Optimization of nonlinear dynamical systems is challenging, and one typically exploits specific structural properties of the dynamics to analyze their evolution and design control laws. A particular class of systems that is central to this thesis is that of quasi-dissipative dynamical systems, which can be viewed as systems with a dissipative component perturbed by an additional term. Given the limitations of classical optimization methods for such dynamical systems, this thesis develops a unified framework for their analysis and optimization using measure relaxation techniques. The moment–sums-of-squares (SOS) hierarchy is the main computational motivation for this approach, as it provides globally optimal solutions with convergence guarantees. Classical control and optimization methods become difficult to apply when system dynamics involve discontinuities, unilateral constraints, or set-valued mappings. We extend the measure-theoretic relaxation formalism to these nonsmooth settings. The first contribution establishes a rigorous formulation of measure evolution for first-order sweeping processes, proving existence, uniqueness, and a superposition representation of measure-valued solutions. A functional regularization approach and a time-discretized optimal transport scheme are developed to approximate these solutions, with convergence guarantees in Wasserstein metrics. Within this framework, we formulate a moment–SOS based semidefinite relaxation to solve the measure evolution problem. The second contribution addresses optimal control of sweeping processes using measure relaxation techniques. We show that relaxing to a linear program in the space of measures introduces no relaxation gap in both continuous and discrete time. Using tools from optimal transport theory, we prove convergence of the discretized problem to the continuous one as the sampling interval tends to zero. A moment-SOS based semidefinite relaxation is proposed to solve the measure-valued optimal control problem. Finally, the framework is extended to quasi-dissipative nonlinear evolution equations on Hilbert spaces, including semilinear and reaction diffusion partial differential equations. We show that the measure formulation of these infinite dimensional problems remains exact, with no relaxation gap, and we propose a convergent moment-SOS hierarchy to obtain certified numerical approximations. Overall, the thesis combines variational analysis, optimal transport, and semidefinite programming to provide globally convergent convex formulations for a broad class of nonsmooth optimal control and PDE problems.