# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

December 2019 , Volume 9 , Issue 4

Special issue on control of infinite dimensional systems

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2019, 9(4): i-ii doi: 10.3934/mcrf.2019042 +[Abstract](1618) +[HTML](251) +[PDF](148.77KB)
Abstract:
2019, 9(4): 607-621 doi: 10.3934/mcrf.2019043 +[Abstract](1417) +[HTML](153) +[PDF](669.86KB)
Abstract:

We investigate two optimal harvesting problems related to age-dependent population dynamics; namely we consider two problems of maximizing the profit for age-structured population dynamics with respect to a size-dependent harvesting effort. We evaluate the directional derivatives for the cost functionals. The structure of the harvesting effort is uniquely determined by its intensity (magnitude) and by its area of action/distribution. We derive an iterative algorithm to increase at each iteration the profit by changing the intensity of the harvesting effort and its distribution area. Some numerical tests are given to illustrate the effectiveness of the theoretical results for the first optimal harvesting problem.

2019, 9(4): 623-642 doi: 10.3934/mcrf.2019044 +[Abstract](2024) +[HTML](140) +[PDF](23271.08KB)
Abstract:

Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [12] provided a theoretical and numerical analysis of nonlinear poro-elastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [3]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

2019, 9(4): 643-671 doi: 10.3934/mcrf.2019045 +[Abstract](1323) +[HTML](238) +[PDF](449.57KB)
Abstract:

The plant to be stabilized is a system node \begin{document}$\Sigma$\end{document} with generating triple \begin{document}$(A,B,C)$\end{document} and transfer function \begin{document}${\bf G}$\end{document}, where \begin{document}$A$\end{document} generates a contraction semigroup on the Hilbert space \begin{document}$X$\end{document}. The control and observation operators \begin{document}$B$\end{document} and \begin{document}$C$\end{document} may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator \begin{document}$E$\end{document} such that, if we replace \begin{document}${\bf G}(s)$\end{document} by \begin{document}${\bf G}(s)+E$\end{document}, the new system \begin{document}$\Sigma_E$\end{document} becomes impedance passive. An easier case is when \begin{document}${\bf G}$\end{document} is already impedance passive and a special case is when \begin{document}$\Sigma$\end{document} has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback \begin{document}$u = - {\kappa} y+v$\end{document}, where \begin{document}$u$\end{document} is the input of the plant and \begin{document}${\kappa}>0$\end{document}, stabilizes \begin{document}$\Sigma$\end{document}, strongly or even exponentially. Here, \begin{document}$y$\end{document} is the output of \begin{document}$\Sigma$\end{document} and \begin{document}$v$\end{document} is the new input. Our main result is that if for some \begin{document}$E\in {\mathcal L}(U)$\end{document}, \begin{document}$\Sigma_E$\end{document} is impedance passive, and \begin{document}$\Sigma$\end{document} is approximately observable or approximately controllable in infinite time, then for sufficiently small \begin{document}${\kappa}$\end{document} the closed-loop system is weakly stable. If, moreover, \begin{document}$\sigma(A)\cap i {\mathbb R}$\end{document} is countable, then the closed-loop semigroup and its dual are both strongly stable.

2019, 9(4): 673-696 doi: 10.3934/mcrf.2019046 +[Abstract](1763) +[HTML](143) +[PDF](411.21KB)
Abstract:

The goal of this article is to provide backward uniqueness results for several models of parabolic equations set on the half line, namely the heat equation, and the heat equation with quadratic potential and with purely imaginary quadratic potentials, with non-homogeneous boundary conditions. Such result can thus also be interpreted as a strong lack of controllability on the half line, as it shows that only the trivial initial datum can be steered to zero. Our results are based on the explicit knowledge of the kernel of each equation, and standard arguments from complex analysis, namely the Phragmén-Lindelöf principle.

2019, 9(4): 697-718 doi: 10.3934/mcrf.2019047 +[Abstract](1306) +[HTML](137) +[PDF](488.69KB)
Abstract:

This work is devoted to establish a bang-bang principle of time optimal controls for a controlled age-structured population evolving in a bounded domain of \begin{document}$\mathbb{R}^n$\end{document}. Here, the bang-bang principle is deduced by an \begin{document}$L^\infty$\end{document} null-controllability result for the Lotka-McKendrick equation with spatial diffusion. This \begin{document}$L^\infty$\end{document} null-controllability result is obtained by combining a methodology employed by Hegoburu and Tucsnak - originally devoted to study the null-controllability of the Lotka-McKendrick equation with spatial diffusion in the more classical \begin{document}$L^2$\end{document} setting - with a strategy developed by Wang, originally intended to study the time optimal internal controls for the heat equation.

2019, 9(4): 719-728 doi: 10.3934/mcrf.2019048 +[Abstract](1341) +[HTML](130) +[PDF](457.25KB)
Abstract:

In this work, we study null-controllability of the Lotka-McKendrick system of population dynamics. The control is acting on the individuals in a given age range. The main novelty we bring in this work is that the age interval in which the control is active does not necessarily contain a neighbourhood of \begin{document}$0.$\end{document} The main result asserts the whole population can be steered into zero in large time. The proof is based on final-state observability estimates of the adjoint system with the use of characteristics.

2019, 9(4): 729-758 doi: 10.3934/mcrf.2019030 +[Abstract](2949) +[HTML](568) +[PDF](1884.8KB)
Abstract:

This paper deals with the numerical approximation of null controls for the wave equation posed in a bounded domain of \begin{document}$\mathbb{R}^n$\end{document}. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & Münch, A mixed formulation for the direct approximation of the control of minimal \begin{document}$L^2$\end{document}-norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized observability inequality, requires \begin{document}$H^2$\end{document}-finite element approximation both in time and space. Following a similar approach, we present and analyze a variational method still leading to strong convergent results but using simpler \begin{document}$H^1$\end{document}-approximation. The main point is to preliminary restate the second order wave equation into a first order system and then prove an appropriate observability inequality.

2019, 9(4): 759-791 doi: 10.3934/mcrf.2019049 +[Abstract](2233) +[HTML](141) +[PDF](491.83KB)
Abstract:

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)).

2019, 9(4): 793-836 doi: 10.3934/mcrf.2019050 +[Abstract](1566) +[HTML](129) +[PDF](476.32KB)
Abstract:

In this paper, we study the controllability of a fluid-structure interaction system. We consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton's laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local null controllability for the velocity and temperature of the fluid and for the position and velocity of rigid body for a control acting only on the temperature equation on a fixed subset of the fluid domain.

2019  Impact Factor: 0.857