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Le but de cet article est d’etudier les equations d’evolution non lineaires monotones a entrees stochastiques du type suivant: $$\frac{{dy}}{{dt}}\left( {t; } \right) + A\left( t \right)y\left( {; } \right) = g \left( {t; } \right) + \frac{{df}}{{dt}}\left( {t; } \right)$$ ouA(t) est une famille d’operateurs monotones d’un espace de BanachV dans son dualV′. Il s’agit d’une generalisation a des equations aux derivees partielles des travaux de Ito sur les equations differentielles stochastiques.

We consider the problem of existence of solutions for stochastic Navier-Stokes equations. The random elements are the initial value of the solution and forcing terms which may be white noise in time. The method uses results on multi-valued functions. We also give energy type results and present examples.

The functions ur(x), u2(x) characterize the optimal cost of a stochastic control problem, whose decision variables are switching points. The results on existence, uniqueness and regularity of (l), (2) are recalled in Section 1, where we also describe the stochastic control problem which corresponds to it. We are interested here in obtaining regions which belong to the set of points where ur = u2 + k, or u2 = ur + k, , denoted by S, , S, respectively, and called the supports of the solution of (l), (2). Th’ IS p ro bl em is the analog of the problem on the support of solutions of elliptic variational inequalities (v.i.) considered by Brezis [6]. For the q.v.i. arising in the impulse control problem, studied by Bensoussan and Lions [3], the analogous question of support has been considered recently by Bensoussan, Brezis and Friedman [5]. The general methodology that we apply relies on the results on support for vi. However, we also use here the special characteristic of (l), (2) in order to obtain sharp estimates on the support, which is the main objective of the paper. We notice that if g, = g, = g then ur - u2 = w is a solution of the v.i.

Differential games with stopping times are introduced. The upper value satisfies a variational inequality of the first order. Next, stochastic differential games with stopping times are introduced and studied using nonlinear parabolic variational inequalities. The behavior of the value is studied as the coefficient of the diffusion matrix goes to zero.

The objective of this paper is to study the stochastic version of a previous paper of the authors, in which hybrid control for deterministic systems was considered. The modelling is quite similar to the deterministic case. We have a system whose state is composed of a continuous part and a discrete part. They are affected by a continuous type control and an impulse control. The dynamics is moreover perturbed by noise, also a continuous and a discrete noise process. The Markovian character of the sate process is preserved. We develop the model and show how the dynamic programming approach leads to some involved quasi-variational inequality.

Survey of currently available theory for systems the evolution of which can be described by semigroups of operators of class CO. Connection between the concepts of stabilizability and detectability and the problem of existence and uniqueness of solutions to the operator Riccati equation. Examples and Open problems.

Representation theory for finite dimensional systems has been the subject of a great deal of discussion in recent years. In the case of linear systems defined over fields an account of the theory can be found in the books of BROCKETT and KALMAN-FALB-ARBIB and in the case of systems defined over rings in the book of EILENBERG, the papers of ROUCHALEAU-WYMAN-KALMAN, ROUCHALEAU-WYMAN and the thesis of JOHNSTON. For a lucid survey of results on systems defined over commutative rings, see SONTAG.

In serveral papers, H. Albach has stressed the importance of the analysis of growth of firms for the development of management science and the understanding of economic systems (Albach 1967a, 1967b; 1976). In fact, the literature on corporate growth or, more generally, on firm dynamics has considerably increased in the last years and probably for two separate reasons: (1) The diffusion of the knowledge of control theory has given to economists a very convenient tool to extend to dynamics the scope of optimization. (2) The firm is more and more recognized as an anticipating, planning, adapting organism plunged into a changing environment and having to face delays, rigidities and uncertainties.

We consider a class of problems in stochastic control theory involving stochastic systems with small parameters. Using both analytical and probabilistic methods adapted to the special structures of singularly perturbed stochastic control problems, we develop a systematic methodology for their analysis.

Results on phase-locked arrays of InGaAsP/InP index-guided diode lasers grown entirely by low pressure metalorganic chemical vapour deposition are presented. Two different array structures are compared : 1) the evanescently coupled array of inverted channeled substrate planar lasers and 2) the Y-junction coupled array of buried ridge lasers in order to assess their potential as sources of high powers in stable and narrow beam. Single-lobe beam operation is achieved by evanescent coupling, while the device is shown to be sensitive to fabrication and operation conditions and not practical as high power source. Uncoated Y-coupled arrays exhibit output power up to 158 mW CW/facet when bonded on diamond heatsink. Such devices are expected to give enhanced performance by further improvement in design and fabrication and should prove useful for free space optical communications.

We study the Dynamic Programming Equation arising in the problem of stochastic control in discrete time over an infinite horozon. The data are unbounded, in which case the solution is not unique. Our results deal with the structure of the set of solutions.

This note summarizes our larger work on a problem of optimal production planning under a production constraint.

We investigate a differential game where the decision variables of both players are sequences of impulse times and jumps on the system. We give an existence theorem of a saddle point and a complete characterization of the solution.

The problem of observers has been introduced in the control literature by Luenberger. Let us consider a dynamic system which is deterministic, but whose initial state is unknown. An observer is a model which mimics the behavior of the physical system, and in particular its state becomes closer and closer as time evolves to the state of physical system. There is a great deal of freedom in such a design and it is important to investigate various kinds of observers.

We consider in this article non linear elliptic equations of the form $$\eqalign{ & - {\partial \over {\partial {x_i}}}(a_{ij}^ \in (x;\omega ){{\partial {u^ \in }} \over {\partial {x_j}}}) = {H^ \in }(x,D{u^ \in },{u^ \in },\omega ),{\rm{ }}x \in O \cr & {\rm{ u }}_{\rm{|}}^ \in \partial o = 0 \cr} $$ where a ∊ i j(x;ω) is a sequence of stochastic processes, the matrix a ∊ i j being uniformly bounded and coercive. The non linear operator H ∊ has quadratic growth in Du ∊.

We study a stochastic version of the Ramsey problem, which represents the basic model of economic growth. Our objective is to give sensitivity results with respect to the characteristics of the uncertainities.

The stochastic control problem with linear stochastic differential equations driven by Brownian motion processes and as cost functional the exponential of a quadratic form is considered. The solution consists of a linear control law and of a linear stochastic differential equation. The latter has the same structure as the Kalman filter but depends explicitly on the cost functional. The separation property does not hold in general for the solution to this problem.

We consider in this presentation the risk-sensitive control problem under fairly general assumptions, in particular as far growth conditions are concerned. This is a much more delicate issue than in the classical stochastic control problem. We then consider the case of small variance of the noises and study the limit, which is related to robust control. The limit problem corresponds to a differential game. In this paper, the complete observation case is considered.

The objective of this article is to apply an operator splitting method to the time integration of Zakai equation. Using this approach one can decompose the numerical integration into a stochastic step and a deterministic one, both of them much simpler to handle than the original problem. A strong convergence theorem is given, in the spirit of existing results for deterministic problems.

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the concept of a reference inventory position. We show that this position along with the magnitude of the latest observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state-dependent base-stock type with respect to the reference inventory position (or state-dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and (s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.

The author has previously pointed out the specific property of exact controllability that linear dynamic systems with skew symmetric operators possess. This particular property holds for the wave equation. In the previous work, there was an assumption stated in terms of the controllability operator. To a certain extent, this assumption restricted the full extent of the abstract formalism that was used. In this paper, an attempt is made to eliminate this awkward assumption, and to express conditions only in terms of the operators A, B characterizing the system. >

This paper considers an infinite horizon stochastic production planning problem with the constraint that production rate must be nonnegative. It is shown that an optimal feedback solution exists for the problem. Moreover, this solution is characterized and is then compared with the solution of the unconstrained problem. Also obtained, by using a policy iteration procedure, are computational solutions to the related problems with upper bounds on the production rate. which determines production rates over time to minimize an integral representing a discounted quadratic loss function. The loss function is defined in terms of the deviations of production and inventory levels from their rated or factory-optimal values. The model is solved both with and without nonnegative production constraints. A stochastic extension of this paper involving a white noise process (1) is analyzed by Sethi and Thompson (9), (10). Closed-form solutions for optimal feedback production policy for both finite and infinite horizon versions of the model without production constraints are obtained. In particular, the model must allow negative production rates or disposals. In this paper, we consider the stochastic production planning problem with the constraint that production rates must be nonnegative. Only the infinite horizon problem is treated. It is shown that an optimal feedback solution exists for the problem. This solution is characterized. Also obtained, by using a policy iteration procedure, are computational solutions to the related problems with upper bounds on the production rate.

We present in this survey some of the main ideas concerning impulse control theory. We have chosen the following framework. The process is a non degenerate diffusion stopped at the exit of a bounded regular domain of Rn. The control is excited by impulses, which modify the state of the system instantaneously. The cost involves an integral cost and a variable cost (depending on the size of the jump).