An effective algorithm for reachability of linear control problems in minimum time

In this paper minimum time for a class of linear control problems is discussed. It is assumed that , control function, is continuous in minimum time linear control problem (MTLCP). A new problem in calculus of variations form which is equivalent to MTLCP is defined. In this new problem, it is discussed how polynomials with unknown constant coefficients can be substituted with the state and control functions. After this substitution, a sequence of minimum time nonlinear problems (MTNP) which its solution converge to the solution of original problem (MTLCP), is obtained. For solving the sequence of MTNPs an algorithm which give polynomials as the approximations of optimal state and control functions and the approximation of minimum time, is proposed. In this method, the error of approximations and the convergence of algorithm are considered. Finally, the efficiency of this approach is confirmed by some numerical examples.