The use of Machine Learning (ML) and optimization for control applications in enhanc-ing performance, replicating expert behaviors, and addressing other complex challenges has emerged as a focal point in contemporary research. This has become possible due to ML techniques’ ability to find patterns, approximate complex functions and make decisions from data. Although promising, learning-based controllers often encounter difficulties in main- taining stability guarantees due to their reliance solely on data. This data is often noisy or incomplete, which may lead to unpredicted and in many cases unstable system responses under previously unseen scenarios. A typical ML problem is formulated by designing a loss function and the parameters that minimize the loss are used to obtain the optimal model. For control applications, a constraint on the stability needs to be imposed both during train- ing and inference. But, most ML models do not impose such conditions as hard constraints and only encourage through soft constraints in the loss function. These constraints might not be satisfied during inference and the ML control laws can destabilize the system. This dissertation addresses the challenge of designing stabilizing controllers utilizing ML and optimization techniques for two distinct classes of systems. In the first part, we explore de- signing controllers through Neural Network potential functions for fully actuated mechanical systems that evolve on manifolds with well-defined dynamics in the state space. The control laws incorporate the concept of invariance for data efficient training and easy transferability between robots with similar kinematic structure. The design methodology will be empha- sized on an application to variable impedance control of mechanical manipulators. In the second part, we discuss the robust control of Multi-Input Single Output (MISO) systems, with a particular focus on Multi-Actuator Hard Disk Drives (HDDs). A design method- ology in frequency domain based on available frequency response data to ensure stability and robustness against disturbances and model uncertainties is presented. An unsupervised ML technique to cluster the plant transfer functions and frequency responses into subgroups is presented. Clustering helps us design common controllers within each cluster to both maintain robustness and improve performance. Lastly, identification methods for obtaining dynamical models of disturbance processes with colored noises and necessary filters that sat- isfy Strictly Positive Real (SPR) conditions for stability of adaptive control algorithms are presented. Throughout the dissertation, we will delve into the theoretical underpinnings of these methodologies, complemented by simulation results that highlight significant improve- ments in system responsiveness and efficiency achieved through these innovative control strategies. This dissertation shows that integrating learning-based mechanisms into mechanical system controllers is both feasible and effective. It also offers guidance for future research to ad- dress the challenges of these technologies. By combining theoretical analysis and simulation studies, this work demonstrates how data-driven approaches can improve control systems.