Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems... more Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \begin{document}$ \begin{equation} \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} & = & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}& = & c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} & = & (u_{0}, v_{0}) \end{array}\right. \ \ \ \ \ \ \ \ \left( {0.1} \right) \end{equation} $\end{document} posed on the periodic domain \begin{document}$ \mathbb{T} $\end{document} in the following four spaces \begin{document}$ \begin{split} { \mathcal H}^s_1: = H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_2: = H^s_0 ( \mathbb {T})\times H^s(\mathbb{T}), \\ { \mathcal H}^s_3: = H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_4: = H^s (\mathbb{T})\times H^s (\mathbb{T}). \end{split} $\end{document} The coefficients are assumed to satisfy \begin{document}$ a_1 a_2\neq 0 $\end{document} and \begin{document}$ \sum\...
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996
The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gai... more The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iter...
Transactions of the American Mathematical Society, 1996
In this paper, we consider distributed control of the system described by the Korteweg-de Vries e... more In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) ∂ t u + u ∂ x u + ∂ x 3 u = f \begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*} on the interval 0 ≤ x ≤ 2 π , t ≥ 0 0\leq x\leq 2\pi , \, t\geq 0 , with periodic boundary conditions (ii) ∂ x k u ( 2 π , t ) = ∂ x k u ( 0 , t ) , k = 0 , 1 , 2 , \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*} where the distributed control f ≡ f ( x , t ) f\equiv f(x,t) is restricted so that the “volume” ∫ 0 2 π u ( x , t ) d x \int ^{2\pi }_0 u(x,t) dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f f is allowed to act on the whole spatial domain ( 0 , 2 π ) (0,2\pi ) , it is shown that the system is globally exactly controllable, i.e., for given T > 0 T> 0 and fu...
We show that the motions of a linear thermoelastic beam may be controlled exactly to zero in a fi... more We show that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam. The optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted, as in the Euler-Bernoulli beam theory.
ESAIM: Control, Optimisation and Calculus of Variations, 2021
The solutions of the Cauchy problem of the KdV equation on a periodic domain 𝕋, [see formula in P... more The solutions of the Cauchy problem of the KdV equation on a periodic domain 𝕋, [see formula in PDF] possess neither the sharp Kato smoothing property, [see formula in PDF] nor the Kato smoothing property, [see formula in PDF] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain 𝕋, (See Eq. (1) below) [see formula in PDF] where g ∈ C∞(𝕋) is a real value function with the support [see formula in PDF] It is shown that (1) if ω ≠ ∅, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; (2) if g is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property.
The focus of the present study is the BBM equation which models unidirectional propagation of sma... more The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The princi-pal new result is an exact theory of convergence of the two-point boundary-value problem to the initial-value problem posed on an infinite stretch of the medium of propagation. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the initial-value problem posed on the entire line.
Control and Estimation of Distributed Parameter Systems, 1998
In this paper we consider distributed control of the system described by the generalized Boussine... more In this paper we consider distributed control of the system described by the generalized Boussinesq equation on the periodic domain S, the unit circle in the plane utt = uxx -</font > (a(u) + uxx)xx = f {u_{tt}} = {u_{xx}} - {(a(u) + {u_{xx}})_{xx}} = f . In the case of local control, if the control f is allowed to act on the whole domain S, it is shown that the system is globally exactly controllable. In the case of local control where the control f is only allowed to act on a sub-domain of S, we show that the same result holds if the initial and terminal states have “small amplitude” in a certain sense.
Unique continuation problems are considered for the Korteweg–de Vries (KdV) equation \[ u_t + uu_... more Unique continuation problems are considered for the Korteweg–de Vries (KdV) equation \[ u_t + uu_x + u_{xxx} = 0,\quad - \infty \frac{3}{2})$ is a solution of the KdV equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the KdV equation, then u must vanish everywhere if it vanishes on two horizontal half lines in the x-t space. This implies that the solution u must vanish everywhere if it vanishes on an open subset in the x-t space. As a consequence of the Miura transformation, the above results for the KdV equation are also true for the modified Korteweg–de Vries equation \[ v_t - 6v^2 v_x + v_{xxx} = 0,\quad - \infty < x,\quad t < + \infty .\]
Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems... more Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \begin{document}$ \begin{equation} \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} & = & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}& = & c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} & = & (u_{0}, v_{0}) \end{array}\right. \ \ \ \ \ \ \ \ \left( {0.1} \right) \end{equation} $\end{document} posed on the periodic domain \begin{document}$ \mathbb{T} $\end{document} in the following four spaces \begin{document}$ \begin{split} { \mathcal H}^s_1: = H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_2: = H^s_0 ( \mathbb {T})\times H^s(\mathbb{T}), \\ { \mathcal H}^s_3: = H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_4: = H^s (\mathbb{T})\times H^s (\mathbb{T}). \end{split} $\end{document} The coefficients are assumed to satisfy \begin{document}$ a_1 a_2\neq 0 $\end{document} and \begin{document}$ \sum\...
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996
The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gai... more The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iter...
Transactions of the American Mathematical Society, 1996
In this paper, we consider distributed control of the system described by the Korteweg-de Vries e... more In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) ∂ t u + u ∂ x u + ∂ x 3 u = f \begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*} on the interval 0 ≤ x ≤ 2 π , t ≥ 0 0\leq x\leq 2\pi , \, t\geq 0 , with periodic boundary conditions (ii) ∂ x k u ( 2 π , t ) = ∂ x k u ( 0 , t ) , k = 0 , 1 , 2 , \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*} where the distributed control f ≡ f ( x , t ) f\equiv f(x,t) is restricted so that the “volume” ∫ 0 2 π u ( x , t ) d x \int ^{2\pi }_0 u(x,t) dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f f is allowed to act on the whole spatial domain ( 0 , 2 π ) (0,2\pi ) , it is shown that the system is globally exactly controllable, i.e., for given T > 0 T> 0 and fu...
We show that the motions of a linear thermoelastic beam may be controlled exactly to zero in a fi... more We show that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam. The optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted, as in the Euler-Bernoulli beam theory.
ESAIM: Control, Optimisation and Calculus of Variations, 2021
The solutions of the Cauchy problem of the KdV equation on a periodic domain 𝕋, [see formula in P... more The solutions of the Cauchy problem of the KdV equation on a periodic domain 𝕋, [see formula in PDF] possess neither the sharp Kato smoothing property, [see formula in PDF] nor the Kato smoothing property, [see formula in PDF] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain 𝕋, (See Eq. (1) below) [see formula in PDF] where g ∈ C∞(𝕋) is a real value function with the support [see formula in PDF] It is shown that (1) if ω ≠ ∅, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; (2) if g is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property.
The focus of the present study is the BBM equation which models unidirectional propagation of sma... more The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The princi-pal new result is an exact theory of convergence of the two-point boundary-value problem to the initial-value problem posed on an infinite stretch of the medium of propagation. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the initial-value problem posed on the entire line.
Control and Estimation of Distributed Parameter Systems, 1998
In this paper we consider distributed control of the system described by the generalized Boussine... more In this paper we consider distributed control of the system described by the generalized Boussinesq equation on the periodic domain S, the unit circle in the plane utt = uxx -&lt;/font &gt; (a(u) + uxx)xx = f {u_{tt}} = {u_{xx}} - {(a(u) + {u_{xx}})_{xx}} = f . In the case of local control, if the control f is allowed to act on the whole domain S, it is shown that the system is globally exactly controllable. In the case of local control where the control f is only allowed to act on a sub-domain of S, we show that the same result holds if the initial and terminal states have “small amplitude” in a certain sense.
Unique continuation problems are considered for the Korteweg–de Vries (KdV) equation \[ u_t + uu_... more Unique continuation problems are considered for the Korteweg–de Vries (KdV) equation \[ u_t + uu_x + u_{xxx} = 0,\quad - \infty \frac{3}{2})$ is a solution of the KdV equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the KdV equation, then u must vanish everywhere if it vanishes on two horizontal half lines in the x-t space. This implies that the solution u must vanish everywhere if it vanishes on an open subset in the x-t space. As a consequence of the Miura transformation, the above results for the KdV equation are also true for the modified Korteweg–de Vries equation \[ v_t - 6v^2 v_x + v_{xxx} = 0,\quad - \infty < x,\quad t < + \infty .\]
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