##### Abstract

This dissertation provides a complete mathematical framework to compute the sensitivities with respect to system parameters for any second order hybrid Ordinary Differential Equation (ODE) and rank 1 and 3 Differential Algebraic Equation (DAE) systems.
The hybrid system is characterized by discontinuities in the velocity state variables due to an impulsive forces at the time of event. At the time of event, such system may also exhibit a change in the equations of motion or in the kinematic constraints.
The analytical methodology that solves the sensitivities for hybrid systems is structured based on jumping conditions for both, the velocity state variables and the sensitivities matrix. The proposed analytical approach is then benchmarked against a known numerical method.
The mathematical framework is extended to compute sensitivities of the states of the model and of the general cost functionals with respect to model parameters for both, unconstrained and constrained, hybrid mechanical systems.
This dissertation emphasizes the penalty formulation for modeling constrained mechanical systems since this formalism has the advantage that it incorporates the kinematic constraints inside the equation of motion, thus easing the numerical integration, works well with redundant constraints, and avoids kinematic bifurcations.
In addition, this dissertation provides a unified mathematical framework for performing the direct and the adjoint sensitivity analysis for general hybrid systems associated with general cost functions. The mathematical framework computes the jump sensitivity matrix of the direct sensitivities which is found by computing the Jacobian of the jump conditions with respect to sensitivities right before the event. The main idea is then to obtain the transpose of the jump sensitivity matrix to compute the jump conditions for the adjoint sensitivities.
Finally, the methodology developed obtains the sensitivity matrix of cost functions with respect to parameters for general hybrid ODE systems. Such matrix is a key result for design analysis as it provides the parameters that affect the given cost functions the most. Such results could be applied to gradient based algorithms, control optimization, implicit time integration methods, deep learning, etc.