This thesis introduces a set of methods for optimization problems in engineering applications. Some methods focus on modeling and separating signal mixtures, particularly those containing periodic and aperiodic components. These types of signals are often encountered in the monitoring of industrial machinery. Another series of methods propose a new numerical approach for optimal system control, applicable to a broad spectrum of engineering systems described by ordinary differential equations. The unifying principle across all methods is the utilization of a-priori system knowledge, to limit the solution space to outcomes that are physically valid. In signal separation, this involves restricting solutions to periodic and aperiodic mixtures, that have a dominant global periodic component. The optimal control methods guarantee attainable results by integrating the system dynamics through mathematical representations. These methods are presented through a series of peer reviewed papers. The thesis is divided into two main parts - Signal Separation & Data Analysis and Optimal Control. The signal separation algorithms are based on the method of variable projection, due to its accurate modeling and good numerical properties. The algorithms are first designed for handling batch data, and are subsequently extended to work recursively, making them more suitable for processing streaming data. The performance of these algorithms is demonstrated on industrial measurement data. The optimal control problem, in particular optimal reference tracking, is posed as an inverse problem of finding an optimal system input in order to achieve a desired output behaviour. Using the calculus of variations framework, a set of differential equations is derived, that constitute an optimality condition, i.e. the Euler-Lagrange equations. To compute the optimal solution numerically, matrix methods for solving differential equations are introduced. This methodology is first applied to derive a feed-forward optimal control trajectory for a multidimensional tracking problem, for both single and multi-agent systems. It is further extended to a closed-loop control formulation, resembling a form of model predictive control with soft constraints. Additionally, the discretization method is used to cast the optimal control task as a quadratic programming problem, which also supports hard constraints. Furthermore, these methods are applicable not only to tracking problems, but also to general full-state feedback control design.