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Mathematical Control & Related Fields

2014 , Volume 4 , Issue 3

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Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients
Franck Boyer and Guillaume Olive
2014, 4(3): 263-287 doi: 10.3934/mcrf.2014.4.263 +[Abstract](65) +[PDF](477.8KB)
In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms.
    Our analysis is mainly based on a criterion given by Fattorini in [12] (and systematically used in [22] for instance), that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems. This work gives theoretical justifications of some numerical observations described in [9].
Time optimal control problems for some non-smooth systems
Hongwei Lou, Junjie Wen and Yashan Xu
2014, 4(3): 289-314 doi: 10.3934/mcrf.2014.4.289 +[Abstract](81) +[PDF](473.3KB)
Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.
Existence theory of capillary-gravity waves on water of finite depth
Shu-Ming Sun
2014, 4(3): 315-363 doi: 10.3934/mcrf.2014.4.315 +[Abstract](117) +[PDF](749.6KB)
This review article discusses the recent developments on the existence of two-dimensional and three-dimensional capillary-gravity waves on water of finite-depth. The Korteweg-de Vries (KdV) equation and Kadomtsev-Petviashvili (KP) equation are derived formally from the exact governing equations and the solitary-wave solutions and other solution are obtained for these model equations. Recent results on the existence of solutions of the exact governing equations near the solutions of these model equations are presented and various two- and three-dimensional solutions of the exact equations are provided. The ideas and methods to obtain the existence results are briefly discussed.
Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities
Haisen Zhang
2014, 4(3): 365-379 doi: 10.3934/mcrf.2014.4.365 +[Abstract](110) +[PDF](358.2KB)
In optimization problems, it is significant to study the directional derivatives and subdifferentials of objective functions. Using directional derivatives and subdifferentials of objective functions, we can establish optimality conditions, derive error bound properties, and propose optimal algorithms. In this paper, the upper and lower estimates for the Clarke directional derivatives of a class of marginal functions are established. Employing this result, we obtain the exact formulations of the Clarke directional derivatives of the regularized gap functions for nonsmooth quasi-variational inequalities.
Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems
Xiuxiang Zhou
2014, 4(3): 381-399 doi: 10.3934/mcrf.2014.4.381 +[Abstract](88) +[PDF](420.1KB)
This paper is addressed to the disturbance decoupling and almost disturbance decoupling problems in infinite dimensions. We introduce a class of approximate finite dimensional systems, and show that if the systems are disturbance decoupled, so does the original infinite dimensional system. It is also shown that this approach can be employed to solve the almost disturbance decoupling problem. Finally, some illustrative examples are provided.

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