Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

Regularization of A Singularly Perturbed Elliptic PDE

In this paper, we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives. Four types of basic spectral regularization methods such as Showalter’s, Tikhonov’s, Lardy’s and Lavrentiev’s are applied to approximate the solution by introducing another large (or small) parameter. Convergence of the regularized solutions to t...

Interior spikes of a singularly perturbed Neumann problem with potentials

where Ø is a smooth bounded domain of R with external normal ν, N ≥ 3, 1 < p < (N + 2)/(N − 2), J : R → R and V : R → R are C functions. In [5], the first author, extending the classical results by Ni and Takagi, in [3, 4], proved that there exist solutions of (1) that concentrate at maximum and minimum points of a suitable auxiliary function defined on the boundary ∂Ø and depending only on J a...

Stability Region of Singularly Perturbed Systems

 This paper studies the characterization of the stability boundary (the topological boundary of stability region) of nonlinear autonomous singularly perturbed dynamical systems. In particular, a relationship between the stability boundary of the singularly perturbed system and the stability boundary and stability region of the slow and fast systems is investigated. Conditions, in terms of loca...

Control of singularly perturbed hybrid stochastic systems

In this paper, we study a class of optimal stochastic control problems involving two different time scales. The fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process. Under a fundamental “ergodicity” property for a class of “infinitesimal control systems” associated with the fast mode, we show that t...

Controller Design of Decentralized Singularly Perturbed Systems