Observability and Control of Schrödinger Equations

Hardy Inequalities, Observability, and Control for the Wave and Schrödinger Equations with Singular Potentials

We address the question of exact controllability of the wave and Schrödinger equations perturbed by a singular inverse-square potential. Exact boundary controllability is proved in the range of subcritical coefficients of the singular potential and under suitable geometric conditions. The proof relies on the method of multipliers. The key point in the proof of the observability inequality is a ...

Quasilinear Schrödinger equations involving critical exponents in $mathbb{textbf{R}}^2$

‎We study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth‎. ‎This model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

Existence of infinitely many solutions for coupled system of Schrödinger-Maxwell's equations

Analytical Soliton Solutions Modeling of Nonlinear Schrödinger Equation with the Dual Power Law Nonlinearity  

Introduction In this study, we use a newly proposed method based on the software structure of the maple, called the Khaters method, and will be introducing exponential, hyperbolic, and trigonometric solutions for one of the Schrödinger equations, called the nonlinear Schrödinger equation with the dual power law nonlinearity. Given the widespread use of the Schrödinger equation in physics and e...

Resolvent Conditions for the Control of Unitary Groups and Their Approximations

A self-adjoint operator A and an operator C bounded from the domain D(A) with the graph norm to another Hilbert space are considered. The admissibility or the exact observability in finite time of the unitary group generated by iA with respect to the observation operator C are characterized by some spectral inequalities on A and C. E.g. both properties hold if and only if x 7→ ‖(A−λ)x‖+‖Cx‖ is ...