Modeling uncertainty with Measure Differential Equations

Recently, a new type of evolution equations for measures, called Measure Differential Equations (briefly MDE), was introduced, based on the concept of Probability Vector Field. The latter is a map associating to a probability measure on a manifold a probability measure on the tangent bundle, whose projection on the base is the original measure. Such approach allows the modeling of finite-speed diffusion, thus provides a new approach to uncertainty for differential equations. After showing some explicit examples of modeling uncertainty with finite-speed diffusion, the theory of MDEs is extended to the time-varying case. This allows the application to control systems, including basic results on disturbance rejection.