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Mathematical Control & Related Fields

December 2015 , Volume 5 , Issue 4

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Finite-time stabilization of a network of strings
Fatiha Alabau-Boussouira, Vincent Perrollaz and Lionel Rosier
2015, 5(4): 721-742 doi: 10.3934/mcrf.2015.5.721 +[Abstract](786) +[PDF](580.7KB)
Abstract:
We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node. Our results lead to the finite-time stabilization even though the systems may not be dissipative.
Exact controllability for the Lamé system
Belhassen Dehman and Jean-Pierre Raymond
2015, 5(4): 743-760 doi: 10.3934/mcrf.2015.5.743 +[Abstract](964) +[PDF](428.1KB)
Abstract:
In this article, we prove an exact boundary controllability result for the isotropic elastic wave system in a bounded domain $\Omega$ of $\mathbb{R}^{3}$. This result is obtained under a microlocal condition linking the bicharacteristic paths of the system and the region of the boundary on which the control acts. This condition is to be compared with the so-called Geometric Control Condition by Bardos, Lebeau and Rauch [3]. The proof relies on microlocal tools, namely the propagation of the $C^{\infty}$ wave front and microlocal defect measures.
Stabilization of hyperbolic equations with mixed boundary conditions
Xiaoyu Fu
2015, 5(4): 761-780 doi: 10.3934/mcrf.2015.5.761 +[Abstract](788) +[PDF](497.3KB)
Abstract:
This paper is devoted to study decay properties of solutions to hyperbolic equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. Both of the equations are equipped with the mixed boundary conditions. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations. Our decay results rely on an analysis of the size of resolvent operators for hyperbolic equations on the imaginary axis. To derive this kind of resolvent estimates, we employ global Carleman estimates for elliptic equations with mixed boundary conditions.
Sign-error adaptive filtering algorithms involving Markovian parameters
Araz Hashemi, George Yin and Le Yi Wang
2015, 5(4): 781-806 doi: 10.3934/mcrf.2015.5.781 +[Abstract](633) +[PDF](655.2KB)
Abstract:
Motivated by reduction of computational complexity, this work develops sign-error adaptive filtering algorithms for estimating randomly time-varying system parameters. Different from the existing work on sign-error algorithms, the parameters are time-varying and their dynamics are modeled by a discrete-time Markov chain. Another distinctive feature of the algorithms is the multi-time-scale framework for characterizing parameter variations and algorithm updating speeds. This is realized by considering the stepsize of the estimation algorithms and a scaling parameter that defines the transition rate of the Markov jump process. Depending on the relative time scales of these two processes, suitably scaled sequences of the estimates are shown to converge to either an ordinary differential equation, or a set of ordinary differential equations modulated by random switching, or a stochastic differential equation, or stochastic differential equations with random switching. Using weak convergence methods, convergence and rates of convergence of the algorithms are obtained for all these cases. Simulation results are provided for demonstration.
Generalization on optimal multiple stopping with application to swing options with random exercise rights number
Noureddine Jilani Ben Naouara and Faouzi Trabelsi
2015, 5(4): 807-826 doi: 10.3934/mcrf.2015.5.807 +[Abstract](688) +[PDF](451.2KB)
Abstract:
This paper develops the theory of optimal multiple stopping times expected value problems by stating, proving, and applying a dynamic programming principle for the case in which both the reward process and the number of stopping times are stochastic. This case comes up in practice when valuing swing options, which are somewhat common in commodity trading. We believe our results significantly advance the study of option pricing.
Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation
Ruoxia Li, Huaiqin Wu, Xiaowei Zhang and Rong Yao
2015, 5(4): 827-844 doi: 10.3934/mcrf.2015.5.827 +[Abstract](885) +[PDF](983.4KB)
Abstract:
This paper is concerned with the projective synchronization issue for memristive neural networks with time-varying delays and stochastic perturbations. Based on LaSalle-type invariance principle of stochastic functional-differential equations, by applying Lyapunov functional approach, several sufficient conditions are developed to achieve the projective synchronization between the master-slave systems with time-varying delays under stochastic perturbation and adaptive controller. A numerical example and its simulation is given to show the effectiveness of the theoretical results in this paper.
Relative controllability of linear systems of fractional order with delay
Therese Mur and Hernan R. Henriquez
2015, 5(4): 845-858 doi: 10.3934/mcrf.2015.5.845 +[Abstract](666) +[PDF](375.5KB)
Abstract:
In this paper we are concerned with the controllability of control systems governed by a fractional differential equations with delay. Using the Mittag-Leffler function we define the concept of solution, and applying the properties of the Laplace transform we characterize the relative or pointwise controllability of the system. Our results generalize those of Kirillova and Churakova, which were established for first order systems. Finally, we show that functionally controllable fractional systems are rare.
Stochastic recursive optimal control problem with time delay and applications
Jingtao Shi, Juanjuan Xu and Huanshui Zhang
2015, 5(4): 859-888 doi: 10.3934/mcrf.2015.5.859 +[Abstract](1175) +[PDF](507.6KB)
Abstract:
This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.

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