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Mathematical Control & Related Fields

2013 , Volume 3 , Issue 2

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Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws
Mapundi K. Banda and Michael Herty
2013, 3(2): 121-142 doi: 10.3934/mcrf.2013.3.121 +[Abstract](191) +[PDF](654.4KB)
Suitable numerical discretizations for boundary control problems of systems of nonlinear hyperbolic partial differential equations are presented. Using a discrete Lyapunov function, exponential decay of the discrete solutions of a system of hyperbolic equations for a family of first--order finite volume schemes is proved. The decay rates are explicitly stated. The theoretical results are accompanied by computational results.
Stability of the determination of a time-dependent coefficient in parabolic equations
Mourad Choulli and Yavar Kian
2013, 3(2): 143-160 doi: 10.3934/mcrf.2013.3.143 +[Abstract](75) +[PDF](431.5KB)
We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.
Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control
Larissa V. Fardigola
2013, 3(2): 161-183 doi: 10.3934/mcrf.2013.3.161 +[Abstract](62) +[PDF](520.2KB)
In this paper necessary and sufficient conditions of approximate $L^\infty$-controllability at a free time are obtained for the control system $ w_{tt}=w_{xx}-q^2w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, where $q>0$, $T>0$, $u\in L^\infty(0,T)$ is a control. This system is considered in the Sobolev spaces.
Time optimal control for a nonholonomic system with state constraint
Jérome Lohéac and Jean-François Scheid
2013, 3(2): 185-208 doi: 10.3934/mcrf.2013.3.185 +[Abstract](102) +[PDF](360.6KB)
The aim of this paper is to tackle the time optimal controllability of an $(n+1)$-dimensional nonholonomic integrator. In the optimal control problem we consider, the state variables are subject to a bound constraint. We give a full description of the optimal control and optimal trajectories are explicitly obtained. The optimal trajectories we construct, lie in a 2-dimensional plane and they are composed of arcs of circle.
Stock trading rules under a switchable market
Duy Nguyen, Jingzhi Tie and Qing Zhang
2013, 3(2): 209-231 doi: 10.3934/mcrf.2013.3.209 +[Abstract](72) +[PDF](557.0KB)
This work provides an optimal trading rule that allows buying and selling of an asset sequentially over time. The asset price follows a regime switching model involving a geometric Brownian motion and a mean reversion model. The objective is to determine a sequence of trading times to maximize an overall return. The corresponding value functions are characterized by a set of quasi variational inequalities. Closed-form solutions are obtained. The sequence of trading times can be given in terms of various threshold levels. Numerical examples are given to demonstrate the results.
Constrained BSDEs, viscosity solutions of variational inequalities and their applications
Shige Peng and Mingyu Xu
2013, 3(2): 233-244 doi: 10.3934/mcrf.2013.3.233 +[Abstract](148) +[PDF](358.8KB)
In this paper, we study the relation between the smallest $g$-supersolution of constrained backward stochastic differential equation and viscosity solution of constraint semilinear parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constrained BSDE, and prove a uniqueness result with a condition on the constraint. Then we use these results to give a probabilistic interpretation result for reflected BSDE with a discontinuous barrier and other kind of reflected BSDE.

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