Micro-electromechanical systems (MEMS) are rapidly advancing technologies with broad applications in healthcare, environmental sensing, industrial automation, aerospace, consumer electronics, and the Internet of Things (IoT). As MEMS evolve, they open new possibilities across diverse fields but also face complex challenges such as manufacturing precision, material limitations, operational reli- ability, and energy efficiency. A key challenge lies in the nonlinearities inherent in MEMS, which significantly impact their dynamic behavior and performance. These nonlinear effects—stemming from material properties, geometric constraints, and operating conditions—can cause frequency shifts, amplitude distortion, and insta- bility, complicating the design, modeling, and control of high-precision MEMS. Addressing these issues demands advanced analytical and computational models, robust control techniques, and innovative structural and material solutions to en- hance reliability and functionality. This thesis adopts a comprehensive approach that combines semi-analytical meth- ods and numerical analysis techniques to address these nonlinearities challenges. The study is organized into four key stages, each focusing on a critical aspect of MEMS resonator behavior and control. The first stage involves an in-depth ana- lytical examination of the nonlinear response of the MEMS resonator, aiming to understand how nonlinearities affect its performance under various conditions. The second stage extends this analysis by studying the behavior of the MEMS device while considering the influence of multiple modes of vibration, offering a more comprehensive understanding of its dynamics. In the third stage, the research introduces a control strategy specifically designed to prevent pull-in instability, a common challenge in MEMS devices that can compromise their functionality. Finally, the fourth stage focuses on characterizing the mathematical nonlinearities of the MEMS resonator under different forms of excitations, such as mechanical shocks to capture how these excitations influence the system’s overall behavior. Through these stages, the study aims to provide valuable insights into the dynamics and control of MEMS resonators, contributing to the development of more reliable and efficient devices.

Optimal Control Problem (OCP) became apparent naturally in miscellancous areas of science and engineering. Considerable work has been done in the area of Optimal Control Problem of integer order dynamic systems. Recently, it has been demonstrated that fractional derivatives provide more accurate behavior of a dynamic system. Fractional Optimal Control Problem (FOCP) is an Optimal Control Problem whose dynamies is described by Fractional Differential Equations (FDE). The two most common types of fractional derivative are the Riemann-Liouville and Caputo derivatives. The thesis consists of four chapters organized as follows: Chapter 1 contains a brief summary for some basic concepts of Optimization Theory, Vector Optimization Problem (VOP), Scalarization Techniques, Fractional Calculus (FC), Optimal Control Problem (OCP) and Pontryagin Maximum Principle (PMP) to get a direct solution for the Optimal Control Problem as well as the Fractional Optimal Control Problem (FOCP) and the necessary optimality conditions. Chapter 2 involves a parametric study for the Fractional Optimal Control Problem in the sense of Riemann-Liouville with parameter in the constraints. Also, the Karush-Kuhn-Tucker optimality necessary conditions are established in the presence of fractional calculus. Qualitative and quantitative analysis for the solvability set and the stability set of the first kind are defined and analyzed for some classes of FOCP that contains a parameter in the constraints. A procedure is described to determine the stability set of the first kind for the considered FOCP. Finally, an illustrative example to clarify the developed method is presented. Chapter 3 covers a parametric study for the Fractional Optimal Control Problem in the sense of Riemann-Liouville with parameter in the objective function. Also, the Karush-Kuhn-Tucker optimality necessary conditions are established in the presence of fractional calculus. Qualitative and quantitative analysis for the solvability set and the stability set of the first kind are defined and analyzed for some classes of FOCP that contains a parameter in the objective function. A procedure is described to determine the stability set of the first kind for the considered FOCP. Finally, an illustrative example to clarify the developed method is presented. Chapter 4 presents a summary of the results obtained in the thesis and future research points.