Finite-time stabilization of some classes of infinite dimensional systems (Q6608534)

For the bilinear system\N\[\N\dot{x}(t) = A x(t) + u(t) B x(t), \quad x(0)=x_0\N\]\Nwith \(A\) the infinitesimal generator of a (quasi) contration semigroup on the Hilbert space \(H\) and \(B\) a bounded, self-adjoint and positive operator on \(H\), the finite-time stability is studied, i.e., the control objective is to construct a feedback such that for every \(x_0\) the state \(x(t)\) is driven to zero in finite time. Similar as in finite dimensions, this can be achieved by a feedback of the form \(u(t) = -\langle Bx(t), x(t) \rangle^{-\mu}\) when \(\langle Bx(t), x(t) \rangle \neq 0\), and \(u(t) =0\) otherwise. Here \(\mu \in (0,\frac{1}{2})\). It is shown that this construction of the feedback can also be used to contruct a feedback driving the a linear system to zero in finite time. The article applies the developed theory to three examples. Namely, the heat equation, the transport equation, and to the bilinear wave equation.\N\NFor the entire collection see [Zbl 1530.49003].