In this paper, we investigate a problem of optimal control over a finite time interval for a linear systemwith constant coefficients and a small parameter in the initial data in the class of piecewise continuous controlswith smooth geometric constraints. We consider a terminal convex performance index. We substantiate the limit relationsas the small parameter tends to zero for the optimal value of the performance index and for the vector generatingthe optimal control in the problem. We show that the asymptotics of the solution can be of complicated nature. Inparticular, it may have no expansion in the Poincaré sense in any asymptotic sequence of rational functions of thesmall parameter or its logarithms