Impulsive control systems

Abstract

Impulsive control systems arose from classical control systems described by differential equations where the control functions could be unbounded. Passing to the limit of trajectories whose velocities are changing very rapidly leads to the state vector to "jump", or exhibit impulsive behavior. The mathematical model in this thesis uses a differential inclusion and a measure-driven control, and it becomes possible to deal with the discontinuity of movements happening over a small interval. We adopt the formulism of impulsive systems in which the velocities are decomposed by the slow and fast ones. The fast time velocity is expressed as the multiplication of point-mass measure with a state-depended term. Our methodology is deeply grounded in the concept of a graph completion, which is a technique to interpret and make rigorous the multiplication of a discontinuous function with a vector-valued measure. After reviewing how this concept is used to define the trajectory of impulsive system, the thesis works out a sampling method to estimate a solution and simultaneously construct a control measure, which is the first part of my research. The second part studies the stability of systems through invariance properties. Invariance of the system involves evolution properties on a given closed set with respect to the initial state belonging to that set. The third and last part of the thesis considers the Hamilton-Jacobi (HJ) theory of impulsive systems, which is related to the minimal time problem, an optimization topic of considerable interest. The minimal time function is uncovered to be the unique solution of HJ equation. Many discussions have earlier been offered in non-impulsive systems, especially in autonomous case, and we attempt to extend these results to impulsive control system. Final thoughts and considerations are put in the last chapter of conclusions and future work.