Pointwise Jacobson type necessary conditions for optimal control problems governed by impulsive differential systems

This work focuses on an exploration of the pointwise Jacobson-type necessary conditions for optimal control problems governed by differential systems with impulse at fixed times; the pointwise Jacobson-type necessary optimality conditions refer to a type of pointwise second-order necessary optimality conditions for optimal singular control in the classical sense. By introducing an impulsive linear matrix Riccati differential equation, we derive the integral representation of the functional second-order variational. Based on this, the integral form of the second-order necessary conditions and the pointwise Jacobson-type necessary conditions are obtained. Incidentally, we have established the Legendre-Clebsch condition and the pointwise Legendre-Clebsch condition. Finally, an example is provided to illustrate the effectiveness of the main result.