Existence and uniqueness of the solution to a second-order nonlinear dynamical system model with an unbounded variable and a discontinuous input

We consider an initial value problem for a second-order nonlinear dynamical system that allows discontinuous inputs. In this problem, one state variable is restricted in a closed and bounded interval while the other state variable is unlimited. The existence and uniqueness of the solution to this problem is studied in a time interval determined by two adjacent instants at which the input may be discontinuous. Some results are obtained on the maximal interval of existence of the unique solution in this time interval. The results are then extended to the entire time domain. A robot example is given to verify our results.