New discussion concerning to optimal control for semilinear population dynamics system in Hilbert spaces

The objective of our paper is to investigate the optimal control of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the optimal control results of the system are obtained using the C0-semigroup approach, fixed point theorem, and some other simple conditions on the nonlinear term as well as on operators involved in the model

On the approximate controllability of some semilinear partial functional integrodifferential equations with unbonded delay

This work concerns the study of the approximate controllability for some nonlinear partial functional integrodifferential equation with infinite delay arising in the modelling of materials with memory, in the framework of Hilbert spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed is part approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator. An example of applications is given for illustration

Global existence and stability for second order functional evolution equations with infinite delay

In this article, the authors give sufficient conditions for existence and attractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem. An example is provided to illustrate the result

Controllability Problem of Fractional Neutral Systems: A Survey

The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces, controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, controllability for a class of fractional neutral integrodifferential equations with unbounded delay, controllability of neutral fractional functional equations with impulses and infinite delay, and controllability for a class of fractional order neutral evolution control systems

Uniform Stability of a Class of Fractional-Order Nonautonomous Systems with Multiple Time Delays

In mathematics, to a large extent, control theory addresses the stability of solutions of differential equations, which can describe the behavior of dynamic systems. In this paper, a class of fractional-order nonautonomous systems with multiple time delays modeled by differential equations is considered. A sufficient condition is established for the existence and uniqueness of solutions for such systems involving Caputo fractional derivative, and the uniform stability of solution is studied. At last, two examples are given to demonstrate the applicability of our results

Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay

In this paper we study the controllability results of impulsive neutral stochastic functional integrodifferential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained

Differentiable positive definite kernels on two-point homogeneous spaces

In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5