Well-posedness and sharp uniform decay rates at the L2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

Abstract

We prove that the Schrödinger equation defined on a bounded open domain of R n and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L 2 ( ) for any n = 1, 2, 3, . . . , and, moreover, stable on L 2 ( ) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L 2 ( )-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely-at the outset-on a far general result of interest in its own right: an energy estimate at the L 2 ( )-level for a fully general Schrödinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H 1 ( )-level energy estimate to the L 2 ( )-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.

Conservative Control Systems Described by the Schrödinger Equation

“…Following Lasiecka et al [9] and Lasiecka and Triggiani [8], the assertions of Remark 5 hold also for the system (52)-(56).…”

“…The equations (52)-(56) can be written as an abstract system of the form (see Lasiecka et al [9] and Lasiecka and Triggiani [8]) Now, the theorem can be established by following the steps of the proof of Theorem 4.…”

Sobolev Spaces in Mathematics III

“…Third, it provides an explicit link between two open-loop controls-the one for the original conservative system and the one for the dissipative systemthat steer the same initial condition to rest, along their respective dynamics. Finally, when accompanied by exact controllability (equivalently, continuous observability) of the corresponding linear model, it implies uniform stabilization with optimal decay rates-according to the strategy laid out in [24] in the case of wave equations and exported to many other dynamics [47] (shells), [51] (Schrödinger equations)-when a nonlinear function of the "open-loop dissipative" boundary observation closes up the loop, to generate a corresponding boundary feedback, closed-loop, dissipative nonlinear problem. A distinctive feature of said uniform stabilization strategy of the nonlinear boundary problem is that optimal decay rates for the energy of the closed-loop boundary nonlinear feedback system can be derived via an explicitly constructed, nonlinear, monotone, first order, separable ordinary differential equation, without any a priori knowledge of the behavior of the dissipation at, or near, the origin (which is the region responsible for the decay rates).…”

Transport Equations and Multi-D Hyperbolic Conservation Laws

“…In particular, the school held in Bressanone offered two courses that provided an introduction to the theory of control problems for hyperbolic-like PDEs (delivered by Roberto Triggiani), and to the study of transport equations with irregular coefficients (delivered by Francois Bouchut), while the conference hosted in Trieste was organized in two courses (delivered by Laure Saint-Raymond and Cedric Villani) and in a series of invited lectures devoted to the main recent advancements in the study of Boltzmann equation. Some of the material covered by the course of Triggiani can be found in [17,18,20], while the main contributions of the conference on Boltzmann will be collected in a forthcoming special issue of the journal DCDS, of title "Boltzmann equations and applications".…”