# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete and Continuous Dynamical Systems - S

June 2022 , Volume 15 , Issue 6

Issue on control theory and inverse problems. Part II

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2022, 15(6): 1317-1338 doi: 10.3934/dcdss.2022098 +[Abstract](209) +[HTML](49) +[PDF](409.59KB)
Abstract:

In this work, we consider a coupled wave equations with partially and locally distributed Kelvin-Voigt damping, where only one equation is dissipative.

Under the assumption that the damping coefficient changes smoothly near the interface of the damped and undamped regions, we investigate the effectiveness of the indirect control, and we prove that the associated semigroup is eventually differential.

2022, 15(6): 1339-1354 doi: 10.3934/dcdss.2022090 +[Abstract](177) +[HTML](61) +[PDF](407.06KB)
Abstract:

In this paper, we consider a system of two wave equations coupled through zero order terms. One of these equations has an internal damping, and the other has a boundary damping. We investigate stability properties of the system according to the variable strings densities. Indeed, our main result is to show that the corresponding model is not exponentially stable using a spectral theory which forms the center of this work. Otherwise, we establish a polynomial energy decay rate of type \begin{document}$\frac{1}{\sqrt{t}}.$\end{document}

2022, 15(6): 1355-1376 doi: 10.3934/dcdss.2022020 +[Abstract](326) +[HTML](106) +[PDF](497.29KB)
Abstract:

The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).

2022, 15(6): 1377-1401 doi: 10.3934/dcdss.2022011 +[Abstract](423) +[HTML](137) +[PDF](467.81KB)
Abstract:

Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.

By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.

2022, 15(6): 1403-1420 doi: 10.3934/dcdss.2022086 +[Abstract](305) +[HTML](56) +[PDF](471.14KB)
Abstract:

This paper treats the stationary Stokes problem in exterior domain of \begin{document}${{\mathbb{R}}}^3$\end{document} with Navier slip boundary condition. The behavior at infinity of the data and the solution are determined by setting the problem in \begin{document}$L^p$\end{document}-spaces, for \begin{document}$p> 2$\end{document}, with weights. The main results are the existence and uniqueness of strong solutions of the corresponding system.

2022, 15(6): 1421-1438 doi: 10.3934/dcdss.2022073 +[Abstract](198) +[HTML](64) +[PDF](411.27KB)
Abstract:

In this paper we consider star-shaped viscoelastic networks, and study the large-time behaviour of these networks by proving polynomial decay rates. The energy decay rate depends on the irrationality properties of the lengths of the rods.

2022, 15(6): 1439-1454 doi: 10.3934/dcdss.2022017 +[Abstract](291) +[HTML](103) +[PDF](438.3KB)
Abstract:

Let \begin{document}$\left(H, \langle \cdot, \cdot \rangle \right)$\end{document} be a separable Hilbert space and \begin{document}$A_{i}:D(A_i) \to H$\end{document} (\begin{document}$i = 1, \cdots, n$\end{document}) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function \begin{document}$u:[0, T] \to H$\end{document} and \begin{document}$n$\end{document} time-dependent diffusion coefficients \begin{document}$\alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+$\end{document} that fulfill the initial-value problem

Under suitable assumptions on the operators \begin{document}$A_i$\end{document}, \begin{document}$i = 1, \cdots, n$\end{document}, on the initial data \begin{document}$x\in H$\end{document} and on the given functions \begin{document}$\varphi_i$\end{document}, \begin{document}$i = 1, \cdots, n$\end{document}, we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lamé parameters in an elasticity problem on a plate.

2022, 15(6): 1455-1467 doi: 10.3934/dcdss.2022031 +[Abstract](297) +[HTML](111) +[PDF](369.73KB)
Abstract:

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping

where \begin{document}$b(x) = 0$\end{document} on \begin{document}$x\in (-1, 0]$\end{document}, and \begin{document}$b(x) = a(x)>0$\end{document} on \begin{document}$x\in (0, 1)$\end{document}. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that \begin{document}$a'(x)$\end{document} has a singularity at \begin{document}$x = 0$\end{document}, we investigate the decay rate of the solution which depends on the order of the singularity.

When \begin{document}$a(x)$\end{document} behaves like \begin{document}$x^{\alpha}(-\log x)^{-\beta}$\end{document} near \begin{document}$x = 0$\end{document} for \begin{document}$0\le{\alpha}<1, \;0\le\beta$\end{document} or \begin{document}$0<{\alpha}<1, \;\beta<0$\end{document}, we show that the system can achieve a mixed polynomial-logarithmic decay rate.

As a byproduct, when \begin{document}$\beta = 0$\end{document}, we obtain the decay rate \begin{document}$t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}}$\end{document} of solution for arbitrarily small \begin{document}$\varepsilon>0$\end{document}, which improves the rate \begin{document}$t^{-\frac{1}{1-{\alpha}}}$\end{document} obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case \begin{document}$\alpha\to 1^-$\end{document}. This is a step toward the goal of obtaining the optimal decay rate eventually.

2022, 15(6): 1469-1498 doi: 10.3934/dcdss.2022029 +[Abstract](368) +[HTML](114) +[PDF](815.53KB)
Abstract:

This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.

2022, 15(6): 1499-1523 doi: 10.3934/dcdss.2022096 +[Abstract](307) +[HTML](62) +[PDF](471.78KB)
Abstract:

We prove in this paper the global approximate controllability of the 1-D Boussinesq equation-subjected to internal control and free boundary conditions-on a bounded domain. The key ingredients of the proof relies Coron's return method for the exact global controllability of the nonlinear control system \begin{document}$y_{tt}+(y^2)_{xx} = u(t)$\end{document}, combined with some priori estimates for nonlinear weak-hyperbolic systems defined respectively in Gevrey class of functions, and in Sobolev spaces.

2022, 15(6): 1525-1546 doi: 10.3934/dcdss.2022087 +[Abstract](262) +[HTML](69) +[PDF](480.94KB)
Abstract:

This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function \begin{document}$F$\end{document} that behaves at infinity like \begin{document}$|s| \ln ^{p}(1+|s|),$\end{document} with \begin{document}$p > 2$\end{document}. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties \begin{document}$F(s, \xi)$\end{document} and \begin{document}$G(s, \xi)$\end{document} growing slower than \begin{document}$|s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|)$\end{document} at infinity.

2022, 15(6): 1547-1560 doi: 10.3934/dcdss.2022007 +[Abstract](293) +[HTML](157) +[PDF](374.37KB)
Abstract:

The purpose of this paper is to stabilize the annular pressure profile throughout the well bore continuously while drilling. A new nonlinear dynamical system is developed and a controller is designed to stabilize the annular pressure and achieve asymptotic tracking by applying feedback control of the main pumps. Hence, the paper studies the control design for the well known Managed Pressure Drilling system (MPD). MPD provides a closed loop drilling process in which pore pressure, formation fracture pressure, and bottomhole pressure are balanced and managed at surface. Although, responses must provide a solution for critical downhole pressures to preserve drilling efficiency and safety. Our MPD scheme is elaborated in reference to a nontrivial backstepping control procedure, and the effectiveness of the proposed control laws is shown by simulations.

2022, 15(6): 1561-1571 doi: 10.3934/dcdss.2022049 +[Abstract](213) +[HTML](68) +[PDF](316.18KB)
Abstract:

In this paper, we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belongs to \begin{document}$L^1_{loc}([0, +\infty)).$\end{document} Under suitable assumptions, we show well-posedness and exponential stability for small initial data. Our strategy combines careful energy estimates and continuity arguments. Some examples illustrate the abstract results.

2022, 15(6): 1573-1597 doi: 10.3934/dcdss.2022099 +[Abstract](159) +[HTML](34) +[PDF](748.68KB)
Abstract:

In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.

2022, 15(6): 1599-1614 doi: 10.3934/dcdss.2022024 +[Abstract](304) +[HTML](83) +[PDF](381.3KB)
Abstract:

This article is concerned with a strong unique continuation property of solutions for a diffusive SIS (Susceptible - Infected - Susceptible, or SI) model, which belongs to a type of observability inequalities in a time interval \begin{document}$[0, T]$\end{document}. That is, if one can observe solution on a convex and connected bounded open set \begin{document}$\omega$\end{document} in a bounded domain \begin{document}$\Omega$\end{document} at time \begin{document}$t = T$\end{document}, then the norms of solution on \begin{document}$[0,T)$\end{document} on \begin{document}$\Omega$\end{document} are observable. In our discussion, boundary condition is a homogeneous Dirichlet one (hostile boundary condition).

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6