Optimal distributed control problem for cubic nonlinear Schrödinger equation (Q1787188)

The nonhomogeneous NLS equation \[ \begin{cases} \partial_zu=i\partial_t^2u+i|u|^2u+g,\quad z\in[0,\zeta],\quad t\in\mathbb R\\ u(0,t)=u_0(t) \end{cases}\leqno{(\ast)} \] models the propagation of pulses in an optical fiber influenced by the presence of noise generation through the term \(g\in L^2([0,\zeta];L^2(\mathbb R))\). The authors prove that the problem \((*)\) is well posed, with its unique solution (in an adequate class) denoted by \(u[g]\). Furthermore, they prove the existence of an optimal noise configuration \(g_*\) which minimizes \[ \mathcal J(g)=\|g\|_{L^2([0,\zeta];L^2)}^2+\kappa\|\sigma(u[g](\zeta)-v_\zeta)\|_{L^2(\mathbb R)}^2 \] over the configurations \(g\) obeying an admissible condition of the form, \[ \|\sigma(u[g](\zeta)-v_\zeta)\|_{L^2(\mathbb R)}^2\leq \eta \] where \(\sigma(t)=k e^{-\frac{(t-t_r)^2}{\tau^2}}\), for a given \((t_r,\tau)\in\mathbb R^2\), is normalized in \(L^2(\mathbb R)\), and \(v_\zeta=v(\cdot,\zeta)\) with \(v(z,t)\) the solution of the noise free problem \[ \begin{cases} \partial_zv=i\partial_t^2v+i|v|^2v,\quad z\in[0,\zeta],\quad t\in\mathbb R\\ v(0,t)=v_0(t). \end{cases} \]