Motivated by the eigenvalue problems for ordinary and partial differential operators, we shall develop the spectral theory for linear operators in Hilbert spaces. Here we transform the unbounded differential operators into singular integral operators which are completely continuous. With his study of integral equations D. Hilbert, together with his students E. Schmidt, I. Schur, and H. Weyl, opened a new era for the analysis.

Linear Operators in Hilbert Spaces

Left and right generalized inverses

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space We show that the left (resp the right) generalized Drazin inverse is a sum of a left invertible (resp a right invertible) operator and a quasi-nilpotent one In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory

Left and right generalized Drazin invertible operators

We investigate a mathematical model in crop pest management, considering plant biomass, pest, and the effect of farming awareness. The pest population is divided into two compartments: susceptible pest and infected pest. We assume that the growth rate of self-aware people is proportional to the density of healthy pests present in the crop field. Impacts of awareness is modeled via a saturated term. It is further assumed that self-aware people will adopt biological control methods, namely integrated pest management. Susceptible pests are detrimental to crops and, moreover, there may be some time delay in measuring the healthy pests in the crop field. A time delay may also take place while becoming aware of the control strategies or taking necessary steps to control the pest attack. In agreement, we develop our model incorporating two time delays into the system. The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays. Stability switches occur through Hopf-bifurcation when time delays cross critical values. Optimal control theory has been applied for the cost-effectiveness of the delayed system. Numerical simulations illustrate the obtained analytical results.

Pest control using farming awareness: Impact of time delays and optimal use of biopesticides

We study, from an optimal control perspective, Noether currents for higher-order problems of Herglotz type with time delay. Main result provides new Noether currents for such generalized variational problems, which are particularly useful in the search of extremals. The proof is based on the idea of rewriting the higher-order delayed generalized variational problem as a first-order optimal control problem without time delays.

Noether currents for higher-order variational problems of Herglotz type with time delay

On the generalized Kato spectrum

In this paper, we give some proprieties  of the  semi-regular, essentially semi-regular and the operators of Kato type   on a Banach space We also show that the essentially semi-regular spectrum of closed, densely defined linear operator is stable under commuting compact perturbation and  its  Kato  spectrum  is stable subjected to additive commuting nilpotent perturbations

/pdf/on-the-kato-semi-regular-and-essentially-semi-regular-13z446x34r.pdf

On the Kato, semi-regular and essentially semi-regular spectra

A classical result of J.Ph. Labrousse (Rev. Roum. Math. Pures Et Appl. XXV(9):1391–1394, 1980) concerning the symmetric difference between the Kato spectrum and the Goldberg spectrum of closed operators in Hilbert spaces is extended to the case of multivalued linear operators in Banach spaces.

Relationship Between the Kato Spectrum and the Goldberg Spectrum of a Linear Relation

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing us to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

/pdf/optimal-control-with-time-delays-via-the-penalty-method-4osz099ez7.pdf

Mohammed Benharrat

Papers

Left and right generalized Drazin invertible operators

On the generalized Kato spectrum

On the Kato, semi-regular and essentially semi-regular spectra

Relationship Between the Kato Spectrum and the Goldberg Spectrum of a Linear Relation

Optimal Control with Time Delays via the Penalty Method