 students, schoolteachers, undergraduate students, and graduate students, and to establish a foundation for lasting contributions to robotics theory, system design, and STEM education through layered mentorship and interdisciplinary learning in the United States. Mobile robotic systems involve complex dynamics with high degrees of freedom, hybrid transitions, and sensitivity to contact and the environment. These challenges are magnified in morphable systems, where the configuration space is combinatorially large and time-varying. Overcoming them requires new representations, numerical methods, and control strategies that generalize across shapes and tasks. This research aims to develop a systematic framework for modeling, analyzing, and controlling hybrid dynamical systems with structured morphological variability and to provide theoretical and algorithmic tools that enable scalable co-design of physical form and control across diverse tasks. The research encompasses three thrusts: (1) constructing a unified framework for mode