Volume 12, Issue 1

Modeling of large lakes or oceans for environmental studies or deformable blood vessels for mini-invasive surgery involve dynamical free boundaries governed by their interaction with their immediate environment. The object of this paper is to further Shape Analysis techniques in the formulation and analysis of dynamical free boundary problems defined over a time-varying domain ( noncylindrical problems ). The generic example of a potential incompressible fluid flow with a pressure condition (Bernouilli type) on the dynamical free boundary is presented along with a discrete time algorithm to follow the evolution of the time-varying free boundary. The dynamical free boundary condition yields a new time-space differential equation for the normal component of the velocity field on the time-varying free boundary. These ideas and constructions can be extended to more complex and realistic models.

In this paper a support vector machine (SVM) approach for characterizing the feasible parameter set (FPS) in non-linear set-membership estimation problems is presented. It iteratively solves a regression problem from which an approximation of the boundary of the FPS can be determined. To guarantee convergence to the boundary the procedure includes a no-derivative line search and for an appropriate coverage of points on the FPS boundary it is suggested to start with a sequential box pavement procedure. The SVM approach is illustrated on a simple sine and exponential model with two parameters and an agro-forestry simulation model.

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C 2 -boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ 0 and Γ 1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level ( H 1 (Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H 1 -level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C 2 -solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H 2 -solutions, and ultimately for H 1 -solutions. The passage from H 2 - to H 1 -solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L 2 -normal traces in the case of Dirichlet B.C., but by contrast do not produce H 1 -traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H 1 -solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ 0 ,Γ 1 } that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L 2 (Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L 2 (Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L 2 (Ω), by an attractive boundary feedback.