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Bhandari Kuntal - One of the best experts on this subject based on the ideXlab platform.
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Contrôlabilité par le bord de quelques systèmes paraboliques couplés avec conditions de Robin ou de Kirchhoff
HAL CCSD, 2020Co-Authors: Bhandari KuntalAbstract:In this thesis, we study the boundary null-controllability of some linear parabolic systems coupled through interior and/or boundary. We begin by giving an overall introduction of the thesis in Chapter 1 and we discuss some essentials about the notion of parabolic controllability in the second chapter. In Chapter 3, we investigate the boundary null-controllability of some 2 × 2 coupled parabolic systems in the cascade form where the boundary conditions are of Robin type. This case is considered mainly in space dimension 1 and in the cylindrical geometry. We prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, which let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popularpenalization method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems. Coming to the Chapter 4, we first discuss the boundary null-controllability of some 2 × 2 parabolic systems in 1-D that contains a linear interior coupling with Real Constant Coefficient and a Kirchhoff-type condition through which the boundary coupling enters in the system. The control is exerted on a part of the boundary through a Dirichlet condition on either one of the two state components. We show that the controllability properties vary depending on which component the control is being applied; the choices of interior coupling Coefficient and the Kirchhoff parameter play a crucial role to deduce positive or negative controllability results. Thereafter, we study a 3 × 3 model with one or two Dirichlet boundary control(s) at one end and a Kirchhoff-type boundary condition at the other; here the third equation is coupled (interior) through the first component. In this case we obtain the following: treating the control on the first component, we have conditional controllability depending on the choices of interior coupling Coefficient and the Kirchhoff parameter, while considering a control on the second component always provides positive result. But in contrast, putting a control on the third entry yields a negative controllability result. In this situation, one must need two boundary controls on any two components to recover the controllability. Further in the thesis, we pursue some numerical studies based on the penalized Hilbert Uniqueness Method (HUM) to illustrate our theoretical results and test other examples in the framework of interior-boundary coupled systems.Dans cette thèse, on étudie la contrôlabilité à zéro par le bord de quelques systèmes paraboliques linéaires couplés par des termes de couplage intérieur et/ou au bord. Le premier chapitre est une introduction à l’ensemble du manuscrit. Dans le deuxième chapitre, on rappelle les principaux concepts et résultats autour des notions de contrôlabilité qui seront utilisés dans la suite. Dans le troisième chapitre, on étudie principalement la contrôlabilité par le bord d’un système couplé 2×2 de type cascade avec des conditions au bord de Robin. En particulier, on prouve que les contrôles associés satisfont des bornes uniformes par rapport aux paramètres de Robin et convergent vers un contrôle de Dirichlet lorsque les paramètres de Robin tendent vers l’infini. Cette étude fournit une justification, dans le contexte du contrôle, de la méthode de pénalisation qui est couramment utilisée pour prendre en compte des données de Dirichlet peu régulières en pratique. Dans le quatrième et dernier chapitre, on étudie d’abord la contrôlabilité à zéro d’un système 2 × 2 en dimension 1 contenant des termes de couplage à la fois à l’intérieur et au bord du domaine. Plus précisément, on considère une condition de type Kirchhoff sur l’un des bords du domaineet un contrôle de Dirichlet sur l’autre bord, dans l’une ou l’autre des équations. En particulier, on montre que les propriétés de contrôle du système diffèrent selon que le contrôle agisse sur la première ou sur la seconde équation, et selon les valeurs du Coefficient de couplage intérieur et du paramètre de Kirchhoff. On étudie ensuite un modèle 3 × 3 avec un ou deux contrôle(s) aux limites de Dirichlet à une extrémité et une condition de type Kirchhoff à l’autre extrémité ; ici la troisième équation est couplée (couplage intérieur) avec la première. Dans ce cas, on obtient ce qui suit : en considérant le contrôle sur la première équation, on a contrôlabilité conditionnelle dépendant des choix du Coefficient de couplage intérieur et du paramètre de Kirchhoff, et en considérant le contrôle sur la deuxième équation, on obtient toujours une contrôlabilité positive. En revanche, considérer un contrôle sur la troisième équation conduit à un résultat de contrôlabilité négative. Dans cette situation, on a besoin de deux contrôles aux limites sur deux destrois équations pour retrouver la contrôlabilité. Enfin, on expose quelques études numériques basées sur l’approche pénalisée HUM pour illustrer les résultats théoriques, ainsi que pour tester d’autres exemples
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Boundary controllability of some coupled parabolic systems with Robin or Kirchhoff conditions
2020Co-Authors: Bhandari KuntalAbstract:Dans cette thèse, on étudie la contrôlabilité à zéro par le bord de quelques systèmes paraboliques linéaires couplés par des termes de couplage intérieur et/ou au bord. Le premier chapitre est une introduction à l'ensemble du manuscrit. Dans le deuxième chapitre, on rappelle les principaux concepts et résultats autour des notions de contrôlabilité qui seront utilisés dans la suite. Dans le troisième chapitre, on étudie principalement la contrôlabilité par le bord d'un système couplé 2x2 de type cascade avec des conditions au bord de Robin. En particulier, on prouve que les contrôles associés satisfont des bornes uniformes par rapport aux paramètres de Robin et convergent vers un contrôle de Dirichlet lorsque les paramètres de Robin tendent vers l'infini. Cette étude fournit une justification, dans le contexte du contrôle, de la méthode de pénalisation qui est couramment utilisée pour prendre en compte des données de Dirichlet peu régulières en pratique. Dans le quatrième et dernier chapitre, on étudie d'abord la contrôlabilité à zéro d'un système 2x2 en dimension 1 contenant des termes de couplage à la fois à l'intérieur et au bord du domaine. Plus précisément, on considère une condition de type Kirchhoff sur l'un des bords du domaine et un contrôle de Dirichlet sur l'autre bord, dans l'une ou l'autre des équations. En particulier, on montre que les propriétés de contrôle du système diffèrent selon que le contrôle agisse sur la première ou sur la seconde équation, et selon les valeurs du Coefficient de couplage intérieur et du paramètre de Kirchhoff. On étudie ensuite un modèle 3x3 avec un ou deux contrôle(s) aux limites de Dirichlet à une extrémité et une condition de type Kirchhoff à l'autre extrémité ; ici la troisième équation est couplée (couplage intérieur) avec la première. Dans ce cas, on obtient ce qui suit : en considérant le contrôle sur la première équation, on a contrôlabilité conditionnelle dépendant des choix du Coefficient de couplage intérieur et du paramètre de Kirchhoff, et en considérant le contrôle sur la deuxième équation, on obtient toujours une contrôlabilité positive. En revanche, considérer un contrôle sur la troisième équation conduit à un résultat de contrôlabilité négative. Dans cette situation, on a besoin de deux contrôles aux limites sur deux des trois équations pour retrouver la contrôlabilité. Enfin, on expose quelques études numériques basées sur l'approche pénalisée HUM pour illustrer les résultats théoriques, ainsi que pour tester d'autres exemples.In this thesis, we study the boundary null-controllability of some linear parabolic systems coupled through interior and/or boundary. We begin by giving an overall introduction of the thesis in Chapter 1 and we discuss some essentials about the notion of parabolic controllability in the second chapter. In Chapter 3, we investigate the boundary null-controllability of some 2x2 coupled parabolic systems in the cascade form where the boundary conditions are of Robin type. This case is considered mainly in space dimension 1 and in the cylindrical geometry. We prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, which let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalization method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems. Coming to the Chapter 4, we first discuss the boundary null-controllability of some 2x2 parabolic systems in 1-D that contains a linear interior coupling with Real Constant Coefficient and a Kirchhoff-type condition through which the boundary coupling enters in the system. The control is exerted on a part of the boundary through a Dirichlet condition on either one of the two state components. We show that the controllability properties vary depending on which component the control is being applied; the choices of interior coupling Coefficient and the Kirchhoff parameter play a crucial role to deduce positive or negative controllability results. Thereafter, we study a 3x3 model with one or two Dirichlet boundary control(s) at one end and a Kirchhoff-type boundary condition at the other; here the third equation is coupled (interior) through the first component. In this case we obtain the following: treating the control on the first component, we have conditional controllability depending on the choices of interior coupling Coefficient and the Kirchhoff parameter, while considering a control on the second component always provides positive result. But in contrast, putting a control on the third entry yields a negative controllability result. In this situation, one must need two boundary controls on any two components to recover the controllability. Further in the thesis, we pursue some numerical studies based on the penalized Hilbert Uniqueness Method (HUM) to illustrate our theoretical results and test other examples in the framework of interior-boundary coupled systems
Kuntal Bhandari - One of the best experts on this subject based on the ideXlab platform.
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Boundary controllability of some coupled parabolic systems with Robin or Kirchhoff conditions
2020Co-Authors: Kuntal BhandariAbstract:In this thesis, we study the boundary null-controllability of some linear parabolic systems coupled through interior and/or boundary. We begin by giving an overall introduction of the thesis in Chapter 1 and we discuss some essentials about the notion of parabolic controllability in the second chapter. In Chapter 3, we investigate the boundary null-controllability of some 2 × 2 coupled parabolic systems in the cascade form where the boundary conditions are of Robin type. This case is considered mainly in space dimension 1 and in the cylindrical geometry. We prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, which let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalization method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems. Coming to the Chapter 4, we first discuss the boundary null-controllability of some 2 × 2 parabolic systems in 1-D that contains a linear interior coupling with Real Constant Coefficient and a Kirchhoff-type condition through which the boundary coupling enters in the system. The control is exerted on a part of the boundary through a Dirichlet condition on either one of the two state components. We show that the controllability properties vary depending on which component the control is being applied; the choices of interior coupling Coefficient and the Kirchhoff parameter play a crucial role to deduce positive or negative controllability results. Thereafter, we study a 3 × 3 model with one or two Dirichlet boundary control(s) at one end and a Kirchhoff-type boundary condition at the other; here the third equation is coupled (interior) through the first component. In this case we obtain the following: treating the control on the first component, we have conditional controllability depending on the choices of interior coupling Coefficient and the Kirchhoff parameter, while considering a control on the second component always provides positive result. But in contrast, putting a control on the third entry yields a negative controllability result. In this situation, one must need two boundary controls on any two components to recover the controllability. Further in the thesis, we pursue some numerical studies based on the penalized Hilbert Uniqueness Method (HUM) to illustrate our theoretical results and test other examples in the framework of interior-boundary coupled systems.
Youngmok Jeon - One of the best experts on this subject based on the ideXlab platform.
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A quadrature method for Constant-Coefficient Cauchy singular integral equations on an interval
Anziam Journal, 2000Co-Authors: Youngmok JeonAbstract:We consider a mesh grading quadrature method for Real Constant-Coefficient Cauchy singular integral equations of index 0. The quadrature method is based on the trapezoidal rule. A complete stability and convergence analysis is given by the use of the noncompact perturbation analysis as in Jeon [ 10] and Elschner and S tephan [7]. The order of convergence can be arbitrarily high if the order of mesh grading is high enough. We also provide an efficient way of evaluating asymptotics of the solution at the end points. Experimentally, we observe that the method also works well for Cauchy singular integral equations with variable Coefficients.
B D Hassard - One of the best experts on this subject based on the ideXlab platform.
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counting roots of the characteristic equation for linear delay differential systems
Journal of Differential Equations, 1997Co-Authors: B D HassardAbstract:Abstract A formula is given that counts the number of roots in the positive half plane of the characteristic equation for general Real, Constant Coefficient, linear delay-differential systems. The formula is used to establish necessary and sufficient conditions for asymptotic stability of the zero solution of linear delay-differential systems. The formula is potentially useful in verifying stability hypotheses that arise in bifurcation analysis of autonomous delay-differential systems. Application of the formula to Hopf bifurcation theory for delay-differential systems is discussed, and an example application to an equation with two delays is given.
Ting-ting Jia - One of the best experts on this subject based on the ideXlab platform.
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Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics
Nonlinear Dynamics, 2019Co-Authors: Gao-fu Deng, Yi-tian Gao, Cui-cui Ding, Ting-ting JiaAbstract:Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the ($$2+1$$)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the Real Constant $$\alpha $$, the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to $$\gamma $$, a Real Constant Coefficient in the equation, velocity along the x direction is independent of $$\gamma $$, while velocity along the y direction is proportional to $$\gamma $$; (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.
