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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Applied mathematics & optimization 28 (1993), S. 49-56 
    ISSN: 1432-0606
    Keywords: Optimal stochastic control ; Controlled diffusions ; Extremal solutions ; Pre-Markov controls ; Marginal classes ; Primary 93E20 ; Secondary 60H10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The set of attainable laws of the joint state-control process of a controlled diffusion is analyzed from a convex analytic viewpoint. Various equivalence relations depending on one-dimensional marginals thereof are defined on this set and the corresponding equivalence classes are studied.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0606
    Keywords: Maximal monotone operators ; Convex programming ; Variational inequalities ; 49A29 ; 49A50 ; 49A52 ; 90C25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetT be a maximal monotone operator defined on ℝ N . In this paper we consider the associated variational inequality 0 εT(x *) and stationary sequences {x k * for this operator, i.e., satisfyingT(x k * → 0. The aim of this paper is to give sufficient conditions ensuring that these sequences converge to the solution setT −1(0) especially when they are unbounded. For this we generalize and improve the directionally local boundedness theorem of Rockafellar to maximal monotone operatorsT defined on ℝ N .
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0606
    Keywords: The minimax inequality ; Variational inequalities ; The FKKM theorem ; Noncompact and nonconvex sets ; Equivalence ; 49A29 ; 90C33 ; 90C50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this note is to give further generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional and to show that our minimax inequalities are equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and a modified FKKM theorem given in this note.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1432-0606
    Keywords: Rapid changes ; Chattering ; Harvesting ; Maximal sustainable catch ; Primary 49J45 ; 49J40 ; 33D15 ; Secondary 92D25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An example of harvesting in an environment that changes rapidly and frequently is examined. The changes are unpredictable, yet their statistical distribution is known. The parameters, however, can be measured on line, and used in the control policy. We show that then chattering systems adequately serve as nominal systems, optimal solutions of which provide near optimal solutions for the system with rapid changes.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-0606
    Keywords: Elliptic systems ; Nonlinear competitive interactions ; Game theory ; Optimal control ; 49A45 ; 35K10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A two-sided game for the control of a stationary semilinear competitive system with autonomous sources is considered, where the controls are the kernels of the nonlocal interaction terms. The saddle point (the optimal solution of the game) is characterized as the unique solution of the associated optimality system, which is solved by an iterative scheme.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1432-0606
    Keywords: Controlled diffusion ; Running maximum ; Wear model ; Policy improvement ; Invariant measure ; 93E20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyze optimal control problems for systems subject to random deterioration and failure. The system is replaced at failure and our objective is to optimize the utilization of the system between failures. The problems are new in that the payoff depends on the running maximum of a diffusion. This provides an intuitively appealing model for naturally monotone phenomena such as wear. The long-term average control problem is reduced to a family of simpler, single-cycle problems, a formula for the invariant measure for the (controlled) process is determined and a computational scheme based on the decomposition and formula is given.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Applied mathematics & optimization 1 (1974), S. 97-120 
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.
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  • 9
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Abstract In this work, we study the problem of the determination of the non-linearity in a parabolic equation from measurements over its solution. This corresponds to an usual physical situation, in which the parameter depends on the state of the system (for instance, the heat conduction coefficient depends on the temperature). We first state the problem as a control problem, for which we show the existence of a solution, and we calculate the derivative of the criterion with respect to the nonlinearity without any assumption on the algebraic form of this latter. We finally give a numerical application of the algorithm, which shows that the situation which is here taken in account (estimation of a parameter function depending on the state) is much better, from the identification point of view, than the usually studied situation for distributed system (estimation of a parameter function depending on space and/or time variables).
    Notes: Resume On étudie dans cet article le problème de la détermination de la nonlinéarité dans une équation parabolique à partir de mesures ponctuelles sur la solution. Ceci correspond à la situation courante en physique, dans laquelle un paramètre dépend de l'état du système (par exemple un coefficient de conductibilité thermique dépend de la température). Après avoir posé le problème comme un problème de contrôle dont on montre l'éxistence d'une solution, on calcule la dérivée du critère par rapport à la non-linéarité sans hypothèse sur la forme analytique de cette dernière. Enfin on donne une application numérique de l'algorithme obtenu, qui montre que la situation considérée ici (estimation d'un paramètre dépendant de l'état) est plus favorable, du point de vue identification, que la situation où l'on se place habituellement (estimation d'un paramètre dépendant des variables d'espace et/ou de temps).
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  • 10
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.
    Type of Medium: Electronic Resource
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