We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin’s Invariant Subspace theorem...

Here we study an inverse problem for a quasilinear hyperbolic equation. We start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function. Then we use a Lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation. Despite the fact that the Lagrangian formulation is formal and tha...

The problem on propagation of long waves in a domain of arbitrary form with the sufficiently smooth boundary on a sphere is considered. The boundary consists of ”solid” parts passing along the coastline and ”liquid” parts passing through the water area. We assume, free surface level observation data on ”liquid” boundary is known. In general case the boundary condition on ”liquid” part of bounda...

This paper discusses key implementation details required for computing the solution of a continuous-time optimal control problem on a Lie group using the projection operator approach. In particular, we provide the explicit formulas to compute the time-varying linear quadratic problem which defines the search direction step of the algorithm. We also show that the projection operator approach on ...

Bioluminescence tomography (BLT) is a promising new method in biomedical imaging. The BLT problem is an ill-posed inverse source problem, usually studied through a regularization technique. A new approach is proposed for solving the BLT problem based on an adjoint equation. Numerical examples show that the new formulation allows us to obtain accurate solutions.

The adjoint method was first developed for aerodynamic shape optimization applications through the use of control theory by Jameson (1998) in the late 1980s and early 1990s, using ideas adapted from more general work by Lions (1971) on optimal control of systems governed by partial differential equations (PDEs). Over the past two decades, adjoint methods have been used in a variety of applicati...

We study null controllability properties for parabolic coupled systems with degeneracy and singularity occurring in the interior of the spatial domain. This article is the first to consider a problem with singular coupling terms; previous result cannot be adapted to this situation. In particular, we focus on the well posedness of the problem and then we prove Carleman estimates for the associat...

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the strength of the time- dependent heat source using the temperature distribution at a point in a three layer system. Also, the effect of noisy data on final solution is studied. The numerical solution of the governing equations is obtained b...