The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

A differential game of incomplete information

In this paper, we study a differential game of incomplete information. In such a game, the cost function depends on parameter . At the start of the game, only one of the players knows the value of this parameter, while the other player has only a (subjective) probability distribution for the parameter. We obtain explicit expressions for both the value of the game and the two players' optimal strategies.     

Nonlocal second-order geometric equations arising in tomographic reconstruction

In this paper, we study new nonlocal geometric equations which are related to tomographic reconstruction when using the level-set approach. We treat two additional difficulties which make the work original. On one hand, the level lines do not evolve along normal directions and the nonlocal term is not of “convolution type”. On the other hand, the speed is not necessarily bounded compared to the nonlocal term. We prove an existence and uniqueness result for our equation.     

A class of differential games with state-dependent closed-loop feedback solutions

In this paper, a class ofN-person, nonzero-sum differential games with state-dependent closed-loop feedback solutions is presented. In particular, the system of Hamilton-Jacobi equations for a closed-loop feedback solution turns out to be a set ofN-independent linear equations with viscosity solutions.The helpful comments of an anonymous referee are gratefully acknowledged.     

Optimal allocation of drug control efforts: A differential game analysis

The present paper considers a dynamic nonzero-sum game between drug dealers and the authorities. Although the game is neither linear-quadratic nor degenerate, in the sense that the closed-loop equilibria coincide with the open-loop equilibria, we are able to calculate explicitly a stationary feedback Nash equilibrium of that game. In a numerical example, we determine the optimal allocation of governmental efforts between treatment and law enforcement minimizing the total discounted cost stream in the equilibrium. Moreover, we provide sensitivity analyses with respect to the efficiency parameters of both competitors. Our results show that a farsighted authority should attack the drug problem from the demand side and put much effort in treatment measures and the improvement of the efficiency of the treatment.This research was supported by the Austrian Science Foundation under Contract P9112-SOZ. We would like to thank A. Luhmer and E. J. Dockner for helpful comments.     

Direct Method Comparing Efficient and Nonefficient Payoffs in Differential Games

In this note, we present a method that allows us to decide when a Markov-perfect Nash equilibrium is not Pareto optimum, without the explicit knowledge of the respective solutions. For that purpose, we establish a sufficient condition in terms of an algebraic inequality where the gradient of the value functions of the cooperative and noncooperative games as well as the state and control variables are involved.     

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.     

Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view

We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small >0. The o.d.e. limit corresponding to =0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the ↓0 limit, the invariant density has the form ≈exp(−W(x)/²), where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given.     

Parental care as a differential game: A dynamic extension of the Houston–Davies game

An interpretation of the conflict between male and female parents during the process of caring for their common offspring by means of Game Theory was given in Houston and Davies. [A.I. Houston, N.B. Davies, The evolution of cooperation and life history in the dunnock Prunella modularis, in: R.M. Sibly, R.H. Smith (Eds.), Behavioral Ecology, Blackwell Scientific Publications, 1985, pp. 471–487]. Mathematically, this model represents a static game with continuous strategy sets. Recently, this model was reconsidered in a dynamic discrete time framework which also included state dependencies [J.M. McNamara et al., A dynamic game-theoretic model of parental care, J. Theor. Biol. 205 (2000) 605–623]. In this article, we give an interpretation of the parental care conflict in continuous time by means of a differential game with state dependent strategies.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

A differential game of evasion with delays

A game of evasion (or contact avoidance problem) described by a system of differential equations containing delays in state and control is considered. There is given a sufficient condition for the existence of a strategy which ensures the contact avoidance. A method of constructing such a strategy is presented.     

A zero sum differential game in a Hilbert space

We study a zero sum differential game of fixed duration in a separable Hilbert space. We prove a minimax principle and establish the equivalence between the dynamic programming principle and the existence of a saddle point equilibrium. We also prove sufficient conditions for optimality.     

Limit Theorem for Controlled Backward SDEs and Homogenization of Hamilton–Jacobi–Bellman Equations

We prove a convergence theorem for a family of value functions associated with stochastic control problems whose cost functions are defined by backward stochastic differential equations. The limit function is characterized as a viscosity solution to a fully nonlinear partial differential equation of second order. The key assumption we use in our approach is shown to be a necessary and sufficient assumption for the homogenizability of the control problem. The results generalize partially homogenization problems for Hamilton–Jacobi–Bellman equations treated recently by Alvarez and Bardi by viscosity solution methods. In contrast to their approach, we use mainly probabilistic arguments, and discuss a stochastic control interpretation for the limit equation.     

Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.     

A Geometric Characterization of Viable Sets for Controlled Degenerate Diffusions

For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a deterministic second-order normal cone to K and the data of the diffusion process. We also give an extension to stochastic differential games.     

Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.     

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Jacobi interpolation approximations and their applications to singular differential equations

Jacobi–Gauss-type interpolations are considered. Some approximation results in certain Hilbert spaces are established. They are used for numerical solutions of singular differential equations and other related problems. The numerical results are illustrated.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.