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.
Nonlinear systems with unbounded controls and state constraints: a problem of proper extension
MOTTA, MONICA;RAMPAZZO, FRANCO
1996
Abstract
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.Pubblicazioni consigliate
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