The Geometric Structure Of Quenching Time Optimal Control Problems For Ordinary Differential Equations

This paper studies the geometric structure of quenching time optimal control problems for ordinary differential equations. Consider the control systems of the following ordinary differential equations:First of all, variation is applied to the initial data of the control systems gov-ered by the above ordinary differential equations after the new variable y0 is added. Then, the corresponding variational equations of the above ordinary differential e-quations will be deduced. Transfer matrix Φt,0 and several important properties of Φt,0 can be obtained through variational equations. Next, we can define the vari-ation of control uε in the form of references. Then, we can get the existence of the variation path whose control variable is the variation control in a certain time. Further, we can obtain the identical equation which the variation path satisfies. On this basis, we obtain some geometric properties of solutions to quenching time optimal control problems for the above ordinary differential equations, which is the main conclusion of the problems studied in this paper. Finally, we will explain the association and differences between the geometric structure problem of quenching time optimal control problems for ordinary differential equations which hold the property of quenching and that of the classical time optimal control problems; and we also discuss the problems that can be further studied by utilizing the relevant methods of this paper, for example, the geometric structure of blowup time optimal control problems for ordinary differential equations. We first identify the differences of the solutions to ordinary differential equations which hold the property of quench-ing and the solutions to ordinary differential equations which have the property of blowup. Morover, for a one-dimentional ordinary differential equation which has the property of blowup, we briefly illustrate geometric structure of its blowup time optimal control problems by the methods of this paper.