American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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Discrete and Continuous Dynamical Systems - S

May 2022 , Volume 15 , Issue 5

Issue on control theory and inverse problems. Part I

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2022, 15(5): ⅰ-ⅰ doi: 10.3934/dcdss.2022084 +[Abstract](289) +[HTML](87) +[PDF](75.66KB)
Abstract:
2022, 15(5): 983-993 doi: 10.3934/dcdss.2021106 +[Abstract](625) +[HTML](293) +[PDF](367.5KB)
Abstract:

In the paper under study, we consider the following coupled non-degenerate Kirchhoff system

where \begin{document}$\Omega$\end{document} is a bounded open subset of \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$\mathtt{α}$\end{document} and \begin{document}$\mathtt{β}$\end{document} be two nonzero real numbers with the same sign and \begin{document}$\mathtt{φ}$\end{document} is given by \begin{document}$\mathtt{φ}(s) = \mathfrak{m}_0+\mathfrak{m}_1s$\end{document} with some positive constants \begin{document}$\mathfrak{m}_0$\end{document} and \begin{document}$\mathfrak{m}_1$\end{document}. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik[5,8].

2022, 15(5): 995-1014 doi: 10.3934/dcdss.2021086 +[Abstract](868) +[HTML](368) +[PDF](435.9KB)
Abstract:

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory acting in the second equation (the shear angle equation) of the system. We prove that the asymptotic stability of the system holds under some general condition imposed into the relaxation function, precisely,

The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data \begin{document}$\eta{0x}$\end{document}. This study generalizes and improves previous literature outcomes.

2022, 15(5): 1015-1043 doi: 10.3934/dcdss.2021147 +[Abstract](452) +[HTML](267) +[PDF](719.44KB)
Abstract:

In this paper, we give some qualitative results on the behavior of a nonsimple elastic plate with memory corresponding to anti-plane shear deformations. First we describe briefly the equations of the considered plate and then we study the well-posedness of the resulting problem. Secondly, we perform the spectral analysis, in particular, we establish conditions on the physical constants of the plate to guarantee the simplicity and the monotonicity of the roots of the characteristic equation. The spectral results are used to prove the exponential stability of the solutions at an optimal decay rate given by the physical constants. Then we present an approximate controllability result of the considered control problem. Finally, we give some numerical experiments based on the spectral method developed with implementation in MATLAB for one and two-dimensional problems.

2022, 15(5): 1045-1059 doi: 10.3934/dcdss.2021164 +[Abstract](329) +[HTML](149) +[PDF](497.08KB)
Abstract:

In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.

2022, 15(5): 1061-1084 doi: 10.3934/dcdss.2021158 +[Abstract](398) +[HTML](172) +[PDF](522.25KB)
Abstract:

In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension \begin{document}$n\geq 2$\end{document} that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.

2022, 15(5): 1085-1102 doi: 10.3934/dcdss.2021092 +[Abstract](795) +[HTML](348) +[PDF](363.38KB)
Abstract:

In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and \begin{document}$1-d$\end{document} unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.

2022, 15(5): 1103-1125 doi: 10.3934/dcdss.2021139 +[Abstract](675) +[HTML](202) +[PDF](522.67KB)
Abstract:

This study is concerned with the pointwise stabilization for a star-shaped network of \begin{document}$N$\end{document} variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by Hansen and Zuazua [SIAM J. Control Optim., 33 (1995), 1357-1391] in a very particular case.

2022, 15(5): 1127-1141 doi: 10.3934/dcdss.2021090 +[Abstract](737) +[HTML](316) +[PDF](394.64KB)
Abstract:

This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.

2022, 15(5): 1143-1164 doi: 10.3934/dcdss.2021087 +[Abstract](808) +[HTML](432) +[PDF](787.34KB)
Abstract:

We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.

2022, 15(5): 1165-1181 doi: 10.3934/dcdss.2021089 +[Abstract](844) +[HTML](348) +[PDF](404.76KB)
Abstract:

Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation

which models the deformation of the deck of either a footbridge or a suspension bridge.

2022, 15(5): 1183-1220 doi: 10.3934/dcdss.2021142 +[Abstract](365) +[HTML](173) +[PDF](578.97KB)
Abstract:

In this paper, we investigate a network of elastic and thermo-elastic materials. On each thermo-elastic edge, we consider two coupled wave equations such that one of them is damped via a coupling with a heat equation. On each elastic edge (undamped), we consider two coupled conservative wave equations. Under some conditions, we prove that the thermal damping is enough to stabilize the whole system. If the two waves propagate with the same speed on each thermo-elastic edge, we show that the energy of the system decays exponentially. Otherwise, a polynomial energy decay is attained. Finally, we present some other boundary conditions and show that under sufficient conditions on the lengths of some elastic edges, the energy of the system decays exponentially on some particular networks similar to the ones considered in [18].

2022, 15(5): 1221-1232 doi: 10.3934/dcdss.2021093 +[Abstract](833) +[HTML](339) +[PDF](327.58KB)
Abstract:

In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.

2022, 15(5): 1233-1245 doi: 10.3934/dcdss.2021107 +[Abstract](692) +[HTML](272) +[PDF](388.22KB)
Abstract:

This work is concerned with a system of wave equations with variable-exponent nonlinearities acting in both equations. We, first, discuss the well-posedness then prove a blow up result for solutions with negative initial energy.

2022, 15(5): 1247-1268 doi: 10.3934/dcdss.2021085 +[Abstract](1120) +[HTML](602) +[PDF](1120.19KB)
Abstract:

This paper presents an optimal control problem of the general variable-order fractional delay model of advertising procedure. The problem describes the flow of the clients from the unaware people group to the conscious or bought band. The new formulation generalizes the model that proposed by Muller. Two control variables are considered to increase the number of customers who purchased the products. An efficient nonstandard difference approach is used to study numerically the behavior of the solution of the mentioned problem. Properties of the proposed system were introduced analytically and numerically. The proposed difference schema maintains the properties of the analytic solutions as boundedness and the positivity. Numerical examples, for testing the applicability of the utilized method and to show the simplicity, accuracy and efficiency of this approximation approach, are presented with some comprising with standard difference methods.

2022, 15(5): 1269-1305 doi: 10.3934/dcdss.2021091 +[Abstract](764) +[HTML](369) +[PDF](418.04KB)
Abstract:

In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the \begin{document}$N$\end{document}-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (i.e. \begin{document}$a = 1$\end{document} in (1)) and where the coupling parameter \begin{document}$b$\end{document} is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions \begin{document}$a = 1$\end{document} and \begin{document}$b$\end{document} small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter \begin{document}$b$\end{document} and on the arithmetic property of the wave propagation speeds \begin{document}$a$\end{document}.

2022, 15(5): 1307-1316 doi: 10.3934/dcdss.2021129 +[Abstract](676) +[HTML](242) +[PDF](368.55KB)
Abstract:

In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying \begin{document}$g'(t) \leq - \xi(t) H(g(t))$\end{document}, where \begin{document}$H$\end{document} is an increasing, convex function and \begin{document}$\xi$\end{document} is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when \begin{document}$H(t) = t^p, \ t\geq 0, \forall p>1$\end{document}, instead the parameter \begin{document}$p \in [1, \frac{3}{2}[$\end{document}.

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