Optimal control of the blowup time

The problem of optimal control of the blowup time of a system of nonlinear controlled ordinary differential equations is considered in this paper. The blowup time is defined to be the first time that the norm of the trajectory becomes infinite. When one seeks to maximize the blowup time the pair (V(x),Omega) comes under consideration, where x is an element of R(n) --> V(x) is an element of [0, infinity] is the value function and Omega subset of R(n) is the blowup set. This is the set of initial points from which finite time blowup will occur for any control. We prove that (V, Omega) is the unique viscosity solution of the equation 1 + max(z) DxV(x) . f(x, z) = 0, x is an element of Omega and conditions lim(\x\-->infinity) V(x) = 0,lim(x-->partial derivative Omega) V(x) = +infinity. Finally, we derive the Pontryagin maximum principle for an optimal control. Some generalizations are also discussed.