Nonlinear systems with unbounded controls and state constraints: a problem of proper extension

The authors consider the control system described by x˙ = f(t, x, u, v, u˙ ), t 2 (t, T], with initial data (x, u)(t) = (x, u), where f is sublinear in u˙ , and the state variable is constrained in the closure of an open set Rm; u and v take values in a closed subset U Rm and a compact set V Rq, respectively. Here, v is a bounded, measurable control, while no bounds are imposed on u˙ . By introducing a space-time vector field f defined by f(t, x, u, v,w0,w) = f(t, x, u, v,w/w0)w0 if w0 6= 0, f1(t, x, u, v,w) if w0 = 0, where f1(t, x, u, v,w) def = limr!1 r−1f(t, x, u, v, rw), the original system is embedded in the extended system t0 = w0(s), x0 = f(t, x, u(s), v(s),w0(s),w(s)), u0 = w(s), (t, x, u)(0) = (t, u, u), s 2 [0, 1]. The main result consists of conditions which guarantee that each constrained trajectory of the extended system is the limit of trajectories of the original system.