# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2011 , Volume 16 , Issue 4

A special issue
Dedicated to Kok Lay Teo and Jie Sun on the Occasion of their 65th Birthdays

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2011, 16(4): i-i doi: 10.3934/dcdsb.2011.16.4i +[Abstract](52) +[PDF](75.9KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems Series B is dedicated to Professor Kok Lay Teo and Professor Jie Sun for their fundamental contributions to optimization and optimal control and their computational methods and applications. The 2010 International Conference on Optimization and Control (ICOCO2010) was held at Guizhou Park Hotel in Guiyang, China on July 18-23, 2010, in honour of Professors Teo and Sun on their 65th birthdays.

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2011, 16(4): 1039-1053 doi: 10.3934/dcdsb.2011.16.1039 +[Abstract](77) +[PDF](375.9KB)
Abstract:
We study optimal investment problem for a continuous time stochastic market model. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, their distributions are unknown, and they are supposed to be currently observable. To cover fixed income management problems, we assume that the number of risky assets can be larger than the number of driving Brownian motion. The optimal investment problem is stated as a problem with a maximin performance criterion to ensure that a strategy is found such that the minimum of expected utility over all possible parameters is maximal. We show that Mutual Fund Theorem holds for this setting. We found also that a saddle point exists and can be found via minimization over a single scalar parameter.
2011, 16(4): 1055-1069 doi: 10.3934/dcdsb.2011.16.1055 +[Abstract](91) +[PDF](457.3KB)
Abstract:
In handling textile materials, deformation is very common and is unavoidable. When the fabrics are dispatched for further feature extractions, it's necessary to recover the original shape for comparison with a standard template. This recovery problem is investigated in this paper. By introducing a set of recovered functions, the problem is formulated as a combined optimal control and optimal parameter selection problem, governed by the dynamical system of a set of two-dimensional control functions. After parameterization of the control functions, the problem is transformed into a nonlinear optimization problem, where gradient based optimization methods can be applied. We also analyze the convergence of the parameterization method. Several numerical examples are used to demonstrate the method.
2011, 16(4): 1071-1082 doi: 10.3934/dcdsb.2011.16.1071 +[Abstract](98) +[PDF](680.0KB)
Abstract:
In this paper, we study the complete synchronization of a class of time-varying delayed coupled chaotic systems using feedback control. In terms of Linear Matrix Inequalities, a sufficient condition is obtained through using a Lyapunov-Krasovskii functional and differential equation inequalities. The conditions can be easily verified and implemented. We present two simulation examples to illustrate the effectiveness of the proposed method.
2011, 16(4): 1083-1099 doi: 10.3934/dcdsb.2011.16.1083 +[Abstract](64) +[PDF](488.6KB)
Abstract:
A new joint random access scheme is proposed to enable effective uplink access when users are partially aware of channel conditions in multichannel wireless systems. The proposed scheme mitigates packet collisions with joint backoff control in the time and frequency domains, cooperating with a sensing method that exploits the channel conditions. The performance analysis of the joint access scheme is facilitated by a Markov model that provides a closed-form throughput expression. Simulation results show that this channel access scheme, working together with a simple sensing method, offers significant improvements to system throughput.
2011, 16(4): 1101-1117 doi: 10.3934/dcdsb.2011.16.1101 +[Abstract](190) +[PDF](378.5KB)
Abstract:
In this paper, we consider a class of optimal PID control problems subject to continuous inequality constraints and terminal equality constraint. By applying the constraint transcription method and a local smoothing technique to these continuous inequality constraint functions, we construct the corresponding smooth approximate functions. We use the concept of the penalty function to append these smooth approximate functions to the cost function, forming a new cost function. Then, the constrained optimal PID control problem is approximated by a sequence of optimal parameter selection problems subject to only terminal equality constraint. Each of these optimal parameter selection problems can be viewed and hence solved as a nonlinear optimization problem. The gradient formulas of the new appended cost function and the terminal equality constraint function are derived, and a reliable computation algorithm is given. The method proposed is used to solve a ship steering control problem.
2011, 16(4): 1119-1136 doi: 10.3934/dcdsb.2011.16.1119 +[Abstract](81) +[PDF](532.9KB)
Abstract:
In this paper we present a state observer approach for the estimation of effective diffusion coefficients of a drug delivery device. In this approach, we construct estimators for the unknown effective diffusion coefficients characterizing the diffusion process of a drug release device using a combination of state observers from the area of adaptive control and the drug diffusion models developed recently by us. We show that the constructed systems are asymptotically stable and the estimators converge to the exact diffusion coefficients. An algorithm is proposed to recursively compute the estimators using a given time series of a release profile of a device. To demonstrate the efficiency and usefulness of this approach, numerical experiments have been performed using experimentally observed drug release profiles of polymeric spherical devices. The numerical results show that the present approach is about 9 times faster than the conventional least squares method when applied to the test problems.
2011, 16(4): 1137-1155 doi: 10.3934/dcdsb.2011.16.1137 +[Abstract](92) +[PDF](403.0KB)
Abstract:
This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear impulsive differential equation on time scale. The reasonable weak solution of nonlinear impulsive differential equation on time scale is introduced and the existence and uniqueness of the weak solution and its properties are presented. By $L^{1}-$strong$-$weak lower semicontinuity of integral functional on time scale, we give the existence of optimal controls. Using integration by parts formula on time scale, the necessary conditions of optimality are derived. An example on mathematical programming is also presented for demonstration.
2011, 16(4): 1157-1169 doi: 10.3934/dcdsb.2011.16.1157 +[Abstract](68) +[PDF](315.6KB)
Abstract:
In this paper, a new spectral PRP conjugate gradient algorithm is developed for solving nonconvex unconstrained optimization problems. The search direction in this algorithm is proved to be a sufficient descent direction of the objective function independent of line search. To rule out possible unacceptably short step in the Armijo line search, a modified Armijo line search strategy is presented. The obtained step length is improved by employing the properties of the approximate Wolfe conditions. Under some suitable assumptions, the global convergence of the developed algorithm is established. Numerical experiments demonstrate that this algorithm is promising.
2011, 16(4): 1171-1183 doi: 10.3934/dcdsb.2011.16.1171 +[Abstract](76) +[PDF](1588.6KB)
Abstract:
This paper presents a mathematical model and numerical technique for simulating the two-fluid flow and the meniscus interface movement in the electromagnetic continuous steel casting process. The governing equations include the continuity equation, the momentum equations, the energy equation, the level set equation and two transport equations for the electromagnetic field derived from the Maxwell's equations. The level set finite element method is applied to trace the movement of the interface between different fluids. In an attempt to optimize the casting process, the technique is then applied to study the influences of the imposed electromagnetic field and the mould oscillation pattern on the fluid flow, the meniscus shape and temperature distribution.
2011, 16(4): 1185-1195 doi: 10.3934/dcdsb.2011.16.1185 +[Abstract](73) +[PDF](348.4KB)
Abstract:
This paper addresses fundamental stability problems of impulsive switched linear systems, featuring given impulsive switching time intervals and switching rules. First, based on the state dynamical behaviors, we construct a new state transition-like matrix, called an impulsive-type state transition (IST) matrix. Then, based on the IST matrix and Lyapunov stability theory, necessary and sufficient conditions for the uniform stability, uniform asymptotic stability, and exponential stability of impulsive switched linear systems are established. These stability conditions require the testing on the IST matrix of the impulsive switched linear systems. The results can be reduced to those for switched linear systems without impulsive effects, and also to those for impulsive linear systems without switchings.
2011, 16(4): 1197-1211 doi: 10.3934/dcdsb.2011.16.1197 +[Abstract](74) +[PDF](422.7KB)
Abstract:
In this paper the stabilization problem for a class of discrete-time Markovian jump system with partially unknown transition probabilities is investigated via using the time-delayed and impulsive controllers. As some elements in transition matrix are unknown, a new approach is proposed to estimate the unknown elements, in which an impulsive stabilizing controller depending on time delays and system mode is presented in terms of linear matrix inequalities (LMIs) with equality constraints. Especially, if there are no time delays and impulsive effects in the controller, it is derived that the conditions for the existence of $H_\infty$ controller can be expressed by LMIs without equality constraints. Finally, illustrative examples are presented to show the benefits and the validity of the proposed approaches.

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