Geert Jan Olsder

Eigenvalues of dynamic max-min systems

Discrete event dynamic systems are studied in which the time evolution depends on the max-, min-, and the summation operation simultaneously. Specifically, necessary and sufficient conditions are given under which the operator which characterizes the evolution of such a system has an eigenvalue and eigenvector(s). Numerical algorithms to calculate these quantities are also provided.

Brief paper: Team-optimal closed-loop Stackelberg strategies in hierarchical control problems

This paper is concerned with the derivation of closed-loop Stackelberg (CLS) solutions of a class of continuous-time two-player nonzero-sum differential games characterized by linear state dynamics and quadratic cost functionals. Explicit conditions are obtained for both the finite and infinite horizon problems under which the CLS solution is a representation of the optimal feedback solution of a related team problem which is defined as the joint minimization of the leaders cost function. First, a specific class of representations is considered which depend linearly on the current and initial values of the state, and then the results are extended to encompass a more general class of linear strategies that also incorporate the whole past trajectory. The conditions obtained all involve solutions of linear matrix equations and are amenable to computational analysis for explicit determination of CLS strategies.

Cramer and Cayley-Hamilton in the max algebra

Abstract Cramers rule and the Cayley-Hamilton theorem are formulated and provided in the so-called max algebra, which consists of the set of reals provided with two operations: maximization and addition. It is surprising to see that these well-known theorems carry over to the max algebra almost without any changes, provided that the conventional addition and multiplication are replaced by maximization and addition respectively. The definitions of determinant and eigenvalue have to be adjusted to this new formulation. The role of the determinant is taken over by the permanent and a refinement of this latter concept called the dominant.

Introduction to the Modelling, Control and Optimization of Discrete Event Systems

The theory of Discrete Event Systems (DES) is a research area of current vitality. The development of this theory is largely stimulated by discovering general principles which are (or are hoped to be) useful to a wide range of application domains. In particular, technological and/or ‘man-made’ manufacturing systems, communication networks, transportation systems, and logistic systems, all fall within the class of DES. There are two key features that characterize these systems. First, their dynamics are event-driven as opposed to time-driven, i.e., the behavior of a DES is governed only by occurrences of different types of events over time rather than by ticks of a clock. Unlike conventional time-driven systems, the fact that time evolves in between event occurrences has no visible effect on the system. Second, at least some of the natural variables required to describe a DES are discrete. Examples of events include the pushing of a button or an unpredictable computer failure. Examples of discrete variables involved in modelling a DES are descriptors of the state of a resource (e.g., UP, DOWN, BUSY, IDLE) or (integer-valued) counters for the number of users waiting to be served by a resource. Some authors use the acronym DEDS, for a discrete event dynamic system, rather than DES, to emphasize the fact that the behavior of such systems can, and usually will, change as time proceeds.

New Trends in Dynamic Games and Applications

I. Minimax control.- Expected Values, Feared Values, and Partial Information Optimal Control.- H?-Control of Nonlinear Singularly Perturbed Systems and Invariant Manifolds.- A Hybrid (Differential-Stochastic) Zero-Sum Game with Fast Stochastic Part.- H?-Control of Markovian Jump Systems and Solutions to Associated Piecewise-Deterministic Differential Games.- The Big Match on the Integers.- II. Pursuit evasion.- Synthesis of Optimal Strategies for Differential Games by Neural Networks.- A Linear Pursuit-Evasion Game with a State Constraint for a Highly Maneuverable Evader.- Three-Dimensional Air Combat: Numerical Solution of Complex Differential Games.- Control of Informational Sets in a Pursuit Problem.- Decision Support System for Medium Range Aerial Duels Combining Elements of Pursuit-Evasion Game Solutions with AI Techniques.- Optimal Selection of Observation Times in a Costly Information Game.- Pursuit Games with Costly Information: Application to the ASW Helicopter Versus Submarine Game.- Linear Avoidance in the Case of Interaction of Controlled Objects Groups.- III. Solution methods.- Convergence of Discrete Schemes for Discontinuous Value Functions of Pursuit-Evasion Games.- Undiscounted Zero Sum Differential Games with Stopping Times.- Guarantee Result in Differential Games with Terminal Payoff.- IV. Nonzero-sum games, theory.- Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and the Algebraic Riccati Equation of Zero-Sum Games.- A Turnpike Theory for Infinite Horizon Open-Loop Differential Games with Decoupled Controls.- Team-Optimal Closed-Loop Stackelberg Strategies for Discrete-Time Descriptor Systems.- On Independence of Irrelevant Alternatives and Dynamic Programming in Dynamic Bargaining Games.- The Shapley Value for Differential Games.- V. Nonzero-sum games, applications.- Dynamic Game Theory and Management Strategy.- Endogenous Growth as a Dynamic Game.- Searching for Degenerate Dynamics in Animal Conflict Game Models involving Sexual Reproduction.

The power algorithm in max algebra

Abstract It is proved that, under certain conditions, an algorithm resembling the power algorithm in conventional linear algebra can be used to find the eigenvector and eigenvalue of a max-algebra system. If the conditions are not satisfied, the eigenvector can be found using an extension of the power algorithm.

Conic surveillance evasion

A surveillance-evasion differential game of degree with a detection zone in the shape of a two-dimensional cone is posed. The nature of the optimal strategies and the singular phenomena of the value function are described and correlated to subsets of the space of all possible parameter combinations, showing the relation of the singular phenomena in differential game theory and control theory.

The “Princess and Monster” Game on an Interval

A minimizing searcher

Linear-quadratic stochastic pursuit-evasion games

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Applications of the theory of stochastic discrete event systems to array processors and scheduling in public transportation

and a maximizing hider