Andrei Halanay

Stability of Limit Cycles in a Pluripotent Stem Cell Dynamics Model

This paper is devoted to the study of the stability of limit cycles of a nonlinear delay differential equation with a distributed delay. The equation arises from a model of population dynamics describing the evolution of a pluripotent stem cells population. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We then investigate the stability of the limit cycles yielded by the bifurcation using the normal form theory and the center manifold theorem. We illustrate our results with some numerics.

Lack of controllability of the heat equation with memory

Abstract We consider a model for the heat equation with memory, which has infinite propagation speed, like the standard heat equation. We prove that, in spite of this, for every T > 0 there exist square integrable initial data which cannot be steered to hit zero at time T , using square integrable controls. We show that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamics.

Stabilization of some nonlinear controlled electrohydraulic servosystems

For control systems that model electro-hydraulic servo-actuators, the possibility of stabilization by a linear stationary feedback low is analysed. It is also shown that, when this is possible, equilibria exhibit also asymptotic stability with respect to some relevant state variables.

Optimal control analysis of a leukemia model under imatinib treatment

The paper is devoted to the study of a mathematical model of drug therapy for Chronic Myelogenous Leukemia (CML). The disease dynamics are given by a couple of delay differential equations that describe the interaction between a stem-like population of CML cells and a more mature, differentiated one, without self-renewal properties. A molecular targeted therapy, such as Imatinib, is considered. The objective is to minimize the size of the tumoral cells’ mass while minimizing the amount of drug (thus, minimizing both the adverse effects and the costs). The optimal control is calculated using a discretization scheme. The results are supported by numerical simulations.

Stability analysis of equilibria in a delay differentialequations model of CML including asymmetric division and treatment

A two dimensional two-delays differential system modeling the dynamics of stem-like cells and white-blood cells in Chronic Myelogenous Leukemia under treatment is considered. Stability of equilibria is investigated and emergence of periodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventually shown. All three types of stem cell division (asymmetric division, symmetric renewal and symmetric differentiation) are present in the model. The effect of drug resistance is considered through the Goldie-Coldman law.

A Control Delay Differential Equations Model of Evolution of Normal and Leukemic Cell Populations Under Treatment

The dynamics and evolution of leukemia is determined by the interactions between normal and leukemic cells populations at every phase of the development of hematopoietic cells. For both types of cell populations, two subpopulations are considered, namely the stem-like cell population (i.e. with unlimited self-renew ability) and a more mature, differentiated one, possessing only the capability to undergo limited reproduction. Treatment effects are included in the model as functions of time and a cost functional is considered. The optimal control is obtained using a discretization scheme. Numerical results are discussed in relation to the medical interpretation.

Geometric control in a regulator problem for electrohydraulic servos

A five-dimensional nonlinear mathematical model of the electrohydraulic servo(mechanism) is considered. In the system equilibria analysis, the critical case of a zero eigenvalue occurs. The Lyapunov-Malkin theorem and Routh-Hurwitz criterion provide conditions for controllers to stabilize all relevant equilibria in the closed-loop system. Geometric control paradigm is then applied in synthesis and the performance of the obtained controlled system is numerically validated from viewpoint of the regulator classical problem.

Stability analysis for a delay differential equations model of a hydraulic turbine speed governor

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

A Complex Mathematical Model with Competition in Leukemia with Immune Response - An Optimal Control Approach

This paper investigates an optimal control problem associated with a complex nonlinear system of multiple delay differential equations modeling the development of healthy and leukemic cell populations incorporating the immune system. The model takes into account space competition between normal cells and leukemic cells at two phases of the development of hematopoietic cells. The control problem consists in optimizing the treatment effect while minimizing the side effects. The Pontryagin minimum principle is applied and important conclusions about the character of the optimal therapy strategy are drawn.

Numerical Experiment for Stabilization of the Heat Equation by Dirichlet Boundary Control

We use the boundary feedback control introduced in Barbu [Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Automat. Control (accepted)], in order to stabilize an unstable heat equation in two dimensions. We propose two numerical algorithms. The feedback boundary condition is treated explicitly in the first algorithm. At each time step, only one linear system is solved. The second algorithm performs at each time step some subiterations, in order to treat the feedback boundary condition implicitly. The second algorithm can stabilize some problems where the first algorithm fails.