Digital twins are information constructs that mimic physical systems, are dynamically updated from data of their physical twin, have predictive capabilities, and inform decisions. We optimize the path of a mobile sensor to minimize the posterior uncertainty of a Bayesian inverse problem. Along its path, the sensor continuously takes measurements of the state, which is a physical quantity modeled as the solution of a partial differential equation (PDE) with uncertain parameters. We derive the closed-form expression of the posterior covariance matrix of the model parameters as a function of the path, and formulate the optimal experimental design problem for minimizing the posterior's uncertainty. We discretize the problem such that the cost function remains consistent under temporal refinement. Our formulation allows the addition of constraints that model the dynamics of the sensor platform and ensure that the path avoids obstacles and remains physically interpretable through a control parameterization. We present computational results for a convection-diffusion equation with unknown initial condition.