Small cells overlaid with macrocells can increase the capacity of two-tier cellular wireless networks by offloading traffic from macrocells. To motivate the small cell service providers (SSPs) to open portion of the access opportunities to macro users (i.e., to operate in a hybrid access mode), we design an incentive mechanism in which the macrocell service provider (MSP) could pay to the SSPs. According to the price offered by the MSP, the SSPs decide on the open access ratio , which is the ratio of shared radio resource for macro users and the total amount of radio resource in a small cell. The users in this two-tier network can make service selection decisions dynamically according to the performance satisfaction level and cost, which again depend on the pricing and spectrum sharing between the MSP and SSPs. To model this dynamic interactive decision problem, we propose a hierarchical dynamic game framework. In the lower level, we formulate an evolutionary game to model and analyze the adaptive service selection of users. An evolutionary stable strategy (ESS) is considered to be the solution of this game. In the upper level, the MSP and SSPs sequentially determine the pricing strategy and the open access ratio, respectively, taking into account the distribution of dynamic service selection at the lower-level evolutionary game. A Stackelberg differential game is formulated where the MSP and SSPs act as the leader and followers, respectively. An open-loop Stackelberg equilibrium is considered to be the solution of this game. We also extend the hierarchical dynamic game framework and investigate the impact of information delays on the equilibrium solutions. Numerical results show the effectiveness and advantages of dynamic control of the open access ratio and pricing.

Pricing, Spectrum Sharing, and Service Selection in Two-Tier Small Cell Networks: A Hierarchical Dynamic Game Approach

In this paper we study optimal control problems with multiple time delays in control and state and
mixed type control-state constraints. We derive necessary
optimality conditions in the form of a Pontryagin type Minimum Principle.
A discretization method is presented by which the delayed control problem is transformed
into a nonlinear programming problem. It is shown that the associated Lagrange multipliers provide a consistent
numerical approximation for the adjoint variables of the delayed optimal control problem. We illustrate
the theory and numerical approach on an analytical example and an optimal control model from immunology.

https://www.aimsciences.org/article/exportPdf?id=4e5f98f1-774d-4dac-98a2-afedda3ac342

Theory and applications of optimal control problems with multiple time-delays

Device-to-device (D2D) communication technology is a promising add-on component for future wireless networks to provide local area services with increased spectrum efficiency and improved user experience. Three modes (i.e., cellular mode, reuse mode, and dedicated mode) can be used for D2D com- munication. A potential D2D user equipment (UE) can select a communication mode and dynamically adapt the mode selection according to the performance and the cost. This is referred to as the user-controlled mode selection problem. Also, a base station (BS) needs to reserve a spectrum band for the dedicated mode of operation, which we refer to as spectrum partitioning. The optimal spectrum partitioning needs to consider the utility of the BS that depends on the distribution of the users' mode selection, which, in turn, is governed by the spectrum partitioning. To jointly address the problems of spectrum partitioning and user-controlled mode selection (which are cyclically dependent on each other), we propose a dynamic Stackelberg game framework in which the BS and the potential D2D UEs act as the leader and the followers, respectively. Specifically, the adaptive mode selection of potential D2D UEs is formulated as a follower evolutionary game, and an evolutionary stable strategy is considered to be the solution. The dynamic control of spectrum partitioning by the BS is formulated as a leader optimal control problem. We also extend the formulation by considering information delays in control and state. Numerical analysis is performed to evaluate the effectiveness of the proposed framework, which shows that although the mode selection is performed in a distributed and user-controlled manner, the dynamic spectrum partitioning can be viewed as an effective incentive mechanism to drive the user distribution close to the optimal one.

Joint Mode Selection and Spectrum Partitioning for Device-to-Device Communication: A Dynamic

We present a delay-differential equation model with optimal control that describes the interactions between human immunodeficiency virus (HIV), CD4

/pdf/optimal-control-of-a-delayed-hiv-infection-model-with-gdwg4uuynv.pdf

Optimal Control of a Delayed HIV Infection Model with ImmuneResponse Using an Efficient Numerical Method

Device-to-device (D2D) communication technology is a promising add-on component for future wireless networks to provide local area services with increased spectrum efficiency and improved user experience. Three modes (i.e., cellular mode, reuse mode, and dedicated mode) can be used for D2D communication. A potential D2D user equipment (UE) can select a communication mode and dynamically adapt the mode selection according to the performance and the cost. This is referred to as the user-controlled mode selection problem. Also, a base station (BS) needs to reserve a spectrum band for the dedicated mode of operation, which we refer to as spectrum partitioning. The optimal spectrum partitioning needs to consider the utility of the BS that depends on the distribution of the users' mode selection, which, in turn, is governed by the spectrum partitioning. To jointly address the problems of spectrum partitioning and user-controlled mode selection (which are cyclically dependent on each other), we propose a dynamic Stackelberg game framework in which the BS and the potential D2D UEs act as the leader and the followers, respectively. Specifically, the adaptive mode selection of potential D2D UEs is formulated as a follower evolutionary game, and an evolutionary stable strategy is considered to be the solution. The dynamic control of spectrum partitioning by the BS is formulated as a leader optimal control problem. We also extend the formulation by considering information delays in control and state. Numerical analysis is performed to evaluate the effectiveness of the proposed framework, which shows that although the mode selection is performed in a distributed and user-controlled manner, the dynamic spectrum partitioning can be viewed as an effective incentive mechanism to drive the user distribution close to the optimal one.

Joint Mode Selection and Spectrum Partitioning for Device-to-Device Communication: A Dynamic Stackelberg Game

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [ 21 ] describing the tumour-immune response to a chemo-immuno-therapy. Assuming $ L^1$-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [ 7 ]. For $ L^1$-type objectives we obtain extremal controls of bang-bang type.

https://www.aimsciences.org/article/exportPdf?id=c39cb38f-0170-4592-930b-1cd4ed52f6ce

Optimal control problems with time delays: Two case studies in biomedicine

Dieses Kapitel liefert eine Einfuhrung in die mathematische Optimierung. Wir beschreiben, wie sich aus einer praktischen Anwendungsaufgabe zur bestmoglichen Gestaltung eines Prozesses oder Produktes ein Optimierungsproblem entwickeln lasst. Dabei gehen wir insbesondere auf die Unterscheidung zwischen einer mathematischen Optimierungsaufgabe, bei der das Ziel, die variablen Grosen und auftretende Restriktionen quantifizierbar und klar definiert sind, und dem im Alltag haufig eher als kontinuierlichen Verbesserungsprozess verstandenen Begriff der Optimierung ein.Nach einer ausfuhrlichen Vorstellung verschiedener Anwendungsbeispiele besprechen wir typische Klassifizierungen und eine Standardformulierung fur Optimierungsprobleme. Anhand dieser Beschreibung diskutieren wir grundlegende Losungseigenschaften der Aufgabe wie: Hat das Problem eine Losung? Wenn ja, ist diese eindeutig? Wie robust ist sie gegenuber Storungen in der Aufgabenstellung? Die hier eingefuhrten Beispiele werden im gesamten Teil uber Optimierung immer wieder aufgegriffen und ausgearbeitet, um die vorgestellten Resultate und Methoden zu erlautern.

Laurenz Göllmann

Papers

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Theory and applications of optimal control problems with multiple time-delays

Optimal control problems with time delays: Two case studies in biomedicine

Einführung in die Optimierung

Mathematik für Ingenieure: Verstehen, Rechnen, Anwenden