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1531-3492

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1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2010 , Volume 14 , Issue 4

Special Issue

Dedicated to David L. Russell on the occasion of his 70th birthday

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2010, 14(4): i-i
doi: 10.3934/dcdsb.2010.14.4i

*+*[Abstract](365)*+*[PDF](23.8KB)**Abstract:**

Professor David L. Russell just celebrated his 70th birthday in 2009. His numerous important and fundamental contributions to the control theory of partial differential equations have won him recognitions as a true pioneer and giant in this field. And his pleasant personality and ready helpfulness have won our hearts as his admirers, students, and friends.

Prof. Russell's lifetime love and interests are control theory and partial differential equations. We have asked Goong Chen, a former PhD student of Prof. Russell, to write a survey of his career and mathematical work. Moreover, as a way to pay our tribute to him, it is only proper here that we put together a collection of papers by researchers who work on these and closely related subjects of Prof. Russell's. Indeed, the topics collected are extensive, including the theory and applications of analysis, computation, control, optimization, etc., for various systems arising from conservation laws and quasilinear hyperbolic systems, fluids, mathematical finance, neural networks, reaction-diffusion, structural dynamics, stochastic PDEs, thermoelasticity, viscoelasticity, etc. They are covered in great depth by the authors. There are a total of 35 papers finally accepted for publication after going through strict manuscript reviews and revision. Due to the large number of papers on hand, we divide them into Groups I (20 papers) and II (15 papers). Papers in Group I are published in Discrete and Continuous Dynamical Systems-Series B (DCDS-B) Volume 14, No. 4 (2010), while those in Group II are published in Journal of Systems Science and Complexity (JSSC) Volume 23, No. 3 (2010). This division is up to the preference of the authors. We thank all the authors and reviewers for their critical contributions and valuable assistance that make the publication of these two journal issues possible.

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2010, 14(4): 1265-1277
doi: 10.3934/dcdsb.2010.14.1265

*+*[Abstract](379)*+*[PDF](693.7KB)**Abstract:**

N/A

2010, 14(4): 1279-1292
doi: 10.3934/dcdsb.2010.14.1279

*+*[Abstract](494)*+*[PDF](217.2KB)**Abstract:**

The classical Mead-Markus sandwich beam consists of two stiff outer layers modeled under Euler-Bernoulli beam assumptions and a compliant "core layer" that is elastic in shear. In this article we consider a multilayer analog consisting of $n = 2m + 1$ layers of alternating stiff and compliant beam layers ($m+1$ stiff and $m$ compliant) with viscous damping proportional to the shear in the compliant layers. We prove that the associated semigroup is analytic and describe the sector of analyticity. We also consider the problem of how to choose the damping parameters to optimize the angle of analyticity. We obtain an analytical solution to the optimization problem.

2010, 14(4): 1293-1311
doi: 10.3934/dcdsb.2010.14.1293

*+*[Abstract](400)*+*[PDF](250.4KB)**Abstract:**

We study the global approximate controllability properties of a one dimensional reaction-diffusion equation governed via the coefficient of the reaction term. The traditional (linear operator) controllability methods based on the duality pairing do not apply to such a problem. Instead, we focus on the qualitative study of the diffusion and reaction parts of the evolution process at hand. We consider the case when both the initial and target states admit no more than finitely many changes of sign.

2010, 14(4): 1313-1335
doi: 10.3934/dcdsb.2010.14.1313

*+*[Abstract](457)*+*[PDF](1463.3KB)**Abstract:**

The paper studies forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small negative or oscillatory bump on a rigid flat bottom. Such wave motions are determined by a non-dimensional wave speed $F$, called Froude number, and $F=1$ is a critical value of $F$. If $F= 1+ \lambda \epsilon $ with a small parameter $\epsilon > 0$, then a forced Korteweg-de Vries (FKdV) equation can be derived to model the wave motion on the free surface. In this paper, the case $\lambda > 0$ (or $F> 1$, called supercritical case) is considered. The steady and unsteady solutions of the FKdV equation with a negative bump function independent of time are first studied both theoretically and numerically. It is shown that there are five steady solutions and only one of them, which exists for all $\lambda > 0$, is stable. Then, solutions of the FKdV equation with an oscillatory bump function posed on $R$ or a finite interval are considered. The corresponding linear problems are solved explicitly and the solutions are rigorously shown to be eventually periodic as time goes to infinity, while a similar result holds for the nonlinear problem posed on a finite interval with small initial data and forcing functions. The nonlinear solutions with zero initial data for any forcing functions in the real line $R$ or large forcing functions in a finite interval are obtained numerically. It is shown numerically that the solutions will become eventually periodic in time for a small forcing function. The behavior of the solutions becomes quite irregular as time goes to infinity, if the forcing function is large.

2010, 14(4): 1337-1359
doi: 10.3934/dcdsb.2010.14.1337

*+*[Abstract](715)*+*[PDF](322.3KB)**Abstract:**

This article studies a hyperbolic conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. Characteristic features are the nonlocal character of the velocity and that the influx and outflux constitute the control and output signal, respectively. We prove the existence and uniqueness of solutions for $L^1$-data, and study their regularity properties. We also prove the existence of optimal controls that minimizes in the $L^2$-sense the mismatch between the actual and a desired output signal. Finally, the time-optimal control for a step between equilibrium states is identified and proven to be optimal.

2010, 14(4): 1361-1373
doi: 10.3934/dcdsb.2010.14.1361

*+*[Abstract](525)*+*[PDF](190.2KB)**Abstract:**

Some stochastic optimal control problems in a Hilbert space are formulated and solved. The controlled equations are abstract equations in a HIlbert space that can model stochastic partial differential equations and stochastic delay equations. Both linear and semilinear equations are considered where the cylindrical Brownian motion can occur as distributed, boundary, or at discrete points in the domain. For the linear equations, the cost is an ergodic, quadratic functional of the state and the control. An optimal linear feedback control is given explicitly. For the semilinear equations, the cost is an ergodic functional. Some results for the null controllability of a stochastic parabolic equation are given. A control problem for a finite dimension linear stochastic system with an arbitrary fractional Brownian motion and a quadratic cost functional is formulated and explicitly solved.

2010, 14(4): 1375-1401
doi: 10.3934/dcdsb.2010.14.1375

*+*[Abstract](747)*+*[PDF](347.2KB)**Abstract:**

We prove a regularity result for an abstract control problem $z' =A z + Bv$ with initial datum $z(0) = z_0$ in which the goal is to determine a control $v$ such that $z(T)=0$. Under standard admissibility and observability assumptions on the adjoint system, when $A$ generates a $C^0$ group, we develop a method to compute algorithmically a control function $v$ that inherits the regularity of the initial datum to be controlled. In particular, the controlled equation is satisfied in a strong sense when the initial datum is smooth. In this way, the controlled trajectory is smooth as well. Our method applies mainly to time-reversible infinite-dimensional systems and, in particular, to the wave equation, but fails to be valid in the parabolic frame.

2010, 14(4): 1403-1417
doi: 10.3934/dcdsb.2010.14.1403

*+*[Abstract](414)*+*[PDF](183.8KB)**Abstract:**

This work provides an optimal trading rule that allows buying, selling and short selling of an asset when its price is governed by mean-reverting model. The goal is to find the buy and sell prices such that the overall return (with slippage cost imposed) is maximized. The associated HJB equations (variational inequalities) are used to characterize the value functions. This paper shows that the solution of the original optimal stopping problem can be achieved by solving four algebraic equations. Numerical examples are given for demonstration.

2010, 14(4): 1419-1432
doi: 10.3934/dcdsb.2010.14.1419

*+*[Abstract](398)*+*[PDF](185.0KB)**Abstract:**

Based on the local exact boundary controllability for 1-D first order quasilinear hyperbolic systems, by an extension method the author gets the global exact boundary controllability for 1-D first order quasilinear hyperbolic systems of diagonal form with applications to Saint-Venant system with friction.

2010, 14(4): 1433-1444
doi: 10.3934/dcdsb.2010.14.1433

*+*[Abstract](552)*+*[PDF](186.3KB)**Abstract:**

In this paper, we study the energy decay rate for a mixed type II and type III thermoelastic system. The system consists of a wave equation and a heat equation of type III in one part of the domain; a wave equation and a heat equation of type II in another part of the domain, coupled in certain pattern. When the damping coefficient function satisfies certain conditions at the interface, a polynomial type decay rate is obtained. This result is proved by verifying the frequency domain conditions.

2010, 14(4): 1445-1464
doi: 10.3934/dcdsb.2010.14.1445

*+*[Abstract](514)*+*[PDF](215.7KB)**Abstract:**

Second-order necessary conditions for optimal control problems are considered, where the "second-order" is in the sense of that Pontryagin's maximum principle is viewed as a first-order necessary optimality condition. A sufficient condition for a local minimizer is also given.

2010, 14(4): 1465-1485
doi: 10.3934/dcdsb.2010.14.1465

*+*[Abstract](548)*+*[PDF](359.7KB)**Abstract:**

This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces final-observability of an evolution semigroup from a spectral inequality, i.e. some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. words Contrary to the original Lebeau-Robbiano strategy, it does not have recourse to null-controllability and it yields the optimal bound of the cost when applied to the heat equation, i.e. $c_0\exp(c/T)$, or to the heat diffusion in potential wells observed from cones, i.e. $c_0\exp(c/T^\beta)$ with optimal $\beta$. It also yields simple upper bounds for the cost rate $c$ in terms of the spectral rate.

This paper also gives geometric lower bounds on the spectral and cost rates for heat, diffusion and Ginzburg-Landau semigroups, including on non-compact Riemannian manifolds, based on $L^2$ Gaussian estimates.

2010, 14(4): 1487-1510
doi: 10.3934/dcdsb.2010.14.1487

*+*[Abstract](464)*+*[PDF](274.3KB)**Abstract:**

In this paper we study the equation of linear viscoelasticity and we prove that two sequences of functions, naturally associated with this equation, are Riesz systems. These sequences appear naturally when observability and controllability problems are reformulated in terms of suitable interpolation/moment problems.

The key contribution of the paper is to be found in the way used to prove that the two sequences are Riesz systems, an idea already applied to the study of different control problems.

2010, 14(4): 1511-1535
doi: 10.3934/dcdsb.2010.14.1511

*+*[Abstract](441)*+*[PDF](286.4KB)**Abstract:**

Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in [19] that the damping is active on a set $(a_0,+\infty)$ with $a_0>0$, we establish the exponential decay of the solutions in the weighted spaces $L^2((x+1)^mdx)$ for $m\in $N

^{*}and $L^2(e^{2bx}dx)$ for $b>0$ by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived.

2010, 14(4): 1537-1564
doi: 10.3934/dcdsb.2010.14.1537

*+*[Abstract](466)*+*[PDF](344.8KB)**Abstract:**

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with a nonhomogeneous divergence condition. We also prove the existence of global weak solutions to the 3D Navier-Stokes equations when the divergence is not equal to zero. These equations intervene in control problems for the Navier-Stokes equations and in fluid-structure interaction problems.

2010, 14(4): 1565-1579
doi: 10.3934/dcdsb.2010.14.1565

*+*[Abstract](540)*+*[PDF](184.3KB)**Abstract:**

A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with continuous coefficients. A Kneser-type theorem for BDSDEs is obtained. We show that there is either unique or uncountable solutions for this kind of BDSDEs.

2010, 14(4): 1581-1599
doi: 10.3934/dcdsb.2010.14.1581

*+*[Abstract](676)*+*[PDF](210.7KB)**Abstract:**

A singular optimal stochastic control problem is studied. A second-order maximum principle is presented. The second-order adjoint processes are involved, though the diffusion of the control system is control independent. The range theorem of vector-valued measures is used to prove the maximum principle. Examples are given to illustrate the applications.

2010, 14(4): 1601-1620
doi: 10.3934/dcdsb.2010.14.1601

*+*[Abstract](522)*+*[PDF](269.5KB)**Abstract:**

First, we consider a coupled system consisting of the wave equation and the heat equation in a bounded domain. The coupling involves an operator parametrized by a real number $\mu$ lying in the interval [0,1]. We show that for $0\leq\mu<1$, the associated semigroup is not uniformly stable. Then we propose an explicit non-uniform decay rate. For $\mu=1$, the coupled system reduces to the thermoelasticity equations discussed by Lebeau and Zuazua [23], and subsequently by Albano and Tataru [1]; we show that in this case, the corresponding semigroup is exponentially stable but not analytic. Afterwards, we discuss some extensions of our results. Second, we consider partially clamped Kirchhoff thermoelastic plate without mechanical feedback controls, and we prove that the underlying semigroup is exponentially stable uniformly with respect to the rotational inertia. We use a constructive frequency domain method to prove the stabilization result, and we obtain an explicit decay rate by showing that the real part of the spectrum is uniformly bounded by a negative number that depends on the parameters of the system other than the rotational inertia; our approach is an alternative to the energy method applied by Avalos and Lasiecka [6].

2010, 14(4): 1621-1639
doi: 10.3934/dcdsb.2010.14.1621

*+*[Abstract](523)*+*[PDF](232.9KB)**Abstract:**

This paper concerns an optimal control problem governed by a linear evolution system with a small perturbation in the system conductivity. The system without any perturbation is assumed to have such a periodic property that it holds a periodic solution. In general, the perturbed system dose not enjoy this periodic property again, even though the perturbation has a small norm. The goal of this research is to restore the periodic property for the system, with a small perturbation, through utilizing such a control that is optimal in certain sense. It also aims to study characteristics of such an optimal control. The existence and uniqueness of the optimal control is obtained. Furthermore, a necessary and sufficient condition for the optimal control is established.

2010, 14(4): 1641-1670
doi: 10.3934/dcdsb.2010.14.1641

*+*[Abstract](640)*+*[PDF](321.6KB)**Abstract:**

This work is concerned with the finite element solutions for parameter identifications in second order elliptic and parabolic systems. The $L^2$- and energy-norm error estimates of the finite element solutions are established in terms of the mesh size, time step size, regularization parameter and noise level.

2010, 14(4): 1671-1688
doi: 10.3934/dcdsb.2010.14.1671

*+*[Abstract](434)*+*[PDF](237.3KB)**Abstract:**

The existence of a global attractor for the solution semiflow of a three-component reversible Gray-Scott system with Neumann boundary condition on a bounded domain of space dimension $n\le 3$ is proved. The methodology features the re-scaling and grouping estimation to overcome the difficulty of non-dissipative coupling of three variables and the coefficient barrier. It is also shown that the global attractor turns out to be an $(H, E)$ global attractor.

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