A goodwill model with predatory advertising

We investigate the dynamic advertising policies of two competing firms in a duopolistic industry, assuming a predatory phenomenon between their advertising campaigns. The resulting model is a differential game which is not linear-quadratic. We show that there exists a Markovian Nash equilibrium, and that it leads to time constant advertising strategies. According to this model, predatory advertising produces a negative externality: the interference between the advertising campaigns decreases the total demand of the market.     

Optimal advertising strategies with age-structured goodwill

The problem of a firm willing to optimally promote and sell a single product on the market is here undertaken. The awareness of such product is modeled by means of a Nerlove–Arrow goodwill as a state variable, differentiated jointly by means of time and of age of the segments in which the consumers are clustered. The problem falls into the class of infinite horizon optimal control problems of PDEs with age structure that have been studied in various papers either in cases when explicit solutions can be found or using Maximum Principle techniques. Here, assuming an infinite time horizon, we use some dynamic programming techniques in infinite dimension to characterize both the optimal advertising effort and the optimal goodwill path in the long run. An interesting feature of the optimal advertising effort is an anticipation effect with respect to the segments considered in the target market, due to time evolution of the segmentation. We analyze this effect in two different scenarios: in the first, the decision-maker can choose the advertising flow directed to different age segments at different times, while in the second she/he can only decide the activation level of an advertising medium with a given age-spectrum.     

An inventory model for stock with advertising sensitive demand

K. S. Chaudhuri Department of Mathematics, Jadavpur University, Kolkata-700 032, West Bengal, India Corresponding author. Email: shib_sankar{at}yahoo.com Email: k_s_chaudhuri{at}yahoo.com The effect of advertising media and sales effort on future demandof merchandize is considered. In an oligopolistic marketingsystem, it is quite natural to boost sales by using advertisingand sales effort to earn more money from a business sector.Determination of demand and costs due to advertising and saleseffort is quite difficult. The approach in this paper is toconcentrate on investment for the purpose of advertising andincreasing sales effort in order to maximize profit. The paperextends an inventory model over finite and infinite time horizonswhere demand is influenced by the level of stock, advertisingmedia and sales effort. The associated profit maximization problemis solved by the ‘Euler–Lagrange method’.The model is illustrated with a numerical example.     

Permanence of a delayed discrete mutualism model with feedback controls

This paper discusses a two species discrete model of mutualism with delays and feedback controls. Sufficient conditions are obtained for the permanence of the system. The results show that feedback control variables have no influence on the persistence property of the system.     

Quasi-periodic and chaotic relaxation oscillations in a laser model with variable delayed optoelectronic feedback

Doklady Mathematics - We derive asymptotically discrete mappings which determine the dynamics of relaxation spikes in a model of laser with optoelectronic feedback in the pump circuit. The time...     

Optimal stock liquidation in a regime switching model with finite time horizon

This paper is concerned with a finite-horizon optimal selling rule. A set of geometric Brownian motions coupled by a finite-state Markov chain is used to characterize stock price movements. Given a fixed transaction fee, the optimal selling rule can be obtained by solving an optimal stopping problem. The corresponding value function is shown to be the unique viscosity solution to the associated HJB equations. Numerical solutions to these equations and their convergence are obtained. A numerical example is presented to illustrate the results.     

Relaxation cycles in a model of two weakly coupled oscillators with sign-changing delayed feedback

Theoretical and Mathematical Physics - We consider the nonlocal dynamics of a model describing two weakly coupled oscillators with nonlinear compactly supported delayed feedback. Such models are...     

A singular control model with application to the goodwill problem

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system’s controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system’s state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product’s image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second-order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems.     

The dynamics of second-order equations with delayed feedback and a large coefficient of delayed control

The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out.     

State feedback and output feedback control of a class of nonlinear systems with delayed measurements

This paper is concerned with the problem of state feedback and output feedback control of a class of nonlinear systems with delayed measurements. This class of nonlinear systems is made up of continuous-time linear systems with nonlinear perturbations. The nonlinearity is assumed to satisfy a global Lipschitz condition and the time delay is assumed to be time-varying and have no restriction on its derivative. On the basis of the Lyapunov–Krasovskii approach, sufficient conditions for the existence of the state feedback controller and the output feedback controller are derived in terms of linear matrix inequalities. Methods of calculating the controller gain matrices are also presented. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Stability and Hopf bifurcation analysis of a predator-prey model with time delayed incomplete trophic transfer

A new nonlinear predator-prey model with incomplete trophic transfer is introduced. In this model,we assume that the rate of the trophic absorption of the predator is less than the rate of the conversion of consumed prey to predator in the Ivlev-type functional responses. The existence and uniqueness of the positive equilibrium of the model and the stability of the equilibrium of the model are studied under various conditions.Hopf bifurcation analysis of the delayed model is provided.     

Discrete time queues with delayed information

We study the behavior of a single-server discrete-time queue with batch arrivals, where the information on the queue length and possibly on service completions is delayed. Such a model describes situations arising in high speed telecommunication systems, where information arrives in messages, each comprising a variable number of fixed-length packets, and it takes one unit of time (a slot) to transmit a packet. Since it is not desirable to attempt service when the system may be empty, we study a model where we assume that service is attempted only if, given the information available to the server, it is certain that there are messages in the queue. We characterize the probability distribution of the number of messages in the queue under some general stationarity assumptions on the arrival process, when information on the queue size is delayedK slots, and derive explicit expressions of the PGF of the queue length for the case of i.i.d. batch arrivals and general independent service times. We further derive the PGF of the queue size when information onboth the queue length and service completion is delayedK=1 units of time. Finally, we extend the results to priority queues and show that when all messages are of unit length, thec rule remains optimal even in the case of delayed information.     

A dynamic IS-LM business cycle model with two time delays in capital accumulation equation

In this paper, we analyze a augmented IS-LM business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to Kalecki’s idea. Applying stability switch criteria and Hopf bifurcation theory, we prove that time delays cause the equilibrium to lose or gain stability and Hopf bifurcation occurs.     

Optimal advertising policy with the contagion model

This paper considers an optimal control problem for the dynamics of a contagion model, the optimal control being the rate of advertising expenditure that maximizes the present value of net profit streams over an infinite horizon. By using a Green's theorem approach, it is shown that there are multiple optimal stationary equilibria and that the optimal path from any given initial condition is a nearest feasible path to one of these equilibria.This work was partially supported by the National Research Council of Canada, Grant No. A4619.     

Permanent coexistence in chemostat models with delayed feedback control

Feedback control for chemostat model first appears in [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48–70] where the authors consider a dilution rate as a feedback control variable, keeping the input nutrient concentration to be constant. They showed that the coexistence of two organisms can be achieved under single nutrient source. In this paper, we improve the model in consideration of a time delay arising from controlling a dilution rate. It is shown that coexistence of two organisms can also be achieved in the improved model. The coexistence is mathematically insisted by solving permanence problem. We prove the problem by using the theory of “average Liapunov function”.     

Uncertain stock model with periodic dividends

This paper presents an uncertain stock model with periodic dividends based on uncertainty theory. Some option pricing formulae related to the proposed model are investigated and several numerical examples are discussed to illustrate the related formula.