Optimal control problems for some ordinary differential equations with behavior of blowup or quenching (Q2001558)

\noindent This paper is on the differential system \[ y'(t) = f(t, y(t), u(t)) \quad (t > 0) \, , \qquad y(0) = y_0,\tag{1} \] where solutions in an interval $[0, T_1)$ are defined by the integrated version \[ y(t) = y_0 + \int_0^t f(s, y(s), u(s)) ds \, , \quad (0 \le t < T_1). \] A solution of (1) has a blowup at $t = T_1$ if $\|y(t)\|$ is unbounded as $t \to T_1 - ;$ for a regular right side $f(t, y, u)$ this is the only type of singularity possible. Another type of singularity is quenching at $t = T_1,$ where $y(t)$ remains bounded but $\|y'(t)\|$ becomes unbounded as $t \to T_1-.$ Both blowup and quenching describe physical phenomena (for instance, blowup may model the catastrophic increase of temperature in a chemical reaction causing ignition). There has been a number of recent results on minimizing/maximizing blowup or quenching time. \par For the blowup case, the authors consider measurable controls that satisfy a constraint $u(t) \in U$ and a functional \[ J(y(\cdot), u(\cdot)) = \int_0^{T_1} f_0(t, y(t), u(t)) dt \] where $T_1$ is the blowup point of $y(t).$ The optimal problem is that of finding a control $\bar u(t)$ that minimizes $J(y, u).$ The authors show an existence theorem for relaxed controls and a version of Pontryagin's maximum principle based on the regular maximum principle in subintervals $[0, T], T < T_1.$ The results for the quenching case are similar but there is a complication; while it doesn't make sense to continue a solution past a blowup point, continuation past a quenching point may be possible, but there may be loss of uniqueness.