American Institute of Mathematical Sciences

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Evolution Equations & Control Theory

December 2017 , Volume 6 , Issue 4

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2017, 6(4): 487-516 doi: 10.3934/eect.2017025 +[Abstract](2109) +[HTML](115) +[PDF](530.8KB)
Abstract:

We deal with the simultaneous controllability properties of two one dimensional (strongly) coupled wave equations when the control acts on the boundary. Necessary and sufficient conditions for approximate and exact controllability are proved.

2017, 6(4): 517-534 doi: 10.3934/eect.2017026 +[Abstract](2087) +[HTML](150) +[PDF](451.6KB)
Abstract:

The controllability of non-autonomous evolution systems is an important and difficult topic in control theory. In this paper, we study the approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. The theory of linear evolution operators is used instead of \begin{document}$C_0-$\end{document}semigroup to discuss the problem. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. Finally, two examples are provided to illustrate the applications of the obtained results.

2017, 6(4): 535-557 doi: 10.3934/eect.2017027 +[Abstract](1944) +[HTML](108) +[PDF](4973.5KB)
Abstract:

We investigate the effectiveness of a simple finite-dimensional feedback control scheme for globally stabilizing solutions of infinite-dimensional dissipative evolution equations introduced by Azouani and Titi in [7]. This feedback control algorithm overcomes some of the major difficulties in control of multi-scale processes: It does not require the presence of separation of scales nor does it assume the existence of a finite-dimensional globally invariant inertial manifold. In this work we present a theoretical framework for a control algorithm which allows us to give a systematic stability analysis, and present the parameter regime where stabilization or control objective is attained. In addition, the number of observables and controllers that were derived analytically and implemented in our numerical studies is consistent with the finite number of determining modes that are relevant to the underlying physical system. We verify the results computationally in the context of the Chafee-Infante reaction-diffusion equation, the Kuramoto-Sivashinsky equation, and other applied control problems, and observe that the control strategy is robust and independent of the model equation describing the dissipative system.

2017, 6(4): 559-586 doi: 10.3934/eect.2017028 +[Abstract](1991) +[HTML](99) +[PDF](600.4KB)
Abstract:

A game control problems of the Schlögl and FitzHugh-Nagumo equations are considered. The problems are investigated both from the viewpoint of the first player (the partner) and of the second player (the opponent). For both players, their own procedures for forming feedback controls are specified.

2017, 6(4): 587-597 doi: 10.3934/eect.2017029 +[Abstract](2076) +[HTML](157) +[PDF](355.8KB)
Abstract:

We consider the large time behavior of a solution to a drift-diffusion equation for degenerate and non-degenerate type. We show an instability and uniform unbounded estimate for the semi-linear case and uniform bound and convergence to the stationary solution for the case of mass critical degenerate case for higher space of dimension bigger than two.

2017, 6(4): 599-613 doi: 10.3934/eect.2017030 +[Abstract](1945) +[HTML](156) +[PDF](375.2KB)
Abstract:

This paper deals with exact controllability of a class of abstract nonlocal Cauchy problem with impulsive conditions in Banach spaces. By using Sadovskii fixed point theorem and Mönch fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for nonlocal functions. An example is given to illustrate the main results.

2017, 6(4): 615-628 doi: 10.3934/eect.2017031 +[Abstract](1471) +[HTML](87) +[PDF](233.5KB)
Abstract:

We consider the large time behavior of solutions to the following nonlinear wave equation: \begin{document} $\partial_{t}^2 u = c(u)^{2}\partial^2 _x u + λ c(u)c'(u)(\partial_x u)^2$ \end{document} with the parameter \begin{document} $λ ∈ [0,2]$ \end{document}. If \begin{document} $c(u(0,x))$ \end{document} is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if \begin{document} $c(· )$ \end{document} has a zero point, then \begin{document} $c(u(t,x))$ \end{document} can be going to zero in finite time. When \begin{document} $c(u(t,x))$ \end{document} is going to 0, the equation degenerates. We give a sufficient condition so that the equation with \begin{document} $0≤q λ < 2$ \end{document} degenerates in finite time.

2017, 6(4): 629-645 doi: 10.3934/eect.2017032 +[Abstract](1597) +[HTML](97) +[PDF](396.1KB)
Abstract:

In this paper, we investigate the initial boundary value problem for the linearized double dispersion equation on the half space \begin{document}$\mathbb{R}^{n}_{+}$\end{document}. We convert the initial boundary value problem into the initial value problem by odd reflection. The asymptotic profile of solutions to the initial boundary value problem is derived by establishing the asymptotic profile of solutions to the initial value problem. More precisely, the asymptotic profile of solutions is associated with the convolution of the partial derivative of the fundamental solutions of heat equation and the fundamental solutions of free wave equation.

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