On a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

In this paper, we investigate a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. Continuing our previous studies, we consider initial conditions depending on the second small parameter. In the degenerate case, this results in an asymptotic expansion of the solution of a fundamentally different kind. The solvability of the problem is proved. We also construct and justify a complete power asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control in a small parameter at the derivatives in the equations of the systems.