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Volume 1, 2011

Mathematical Control & Related Fields

2011 , Volume 1 , Issue 1

A special issue
Dedicated to Professor Masao Fukushima on the occasion of his 60th birthday

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On the fast solution of evolution equations with a rapidly decaying source term
Alain Haraux
2011, 1(1): 1-20 doi: 10.3934/mcrf.2011.1.1 +[Abstract](617) +[PDF](161.3KB)
Abstract:
If $L$ is the generator of a uniformly bounded group of operators $T(t)$ on a Banach space $X$, the abstract evolution equation $ u' + Lu(t) = h(t) $ has a (weak) solution tending to $0$ as $t\rightarrow +\infty $ if, and only if $\int_0^{+\infty}T(s) h(s) ds $ is semi-convergent, and then this solution is unique. For the semi-linear equation $ u' + Lu(t) + f(u) = h(t) $, if $f$ such that $f(0) = 0$ is Lipschitz continuous on bounded subsets of $X$ and has a Lipschitz constant bounded by $ Cr^\alpha $ in the ball $B(0, r)$ for $r\leq r_0$, for any $h$ satisfiying

$||h(t)|| \leq c(1+t)^{-(1+ \lambda )} $

with $\lambda >\frac{1}{\alpha}$ and $c$ small enough there exists a unique solution tending to $0$ at least like $(1+t)^{- \lambda}.$ When the system is dissipative, this special solution makes it sometimes possible to estimate from below the rate of decay to $0$ of the other solutions.

Numerical methods for dividend optimization using regime-switching jump-diffusion models
Zhuo Jin, George Yin and Hailiang Yang
2011, 1(1): 21-40 doi: 10.3934/mcrf.2011.1.21 +[Abstract](700) +[PDF](397.8KB)
Abstract:
This work develops numerical methods for finding optimal dividend policies to maximize the expected present value of dividend payout, where the surplus follows a regime-switching jump diffusion model and the switching is represented by a continuous-time Markov chain. To approximate the optimal dividend policies or optimal controls, we use Markov chain approximation techniques to construct a discrete-time controlled Markov chain with two components. Under simple conditions, we prove the convergence of the approximation sequence to the surplus process and the convergence of the approximation to the value function. Several examples are provided to demonstrate the performance of the algorithms.
Cesari-type conditions for semilinear elliptic equation with leading term containing controls
Bo Li and Hongwei Lou
2011, 1(1): 41-59 doi: 10.3934/mcrf.2011.1.41 +[Abstract](552) +[PDF](231.0KB)
Abstract:
An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.
Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain
Ivonne Rivas, Muhammad Usman and Bing-Yu Zhang
2011, 1(1): 61-81 doi: 10.3934/mcrf.2011.1.61 +[Abstract](951) +[PDF](455.9KB)
Abstract:
In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg-de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2$- a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially.
A deterministic linear quadratic time-inconsistent optimal control problem
Jiongmin Yong
2011, 1(1): 83-118 doi: 10.3934/mcrf.2011.1.83 +[Abstract](1031) +[PDF](442.3KB)
Abstract:
A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with a quadratic cost functional. A notion of time-consistent equilibrium strategy is introduced for the original time-inconsistent problem. Under certain conditions, we construct an equilibrium strategy which can be represented via a Riccati--Volterra integral equation system. Our approach is based on a study of multi-person hierarchical differential games.
Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential
Jian Zhang, Shihui Zhu and Xiaoguang Li
2011, 1(1): 119-127 doi: 10.3934/mcrf.2011.1.119 +[Abstract](806) +[PDF](364.9KB)
Abstract:
We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Tsutsumi's arguments as well as Carles' transform, the $L^2$-concentration property of radially symmetric blow-up solutions is obtained.

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