Time optimal control problem for second-order linear time-invariant systems

The paper deals with problems governed by second-order linear time-invariant systems. At the same time, some qualitative results regarding the solution of this system are also studied. Then, under the established ‘General Position Condition,’ it is shown that the maximum principle uniquely determines a piecewise constant control function. The method of variation of parameters is used to find the general solution of non-homogeneous second-order linear systems. Then the existence and uniqueness theorems are proved. As an example, the classical problem of time-optimal control–which reflects Newton's second law and is often described as ‘the fastest stop of a train at a station'–is considered, where the minimum time, corresponding to the optimal control, is calculated using the initial data.