# Mathematics Colloquia and Seminars

Many problems in engineering and economics are formulated in the impulse/singular control framework. Compared to regular controls, impulse control provides a more natural mathematical framework when the state space is discontinuous. However, many structural results amount to solving complex algebraic equations that are hard to verify without a priori knowledge of the regularity property, thus the correctness of the solutions'' is dubious. In this talk, we provide sufficient conditions for the smooth-fit $C1$ property of the value function for multi-dimensional controlled diffusions, using a viscosity solution approach. This approach is different from the Quasi-Variational Inequalities (QVI) established by Bensoussan and Lions (1982). We show by simple examples where the regularity property may fail especially in multi-dimensional case.