Time optimal control for some ordinary differential equations with multiple solutions (Q2401509)

The system is driven by the vector equation \[ y'(t) = f(t, y(t), u(t)) \quad (t > 0) \qquad y(0) = y_0 \eqno(1) \] where \(y(t) \in {\mathbb R}^n\) and \(u(t) \in U \subset {\mathbb R}^m\) with \(U\) bounded and closed. The function \(f(t, y, u)\) is measurable in \(t\) and continuous in \((y, u)\) in \([0, \infty) \times {\mathbb R}^n \times U\) and satisfies more stringent smoothness assumptions (including continuous differentiability with respect to \(y)\) in \([0, \infty) \times ({\mathbb R}^n \setminus \{y_1\})\times U,\) where \(y_1\) is a singular point. Under these conditions, solutions of (1) with measurable controls (defined as usual by the integrated version of the equation) may not be unique, as illustrated by the system \[ y'(t) = y(t)^{2/3} + u(t) \] where the singular point is \(y_1 = 0.\) The control problem is the time optimal problem with a target set \(Q.\) The authors obtain a version of Pontryagin's maximum principle when the system is autonomous \((f(t, y, u) = f(y, u))\) and prove also existence results under the assumptions that \(f(t, y, u) = g(t, y) + B(t)u\) and \(U\) is convex. In the introduction, there is a comparison of the results obtained with the existing literature on optimal problems for systems that are not well posed, of which (1) is an instance.