Time Optimal Control Of The Mayer Approximation Algorithm

In this paper, a class of computation methods on the time optimal control problem is introduced. As we know, the dynamic programming approach is a useful method to dealt with the feedback control problem which can't be solved with Pontryagin maximum value principle. But it is difficult to solve the time optimal control problem with the fixed terminal time by the dynamic programming approach.In this paper, we propose to discuss a class of approximation algorithms of time optimal control problems with the dynamic programming approach. First, we use a series of Mayer problems with fixed terminal time to approach time optimal control problem through a correct transformation, then we introduce a correct viscosity factor, thus we can change solving the HJB equation of the dynamic programming approach into solving convection-diffusion equation. Furthermore, we consider using the modified method of characteristics to find the numerical solution of the convection-diffusion equation. At last, a numerical experiment is presented to illustrate the validity of numerical method.The concrete step is as follows: Consider following nonlinear controlled systemwhere V_ad is admissible control set. Then the time optimal control problem corresponding to the system (0.1) is to find an optimal control v^*∈V_ad such that thestate z₀ changes into the state z₁ through a period of time and this period time is the shortest. To solve this class of time optimal control problem (P), we introduce a correct transformation t = ks (0≤s≤1,k∈R⁺). Through this transformation, system (0.1) can be replaced bywhere x(s) = z(ks),u(s) = v(ks) and define Is an admissible control set. Minimize the cost functional given byover W, and x(u,k) is the solution of (0.2) corresponding to w' = (u,k). The solution of problems (P) can be approached to the solutions of a series of Mayer problems ( P_ε) and Mayer problem ( P_ε) has a solution.Then the Mayer problem ( P_ε) is solved by dynamic programming approachthrough introducing a viscosity factor .That is , according to the dynamic programming approach, we can know that the value function corresponding toproblem (P_ε) is:and H-J-B equations corresponding to the problem ( P_ε) are:Then, by intrducing a correct viscosity factor we transfer the problem to the viscosity solution of HJB equation into the problem to the classical solution of convection-diffusion equation. Furthermore, we consider using the modified method of characteristics to find the numerical solution of this convection-diffusion equation.At last we get a numerical method for time optimal control problem.