  In this paper, we propose two discontinuous dynamical systems in continuous
time with guaranteed prescribed finite-time local convergence to strict local
minima of a given cost function. Our approach consists of exploiting a
Lyapunov-based differential inequality for differential inclusions, which leads
to finite-time stability and thus finite-time convergence with a provable bound
on the settling time. In particular, for exact solutions to the aforementioned
differential inequality, the settling-time bound is also exact, thus achieving
prescribed finite-time convergence. We thus construct a class of discontinuous
dynamical systems, of second order with respect to the cost function, that
serve as continuous-time optimization algorithms with finite-time convergence
and prescribed convergence time. Finally, we illustrate our results on the
Rosenbrock function.
