Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Second, the rendezvous problem for kinematic point-mass mobile robots is studied when the robots’ fields of view have a fixed radius. The circumcenter control law of Ando e [IEEE Trans. Robotics Automation, 15 (1999), pp. 818-828] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent.

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State Agreement for Continuous-Time Coupled Nonlinear Systems

Dynamic network flow models describe network-structured, decision-making problems over time. They are of interest because of their numerous applications and intriguing dynamic structure. The dynamic models are specially structured problems that can be solved with known general methods. However, specialized techniques have been developed to exploit the underlying dynamic structure. Here, we present a state-of-the-art survey of the results, applications, algorithms and implementations for dynamic network flows.

A survey of dynamic network flows

Dedicated to R. T. Rockafellar on his 60th Birthday A subset X of a real Hilbert space H is said to be proximally smooth provided that the function dX : H! R (the distance to X) is continuously dieren tiable on an open tube U around X. It is proven that this property is equivalent to dX having a nonempty proximal subgradient at every point of U, and that the (G^ ateaux = Fr echet) derivative is locally Lipschitz on U. The Lipschitz behavior of the derivative is a consequence of the fact that under proximal smoothness, the metric projection onto X is single valued and Lipschitz on U. Alternate characterizations of proximal smoothness are given as well, in terms of properties of the proximal normal cone multifunction on X and on nearby closed neighborhoods of X. In case X is weakly closed, the list of equivalences is extended to include each point of U admitting a unique closest point in X. Further specializations are given in nite dimensions. In that setting, we discuss properties of locally Lipschitz real valued functions whose epigraphs are proximally smooth in a local sense. It is demonstrated that this function class coincides with the lower{C 2 functions studied by

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Proximal Smoothness and the Lower{C 2 Property

1. Introduction.- Basic Vegetation Science Contributions.- 2. Plant synecology in the service of rangeland management.- 3. Ecophysiology of range plants.- 4. Rangeland plant taxonomy.- 5. New plant development in range management.- 6. Successional concepts in relation to range condition assessment.- 7. A role for nonvascular plants in management of arid and semiarid rangelands.- 8. Seedbeds as selective factors in the species composition of rangeland communities.- 9. Modelling rangeland ecosystems for monitoring and adaptive management.- Vegetation Distribution and Organization.- 10. Vegetation-soil relationships on arid and semiarid rangelands.- 11. Vegetation attributes and their applications to the management of Australian rangelands.- 12. The ecology of shrubland/woodland for range use.- 13. Tundra vegetation as a rangeland resource.- 14. Forest rangeland relationships.- 15. Ecological principles and their application to rangeland management practice in South Africa.- 16. Range management from grassland ecology.- 17. Riparian values as a focus for range management and vegetation science.- Vegetation Science Rangeland Applications.- 18. Rangeland vegetation productivity and biomass.- 19. Rangeland vegetation - hydrologic interactions.- 20. Grazing management and vegetation response.- 21. Understanding fire ecology for range management.- 22. Reclamation of drastically disturbed rangelands.- 23. Rangeland vegetation as wildlife habitat.- 24. Revegetation of arid and semiarid rangelands.

Vegetation science applications for rangeland analysis and management.

A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.

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Parallel decomposition of multistage stochastic programming problems

An approach to the solution of max-min problems which takes into account the peculiarities of both the external (max) and the internal (min) operations is considered. 

The solution allows us to develop a set of methods for the solution of different kinds of max-min problems, including multistage max-min problems, max-min problems with linked constraints, etc.

On the Theory of Max-Min

This paper describes a dynamic linear programming model for studying long-range development alternatives of forestry and forest based industries at a national and regional level. The Finnish forest sector is used as an object of implementation and for numerical examples. Our model is comprised of two subsystems, the forestry and the industrial subsystem, which are linked to each other through the wood supply. The forestry submodel describes the development of the volume and age distribution of different tree species within the nation or its subregions. In the industrial submodel we consider various production activities, such as saw mill industry, panel industry, pulp and paper industry, as well as further processing of primary products. For a single product, alternative technologies may be employed. Thus, the production process is described by a small Leontief model with substitution. Besides supply of wood and demand of wood products, production is restricted through labor availability, production capacity, and financial resources. The production activities are grouped into financial units and the investments are made within the financial resources of such units. Objective functions related to GNP, balance of payments, employment, wage income, stumpage earnings, and industrial profit have been formulated. Terminal conditions have been proposed to be determined through an optimal solution of a stationary model for the whole forest sector. 

The structure of the integrated forestry-forest industry model is given in the canonical form of dynamic linear programs for which special solution techniques may be employed. Two versions of the Finnish forest sector models have been implemented for the interactive mathematical programming system called SESAME, and a few numerical runs have been presented to illustrate possible use of the model.

A model for the forest sector

This paper presents an extension of the simplex method, the basic method for solution of dynamic linear programming problems. The paper consists of three parts. Part I, "dual systems of DLP", concerns theoretical properties of the problem, primarily, duality relations; Part II, "the dynamic simplex method: general approach" describes the idea and the theory of the method; and Part III, "a basis factorization approach", gives a complete description of the algorithm, as well as the connection with the basis factorization approach. Part III also includes a numerical example that is not trivial for a general LP algorithm but is solved very easily by using the dynamic simplex method. Part II is written in a language more familiar to control theory specialists, Part III is closer to linear programming. All parts are written as independent papers with their own references and thus can be read independently. However, the whole paper comprises a theory of finite-step methods for DLP. The next development of the research might be first numerical tests on the behavior of the method and thus a judgement of its efficiency, and second extensions of the approach to other classes of strucured linear programming (for example, to DLP problems of the transportation type).

The Simplex Method for Dynamic Linear Programs

This paper considers the dynamic linear programming model for multispecies live-stock farming with a feed production subsystem. 

The problem is to determine the optimal livestock mix with the projected growth rate and corresponding development of feed production in order to obtain the maximum profit for the given planning horizon. 

As examples, the planning model for a dairy farm and the control problem of age-size structure of a biological population are given.

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Dynamic linear programming models for livestock farms

Dynamic linear programming (DLP) can be considered as a new stage of linear programming (LP) development Nowadays it becomes difficult, maybe even impossible, to make decisions in large systems and not take into account the consequences of the decision over a long-range period Thus, almost all problems of optimal decision making become dynamic, multi-stage ones New problems require new approaches With DLP it is difficult to exploit only LP ideas and methods: even having ,found the optimal program, we often do not know how to use it This paper represents in some sense the statement of the problem; although it contains a brief survey of DLP, it is focused on the things to be done, rather than on those already being tackled

A.I. Propoi

Papers

On the Theory of Max-Min

A model for the forest sector

The Simplex Method for Dynamic Linear Programs

Dynamic linear programming models for livestock farms

Problems of Dynamic Linear Programming