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考虑具有Lipschitz非线性项,半线性热方程的最优控制问题.我们将运用观测不等式,证明值函数ψ作为相应Hamilton-Jacobi方程的唯一粘性正解是局部Lipschitz连续的.最后,运用动态规划方法,得到系统最优的反馈控制. 相似文献
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Cabanillas V. R. de Menezes S. B. Zuazua E. 《Journal of Optimization Theory and Applications》2001,110(2):245-264
We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain of
N
with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain such that the uncontrolled region \ is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is
C1
and globally Lipschitz, the system is null controllable. 相似文献
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主要采用了构造辅助系统u′ △u=0的方法,同时也利用了强单调算子 的满射性。在条件(H)的修改下,得到在有限时间内的精确可控性,研究半线性热方程系统ey/et=△y-f(x,t,y,倒△y) h(x,t)的精确内部能控性。 相似文献
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This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces. 相似文献
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A. H. Belghazi 《Acta Appl Math》2014,134(1):87-109
We study in this paper a class of parabolic equations in singular domains. These equations are defined in a singular cylindrical domain whose cross-section contains one reentrant corner or one straight emerging crack. We assume that the diffusion coefficients are non-smooth in the normal direction. We will show some spectral inequality thanks to Carleman type estimates and the construction of a suitable weight function satisfying some properties. As in Benabdallah et al. (C. R. Acad. Sci. Paris 344(6):357–362, 2007), we deduce the null-controllability of these equations with the help of the Lebeau-Robbiano method. 相似文献
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本文讨论非线性项带一个极大单调图的半线性抛物型方程系统的零能控性.文中利用Kakutani不动点定理和线性抛物型方程的能控性得到该系统是零能控的,如果控制作用在内部区域上.由此,还得到该系统是零能控的,如果控制施加在一部分边界上. 相似文献
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Exact Controllability to Trajectories for a Semilinear Heat Equation with a Superlinear Nonlinearity
In this paper, we consider exact controllability to trajectories for a semilinear heat equation with nonlinearity that has superlinear growth at infinity with Dirichlet boundary conditions in a bounded domain of ? N . The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem and generalize the previous results. 相似文献
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Exact Controllability of the Superlinear Heat Equation 总被引:4,自引:0,他引:4
V. Barbu 《Applied Mathematics and Optimization》2000,42(1):73-89
The exact internal and boundary controllability of parabolic equations with superlinear nonlinearity is studied.
Accepted 3 January 2000 相似文献
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Jean-Michel CORON Jes′us Ildefonso D′IAZ Abdelmalek DRICI Tommaso MINGAZZINI 《数学年刊B辑(英文版)》2013,34(3):333-344
The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques. 相似文献
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W. Bian 《Acta Mathematica Hungarica》1998,81(1-2):41-57
The relations between the (strong) reachable sets of the semilinear evolution equation systems x′(t) + A(t)x(t) = f(t, x(t), u(t)) + Hu(t), x′(t) + A(t)x(t) = f(t, x(t), Hu(t)) + Hu(t) on a Banach space, and their corresponding linear systems are studied. Compared with previous results, the systems considered here are more general (f is not independent of the control u), no compactness assumptions on A or f are imposed in some of our main results, and we suppose f is a set-contraction rather than Lipschitz and have less restriction on the contraction coefficient. Other kinds of conditions are involved to guarantee the approximate controllability. 相似文献
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A model representing a coupling between a heat conducting medium and a solid structure is considered. We establish a Carleman
inequality for this model. Next we deduce a null controllability result with an internal control in the conducting medium
and there is no control in the solid part. 相似文献
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In this work, we study an approximate control problem for the heatequation, with a nonstandard but rather natural restriction on thesolution. It is well known that approximate controllability holds. On theother hand, the total mass of the solutions of the uncontrolled system isconstant in time. Therefore, it is natural to analyze whether approximatecontrollability holds supposing the total mass of the solution to be a givenconstant along the trajectory. Under this additional restriction,approximate controllability is not always true. For instance, this propertyfails when is a ball. We prove that the system is genericallycontrollable; that is, given an open regular bounded domain , thereexists an arbitrarily small smooth deformation u, such that the system isapproximately controllable in the new domain +u underthis constraint. We reduce our control problem to a nonstandard uniquenessproblem. This uniqueness property is shown to hold generically with respectto the domain. 相似文献
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In this note, we prove the exact controllability for the semilinear wave equations in any space dimensions under the condition that the nonlinearity behaves like
as s → ∞. 相似文献
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A control system described by the one-dimensional heat equationwith a boundary control is considered. The problem of findinga control which drives the system, in finite time T, to a constantfinal temperature distribution is discussed. A simple boundfor the norm of the control in terms of the final time T isderived, and a numerical example is presented. 相似文献
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本文研究初值问题ut=Δu+g(t)f(u)(t>0),u|t=0=u0(x)和初边值问题ut=Δu+g(t,x)f(u)(t>0,x∈Ω),u|t=0=u|?Ω=0之解的整体存在性。如文献[6]中所作的那样,在非线性项中引进因子g(t)或g(t,x),是为了防止解的爆破或熄灭现象发生。本文的结果表明,文献[6]的两个定理中对f,g和u0的大部分限制可以取消或者减弱;对g可以只要求它在f大时充分小;在一定条件下,控制初始状态即可避免爆破。 相似文献
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In this note, we clarify a technical point of Ref. 1, devoted to studying a constrained approximate controllability for the heat equation. 相似文献