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Mathematical Control & Related Fields

2012 , Volume 2 , Issue 1

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Exponential stability of a general network of 1-d thermoelastic rods
Abdallah Ben Abdallah and Farhat Shel
2012, 2(1): 1-16 doi: 10.3934/mcrf.2012.2.1 +[Abstract](186) +[PDF](401.1KB)
Abstract:
We consider a finite planar network of 1-$d$ thermoelastic rods using Fourier's law or Cattaneo's law for heat conduction, we show that the system is exponentially stable in the two cases.
Eventual regularity of a wave equation with boundary dissipation
Kangsheng Liu, Xu Liu and Bopeng Rao
2012, 2(1): 17-28 doi: 10.3934/mcrf.2012.2.17 +[Abstract](224) +[PDF](346.0KB)
Abstract:
This paper addresses a study of the eventual regularity of a wave equation with boundary dissipation and distributed damping. The equation under consideration is rewritten as a system of first order and analyzed by semigroup methods. By a certain asymptotic expansion theorem, we prove that the associated solution semigroup is eventually differentiable. This implies the eventual regularity of the solution of the wave equation.
Extension of the $\nu$-metric for stabilizable plants over $H^\infty$
Amol Sasane
2012, 2(1): 29-44 doi: 10.3934/mcrf.2012.2.29 +[Abstract](222) +[PDF](412.4KB)
Abstract:
An abstract $\nu$-metric was introduced in [1], with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. Here we give an important concrete special instance of the abstract $\nu$-metric, namely the case when the ring of stable transfer functions is the Hardy algebra $H^\infty$, by verifying that all the assumptions demanded in the abstract set-up are satisfied. This settles the open question implicit in [2].
Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms
Louis Tebou
2012, 2(1): 45-60 doi: 10.3934/mcrf.2012.2.45 +[Abstract](251) +[PDF](140.1KB)
Abstract:
We consider the Euler-Bernoulli equation coupled with a wave equation in a bounded domain. The Euler-Bernoulli has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is distributed everywhere in the domain under consideration acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the Euler-Bernoulli equation. We show that in this case the coupled system is not exponentially stable. Next, using a frequency domain approach combined with the multiplier techniques, and a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this new system is not exponentially stable, and we provide precise polynomial decay estimates for its energy. The results obtained complement those existing in the literature involving the hinged Euler-Bernoulli equation.
Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints
Huaiqiang Yu and Bin Liu
2012, 2(1): 61-80 doi: 10.3934/mcrf.2012.2.61 +[Abstract](289) +[PDF](464.9KB)
Abstract:
This paper deals with the Pontryagin's principle of optimal control problems governed by the 2D Navier-Stokes equations with integral state constraints and coupled integral control--state constraints. As an application, the necessary conditions for the local solution in the sense of $L^r(0,T;L^2(\Omega))$ ($2 < r < \infty$) are also obtained.
Optimal trend-following trading rules under a three-state regime switching model
Jie Yu and Qing Zhang
2012, 2(1): 81-100 doi: 10.3934/mcrf.2012.2.81 +[Abstract](449) +[PDF](697.0KB)
Abstract:
Momentum (or trend-following) trading strategies are widely used in the investment world. To better understand the nature of trend-following trading strategies and discover the corresponding optimality conditions, we consider the cases when the market trends are fully observable. In this paper, the market follows a regime switching model with three states (bull, sideways, and bear). Under this model, a set of sufficient conditions are developed to guarantee the optimality of trend-following trading strategies. A dynamic programming approach is used to verify these optimality conditions. The value functions are characterized by the associated HJB equations and are shown to be either linear functions or infinity depending on the parameter values. The results in this paper will help an investor to identify market conditions and to avoid trades which might be unprofitable even under the best market information. Finally, the corresponding value functions will provide an upper bound for trading performance which can be used as a general guide to rule out unrealistic expectations.

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