Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $\rho$, $k$, $\gamma$ is applied. The results are illustrated by examples

On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment

The control system w tt = w xx − q(x)w, w The necessary and sufficient conditions of null-controllability and approximate nullcontrollability are obtained for this system. The controllability problems are considered in the modified Sobolev spaces. The controls that solve these problems are found explicitly. It is proved that among the solutions of the Markov trigonometric moment problem there are bang-bang controls solving the approximate null-controllability problem

On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment

The control system wtt = wxx - q(x)w, wx(0; t) = u(t), wx(d, t) = 0, x is in (0; d), t is in (0; T), d > 0, 0 0, q'₊(0) = q₋(d) = 0, u is a control, |u(t)| ≤ 1 on (0, T). The necessary and suffcient conditions of null-controllability and approximate null-controllability are obtained for this system. The controllability problems are considered in the modified Sobolev spaces. The controls that solve these problems are found explicitly. It is proved that among the solutions of the Markov trigonometric moment problem there are bang-bang controls solving the approximate null-controllability problem

Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis

In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from these spaces depends on the growth of ρ and k. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system zₜₜ = zξξ − q²z, zξ(0, t) = v(t), ξ > 0, t belongs (0, T), and vise versa. Here q ≥ 0 is a constant and v belongs L∞(0, ∞) is a control. For the main system, necessary and sufficient conditions of the L∞-controllability and the approximate L∞-controllability are obtained from those known for the auxiliary system

On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis

In the paper, the equation of a vibrating non-homogeneous string, whose potential is not equal to a constant, is considered on a half-axis. The Neumann control of the class L∞ is considered at a point x = 0. The control problem is studied in the Sobolev spaces. The suffcient conditions for nullcontrollability and approximate null-controllability at a free time T > 0 are obtained for the given system. The controls solving these problems are found explicitly.Рассмотрено уравнение колебания неоднородной струны на полуоси с потенциалом, не равным константе. На левом конце рассмотрено управление типа Неймана из класса L∞. Задача управляемости изучена в пространствах Соболева. Для заданной системы получены достаточные условия 0-управляемости и ε-управляемости за свободное время T > 0. Управления, которые решают эти задачи, найдены в явном виде

Controllability Problems for the Non-Homogeneous String that is Fixed at the Right End Point and has the Dirichlet Boundary Control at the Left End Point

In the paper, the necessary and su±cient conditions of null-controllability and approximate null-controllability are obtained for the some control system. The problems for the control system are considered in the modified Sobolev spaces. The control that solves these problems is found explicitly. The bang-bang controls solving the approximate null-controllability problem are constructed as the solutions of the Markov trigonometric moment problem

Spectral Analysis of Complex Dynamical Systems

The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that lateral vibrations of such strings can be decoupled from their in-plane vibrations. We then study the eigenvalue problem originating from the lateral vibrations. We show that, unlike the well-known one string vibrations case, the eigenvalues in a multi-string vibrating system do not have to be simple. Moreover we prove that the multiplicities of the eigenvalues depend on the symmetry of the model and on the total number of the connected strings [50]. We also apply Nevanlinna functions theory to characterize the spectra and to solve the inverse problem for a discrete multi-string system in a more general setting than it was done in [71],[73], [22], [69]-[72]. We also represent multi-string vibrating systems using a coupling of non-densely defined symmetric operators acting in the infinite dimensional Hilbert space. This coupling is defined by a special set of boundary operators acting in finite dimensional Krein space (the space with indefinite inner product). The main results of this research are published in [50]. The Hypothalamic Pituitary Adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we first perform rigorous stability analysis of all multi-parametric steady states and secondly, by construction of a Lyapunov functional, we prove nonlinear asymptotic stability for some of multi-parametric steady states. We then take into account the additional effects of the time delay parameter on the stability of the HPA axis system. Finally we prove the existence of periodic solutions for the HPA axis system. The main results of this research are published in [51]