Dynamical Constrained Impulse System and application for designing discontinuous feedbacks

Reaching a target while remaining in a given set for impulse dynamics can be characterized by a non deterministic controlled differential equation and a controlled instantaneous reset equation. The set of initial conditions from which a given objective can be reached and the optimal control ensuring both target capture and viability are computed using the Hybrid Guaranteed Capture Basin Algorithm. After recalling how this can be extended for solving optimal control problems in the presence of impulse and in the presence of uncertainty and giving two applications relevant to engineering and to finance, we apply these approach for designing discontinuous controls selected in the class of linear feedbacks for the double integrator problem which substantially improve the efficiency for controlling systems with the only P.I.D. regulation.