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Mathematical Control & Related Fields

March 2017 , Volume 7 , Issue 1

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Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value
Gongpin Cheng and Lin Xu
2017, 7(1): 1-19 doi: 10.3934/mcrf.2017001 +[Abstract](208) +[HTML](4) +[PDF](479.6KB)
Abstract:

This paper investigates the optimal control problem with a nonlinear capital process attributed to internal competition factors. Suppose that the company can control its capital process by paying dividend, refinancing and changing the size of business. The transaction costs generated by the control processes as well as the liquidation value at ruin are considered. We aim at seeking the optimal control strategies for maximizing the company's value. The results show that the company should expand the business scale when the current capital increases. The refinancing may only happen at the moments when, and only when, the capital is null. The dividends should be paid out according to barrier strategy if the dividend rate is unconstrained or threshold strategy if the dividend rate is bounded, respectively.

Construction of gevrey functions with compact support using the bray-mandelbrojt iterative process and applications to the moment method in control theory
Pierre Lissy
2017, 7(1): 21-40 doi: 10.3934/mcrf.2017002 +[Abstract](168) +[HTML](3) +[PDF](512.4KB)
Abstract:

In this paper, we construct some interesting Gevrey functions of order $α$ for every $α>1$ with compact support by a clever use of the Bray-Mandelbrojt iterative process. We then apply these results to the moment method, which will enable us to derive some upper bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type that partially improve the existing results proved in [P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension SIAM J. Control Optim., 52(4), 2651-2676]. However this construction fails to improve the results of [G. Tenenbaum and M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100] in the precise case of the usual heat and Schrödinger equation.

A discrete hierarchy of double bracket equations and a class of negative power series
Nancy López Reyes and Luis E. Benítez Babilonia
2017, 7(1): 41-52 doi: 10.3934/mcrf.2017003 +[Abstract](268) +[HTML](1) +[PDF](388.0KB)
Abstract:

The space of negative power series of \begin{document} $z$ \end{document} on \begin{document} $\{z\in \mathbb{C}:|z|>1\}$ \end{document} can also be parametrized by means of a system of double bracket differential equations. To show this parametrization we introduce a group factorization for equation system. This work, for the case of a double bracket system, is a continuation of an earlier study discussed in The discrete KP hierarchy and the negative power series on the complex plane. Comp. and App. Math. 32 (2013), 483-493 for the case of one bracket system.

Control and stabilization of 2 × 2 hyperbolic systems on graphs
Serge Nicaise
2017, 7(1): 53-72 doi: 10.3934/mcrf.2017004 +[Abstract](174) +[HTML](3) +[PDF](502.8KB)
Abstract:

We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

Decompositions and bang-bang properties
Gengsheng Wang and Yubiao Zhang
2017, 7(1): 73-170 doi: 10.3934/mcrf.2017005 +[Abstract](160) +[HTML](7) +[PDF](1316.9KB)
Abstract:

We study the bang-bang properties of minimal time and minimal norm control problems (where the target set is the origin of the state space and the controlled system is linear and time-invariant) from a new perspective. More precisely, we study how the bang-bang property of each minimal time (or minimal norm) problem depends on a pair of parameters \begin{document} $(M, y_0)$ \end{document} (or \begin{document} $(T,y_0)$ \end{document}), where \begin{document} $M>0$ \end{document} is a bound of controls and \begin{document} $y_0$ \end{document} is the initial state (or \begin{document} $T>0$ \end{document} is an ending time and \begin{document} $y_0$ \end{document} is the initial state). The controlled system may have neither the \begin{document} $L^∞$ \end{document}-null controllability nor the backward uniqueness property.

2017  Impact Factor: 0.542

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