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## Publications

Publications (53)

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem w...

The paper develops an optimization model of the career of a scientist. Recognizing that research efforts and networking get more efficient if the scientist is more knowledgeable, history dependent solutions are developed. We give a theoretical underpinning of the four different research patterns detected in Way et al. (2017, Proceedings of the Nati...

Flooding events can affect businesses close to rivers, lakes or coasts. This paper provides an economic partial equilibrium model, which helps to understand the optimal location choice for a firm in flood risk areas and its investment strategies. How often, when and how much are firms willing to invest in flood risk protection measures? We apply Im...

In this paper a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. The underlying dynamics is assumed to be affine-linear in control. The crucial idea which is followed in this paper is the choice of a weighted Sobolev space as the state s...

Lanchester (1916) modeled combat situations between two opponents, where mutual attrition occurs continuously in time, by a pair of simple ordinary (linear) differential equations. The aim of the present paper is to extend the model to a conflict consisting of three parties. In particular, Lanchester’s main result, i.e. his square law, is adapted t...

The economics of art and science differs from other branches by the small role of material inputs and the large role of given talent and access to markets. E.g., an African violinist lacks the audience ( = market) to appreciate her talent unless it is so large that it transgresses regional constraints; conversely, a European violinist of equal tale...

In this paper we deal with complex systems and how to handle them. We focus on a well-known class of dynamical systems, namely predator-prey models, firstly by applying this type of model to urban economic development and secondly by testing models in an experimental setting in order to ascertain how successful human decision makers are in managing...

Recently socio-hydrology models have been proposed to analyze the interplay of community risk-coping culture, flooding damage and economic growth. These models descriptively explain the feedbacks between socio-economic development and natural disasters such as floods. Complementary to these descriptive models, we develop a dynamic optimization mode...

For certain goods or services, the quality of the product can be assessed by customers only after consumption. We determine the optimal time paths for pricing, advertising and quality for a profit-maximizing firm facing demand that is influenced both by this experience quality as well as by advertising. In particular, there may exist two optimal tr...

Given the constantly raising world-wide energy demand and the accompanying increase in greenhouse gas emissions that pushes the progression of climate change, the possibly most important task in future is to find a carbon-low energy supply that finds the right balance between sustainability and energy security. For renewable energy generation, howe...

We consider a stock pollution problem where the biosphere can transform from a sink to a source of pollution in the presence of self-regenerating environmental absorption efficiency. We examine the problem of controlling pollution and the capacity of the biosphere to absorb pollution: the regulator can mitigate emissions and can invest to build-up...

In the literature on marketing models, the assumption of mixed word-of-mouth has been limited to the Bass diffusion model. Yet explicit leveraging of the originating factors of such assumption is lacking. Apart from that example, mixed word-of-mouth has been disregarded in contagion sales models. This paper bridges the gap by suggesting a sales mod...

The paper provides a framework that enables us to analyze the important
topic of capital accumulation under technological progress.We describe an algorithm
to solve Impulse Control problems, based on a (multipoint) boundary value problem
approach. Investment takes place in lumps and we determine the optimal timing of
technology adoptions as well as...

Autonomous ‘word of mouth’, as a channel of social influence that is out of firms’ direct control, has acquired particular importance with the development of the Internet. Depending on whether a given product or service is a good or a bad deal, this can significantly contribute to commercial success or failure. Yet the existing dynamic models of sa...

We show that the standard class of optimal control models in OCMat can be
used to analyze 1D spatial distributed systems. This approach is an
intermediate step on the way to the FEM discretization approach presented in
Grass and Uecker (2015). Therefore, the spatial distributed model is
transformed into a standard model by a finite difference discr...

This paper models dynamic interactions between talent and its
market. The economics of art and science differs from other branches,
by the small role of material inputs and the large role of market accessibility.
E.g., an author requires access to (potential) readers as a
complement to his productive capital. We study the problem of
optimal inves...

We present a framework to numerically treat spatially distributed optimal
control problems with an infinite time horizon. The basic idea is to consider
the associated canonical system in two steps. First we perform a bifurcation
analysis of the steady state canonical system using the continuation and
bifurcation package pde2path, yielding a number...

We derive small-noise approximations of the value function of stochastic optimal control problems over an unbounded domain and use these to perform a bifurcation analysis of these problems. The corresponding zero-noise problems may feature indiffierence (shock, Skiba) points, that is, points of non-differentiability of the value function. Small-noi...

Since the end of the seventies Skiba points have been studied in infinite time optimal control problems with multiple steady states. At such a Skiba point the decision maker is indifferent between choosing trajectories that approach different steady states. This paper extends this theory towards free end-time optimal control problems, where the dec...

This pollution accumulation model shows that the environmental absorption capacity is impacted by economic activity. The resulting optimal control problem has two inter-related state variables: the stock of pollution and the absorption capacity of the environment. The stock of pollution decreases with environmental absorption capacity and increases...

The probably biggest challenge for climate change mitigation is to find a secure low-carbon energy supply, which especially is difficult as the supply of renewable sources underlies strong volatility and storage possibilities are limited. We therefore consider the energy sector of a small country that optimizes a portfolio consisting of fossil and/...

T. C. Schelling [Micromotives and macrobehavior. New York: W. W. Norton & Co. (1978)] suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the...

In a general equilibrium model of the world economy, we develop a two-dimensional energy balance climate model featuring heat diffusion and anthropogenic forcing driven by global fossil fuel use across the sphere of the Earth. This introduces an endogenous location dependent temperature function, driving spatial characteristics, in terms of locatio...

Software can be distributed closed source (proprietary) or open source (developed collaboratively). While a firm cannot sell open source software, and so loses potential sales revenue, the open source software development process can have a substantial positive impact on the quality of a software, its diffusion, and, consequently, the demand for a...

We present a novel model of corruption dynamics in the form of a nonlinear optimal dynamic control problem. It has a tipping point, but one whose origins and character are distinct from that in the classic Schelling (1978) model. The decision maker choosing a level of corruption is the chief or some other kind of authority figure who presides over...

In this paper we describe three different algorithms, from which two (as far as we know) are new in the literature. We take both the size of the jump as the jump times as decision variables. The first (new) algorithm considers an Impulse Control problem as a (multipoint) Boundary Value Problem and uses a continuation technique to solve it. The seco...

Numerous optimal control models analyzed in economics are formulated as discounted infinite time horizon problems, where the defining functions are nonlinear as well in the states as in the controls. As a consequence solutions can often only be found numerically. Moreover, the long run optimal solutions are mostly limit sets like equilibria or limi...

This paper combines technology adoption with capital accumulation taking into account technological progress. We model this as a multi-stage optimal control problem and solve it using the corresponding maximum principle. The model with linear revenue can be solved analytically, while the model with market power is solved numerically. We obtain that...

This paper considers a firm that has the option to undertake product innovations. For each product innovation the firm has to install a new production plant. We find that investments are larger and occur in a later stadium when more of the old capital stock needs to be scrapped. Moreover, we obtain that the firm’s investments increase when the tech...

The paper considers the problem of a firm that, while producing a standard product, has the option to introduce an innovative product. The innovative product competes with the standard product and will therefore reduce revenues of the standard product. A distinction is made between innovative products that do or do not become even more relatively a...

This paper considers the problem of how to price a conspicuous product when the economy is in a recession that disrupts capital markets. A conspicuous product in this context is a luxury good for which demand is increasing in brand image. Brand image here means the ability of a consumer to impress observers by conspicuously displaying consumption o...

In this paper we present an optimal control model that addresses a central challenge of modern housing policy: deconcentrating
poverty via “housing mobility programs”, which move poor families into middle-class neighborhoods. The tension is that helping
poor families is clearly an opportunity, but pursuing these programs too aggressively increases...

The so-called DNSS points of indifference are of interest because they give decision makers in optimal control problems a choice between following either of two or more trajectories while still achieving optimality. Usually they are described in terms of initial conditions, so that if the system starts at a DNSS point, the decision maker can procee...

A control model is presented which studies optimal spending for the fight against terrorism. Under the assumptions that economic damages are larger the greater the number of terrorists and that the success of counter terror operations depends on public opinion, it is demonstrated that a so-called DNSS threshold may exist, separating the basin of at...

The goal of this book is to prepare readers to apply the optimal control theory to nonlinear processes beyond the standard applications. The examples investigated in depth are drawn from drug policy, corruption, and counter-terror, so this book will be of particular interest to economists, epidemiologists, and other scholars interested in the econo...

This paper deals dynamically with the question of how recruitment to terror organizations is influenced by counter-terror operations. This is done within an optimal control model, where the key state is the (relative) number of terrorists and the key controls are two types of counter-terror tactics, one (“water”) that does not and one (“fire”) that...

This chapter presents the fundamentals of optimal control theory. In the first section we give a short historical survey, introducing the reader to the main ideas and notions. Subsequently we introduce the standard problem of optimal control theory.
We state Pontryagin's Maximum Principle, distinguishing between the cases without and with mixed pat...

In the last chapter we presented Pontryagin's Maximum Principle and the Hamilton–Jacobi–Bellman equation to solve optimal control problems. Now we deal with its exposition from a different point of view. In this chapter we first describe an intuitive approach to the Maximum Principle and its results, in which we reformulate the continuous-time prob...

The theory of optimal control is wonderful. Now, how does one actually apply it? In particular, how does one compute the solutions numerically, including for higher-dimensional systems (since solving these systems can substantially be harder than solving one-dimensional models)? This third part of the book sheds light on the practical side of apply...

This chapter addresses the interesting and important topics of multiplicity and history-dependence of optimal solutions. Multiplicity means that for given initial states there exist multiple optimal solutions; thus the decision-maker is indifferent about which to chose. This explains why such initial states are called points of indifference. In con...

We aim at keeping the theory presented in this textbook strictly target-oriented and therefore confine ourselves to spaces, terms, and concepts necessary to understand nonlinear dynamics and optimal control theory. For the reader searching for more information about the mathematics behind our straightforward approach we refer to additional literatu...

In the preceding seven chapters we explored the basic principles of optimal control theory applied to problems typically arising in economics and management science. In particular we illustrated how the interplay of analytical and numerical methods can be used to obtain insight when fighting drugs, corruption, and terror. Before concluding our tour...

The importance of continuous dynamical systems for optimal control theory is twofold. First, dynamical systems already occur in the problem formulation, in which the evolution of the states to be controlled is formulated as a differential equation. Second, and more important, the techniques for calculating and analyzing the solutions of optimal con...

This appendix summarizes technicalities and proofs which may be of interest to the reader but would have interrupted the flow of exposition in the main text.

In the previous chapters we presented both the theoretical background of optimal control theory and some interesting examples from the application fields of drugs, corruption, and terror. What remains is the question of how to actually compute the optimal solutions. How should we apply the theoretical results to retrieve algorithms for numerical ca...

A simple optimal control model is introduced, where “bridge building” positions are rewarded. Optimal solutions can be classified
with regard to the two parameters (i)the cost of adjusting one’s position in a social network and (ii)the discount rate.
Acomplete analytical description is derived for the bifurcation lines in parameter space, which sep...

This paper deals with macroeconomic aspects of widespread substance abuse with a reference to illicit drugs as an example. Substance abuse impedes the productivity of the labor force and reduces economic growth. Workers are either nonusers and therefore fully productive, a number of whom are employed by the government in drug-control activities, or...

A central objective of modern US housing policy is deconcentrating poverty through "housing mobility programs" that move poor families into middle class neighborhoods. Pursuing these policies too aggressively risks inducing middle class flight, but being too cautious squanders the opportunity to help more poor families. This paper presents a styliz...

A central objective of modern US housing policy is deconcentrating poverty through "housing mobility programs" that move poor families into middle class neighborhoods. Pursuing these policies too aggressively risks inducing middle class flight, but being too cautious squanders the opportunity to help more poor families. This paper presents a styliz...

A simple model is considered that rewards "moderation" - finding the right balance between sliding down either of two "slippery slopes". Optimal solutions are computed as a function of two key parameters: (1) the cost of resisting the underlying uncontrolled dynamics and (2) the discount rate. Analytical expressions are derived for bifurcation line...

Recently (Troje, Huber, Loidolt, Aust, & Fieder 1999), we found that pigeons discriminated between large sets of photorealistic frontal images of human faces on the basis of sex. This ability was predominantly based on information contained in the visual texture of those images rather than in their configural properties. The pigeons could learn the...