Shape, Velocity, and Exact Controllability for the Wave Equation on a Graph with Cycle

Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails

Shape, Velocity, and Exact Controllability for the Wave Equation

A new method to prove exact controllability for the wave equation is demonstrated and discussed on several examples. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability

Pointwise boundary stabilizability of hyperbolic evolution equations: Two-dimensional hybrid elastic structures

AbstractA two-dimensional hybrid elastic structure of a rectangular membrane linked with two rib strings on the boundary sides that have rigid bodies and pointwise controllers attached at the corner points is modelled as an abstract hyperbolic evolution equation. By an analysis of energy decay and ω-limit sets with a dissipative feedback, the following necessary and sufficient condition for stabilizability in the energy space is obtained: vi2 − vj2 ≠ 2n(πl2)2, for any positive integer n and for any i > j, where v0 = 0 and vi (i = 1, 2,…) are the increasing positive roots of the transcendental equation cot(l1μ) = 12(μa − aμ), in which a > 0 is a structural constant, l1 × l2 is the size of the region. If this condition is satisfied, then the stabilization is achieved by a pointwise boundary damping feedback. Otherwise the stabilization can be achieved by combining an additional boundary damping feedback controller locally distributed in the interior of one boundary string

Exact controllability for wave equation on general quantum graphs with non-smooth controls

In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff-Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and moment method approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph

Solitons in Supersymmety Breaking Meta-Stable Vacua

In recently found supersymmetry-breaking meta-stable vacua of the supersymmetric QCD, we examine possible exsitence of solitons. Homotopy groups of the moduli space of the meta-stable vacua show that there is no nontrivial soliton for SU(N_c) gauge group. When U(1)_B symmetry present in the theory is gauged, we find non-BPS solitonic (vortex) strings whose existence and properties are predicted from brane configurations. We obtain explicit classical solutions which reproduce the predicitions. For SO(N_c) gauge group, we find there are solitonic strings for N = N_f-N_c+4 = 2, and Z_2 strings for the other N. The strings are meta-stable as they live in the meta-stable vacua.Comment: 30 pages, 14 figures, Comments on stability of non-BPS vortices are added, Comments on sigma model solitons are added, An appendix is adde