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The problem of stochastic control of partially observable systems plays an important role in many applications. All real problems are in fact of this type, and deterministic control as well as stochastic control with full observation can only be approximations to the real world. This justifies the importance of having a theory as complete as possible, which can be used for numerical implementation. This book first presents those problems under the linear theory that may be dealt with algebraically. Later chapters discuss the nonlinear filtering theory, in which the statistics are infinite dimensional and thus, approximations and perturbation methods are developed.

We deal in this article with the theory of robust control for infinite dimensional systems in the case of full and partial observation. The first case corresponds to a theorem of [1]. Here the contribution consists in a simplification of the proof showing that all steps reduce to standard LQ problems for which the theory is well known and can be heavily applied. As a side effect an interesting LQ problem is identified, with a nice interpretation. The problem is equivalent to solving a differential game with full information for both players. The second case corresponds to a theorem of [2]. Here again we provide a simplified proof. In this case the problem is not equivalent to a differential game, but includes the solution of three related differential games with full information. We show the role of the “observer” concept in this context.

We consider in this paper the limit behavior of the solutionsuɛ of the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$ whereHɛ has quadratic growth inDuɛ andaɛ(x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$ whereGɛ has quadratic growth inDuɛ and satisfiesGɛ(x, s, ξ)s ≥ 0. Note that in this last modeluɛ is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.

Interaction between Computer Science and Mathematics has evolved considerably. We discuss here some features which have been emerging, among the main ones.

It is well known that the classical homogenization theory of elliptic and parabolic equations can be derived by probabilistic methods, through the limit of diffusion stochastic processes (see A. BENSOUSSAN - J.L. LIONS - G. PAPANICOLAOU [4]).

Nagase has given new results for the existence of solutions of stochastic partial differential equations. The main idea is to use a compactness argument based on a special Hilbert space introduced by J. L. Lions (1961) in the context of parabolic linear partial differential equations. Previously, the author considered a class of nonlinear partial differential equations which generalize those of Nagase as far as the nonlinearity is considered. Here the author considers another class of nonlinearity which covers the Navier-Stokes equation in dimension 2. >

On etudie des systemes d'equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de Nash

The nonlinear filtering problem of a vector diffusion process is considered when several noisy vector observations with possibly different dimension of their range space are available. At each time any number of these observations (or sensors) can be used in the signal processing performed by the nonlinear filter. The problem considered is the optimal selection of a schedule of these sensors from the available set, so as to optimally estimate a function of the state at the final time. Optimality is measured by a combined performance measure that allocates penalties for errors in estimation, for switching between sensor schedules, and for running a sensor. The solution is obtained in the form of a system of quasi-variational inequalities in the space of solutions of certain Zakai equations.

One is given a diffusion process and two payoffs which depend on the process and on two stopping times τ 1 , τ 2 {\tau _1},{\tau _2} . Two players are to choose their respective stopping times τ 1 , τ 2 {\tau _1},{\tau _2} so as to achieve a Nash equilibrium point. The problem whether such times exist is reduced to finding a “regular” solution ( u 1 , u 2 ) ({u_1},{u_2}) of a quasi-variational inequality. Existence of a solution is established in the stationary case and, for one space dimension, in the nonstationary case; for the latter situation, the solution is shown to be regular if the game is of zero sum.

Cet article a pour objet de definir les criteeres generaux de conception des reseaux de diodes lasers a rubans couples par champs evanescents. Une methode generale de conception de ces reseaux est presentee en s’appuyant sur la methode de l’indice effectif et sur la theorie des modes couples. Elle permet de definir les conditions de confinement imposees aux diodes elementaires et elle determine les configurations de reseaux admissibles.

We consider a stochastic control problem for a random evolution. We study the Bellman equation of the problem and we prove the existence of an optimal stochastic control which is Markovian. This problem enables us to approximate the general problem of the optimal control of solutions of stochastic differential equations.

Resume Ce papier est consacre a la recherche d'horizons de planification dans une production faisant intervenir des couts concaves. L'approche choisie est la programmation dynamique de type retrograde. Cette approche, associee a un resultat que nous etablissons, permet: - de simplifier un algorithme de recherche de la solution optimale du probleme a horizon fini propose precedemrrent par les auteurs. - de degager un algorithme qui conduit a un horizon de planification dans un nombre important de cas. Nous presentons des resultats numeriques.

Some unilatral problems arising from dynamic programming are discussed. The main interest is in the number of solutions of the problem. In particular, an example with exactly two solutions is studied.

The paper extends the finite dimensional stochastic maximum principle to a large class of distributed parameter systems. Although this class of systems does not include all actual distributed parameter systems, the approach of this paper can be easily adapted to any particular situation.

We consider in this article a filtering problem for a linear distributed parameter system where the input disturbances are located in points instead of being distributed. The observation process is assumed to be distributed inside the domain. Although formally one can derive equations for the Kalman filter and the Riccati equation, the complete justification of these equations leads to problems which cannot be solved by standard methods, in the sense that some new functional spaces need to be introduced. The objective of the paper is to give a rigorous treatment of this nonclassical filtering problem, which is relevant for applications.

The objective of this article is to provide an axiomatic framework in order to define the concept of value function for risky operations for which there is no market. There is a market for assets, whose prices are characterized as stochastic processes. The method consists of constructing a portfolio of these assets which will mimic the risks involved in the operation. We follow the terminology of the theory of options although the set-up goes beyond that particular problem.

In this paper, we consider the following problem: Here the coefficients aij and bi are smooth, periodic with respect to the second variable, and the matrix (aij)ij is uniformly elliptic. The Hamiltonian H is locally Lipschitz continuous with respect to uϵ and Duϵ, and has quadratic growth with respect to Duϵ. The Hamilton-Jacobi-Beliman equations of some stochastic control problems are of this type. Our aim is to pass to the limit in (0ϵ) as ϵ tends to zero. We assume the coefficients bi to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive L∞, H and W, p0 > 2, estimates for the solutions of (0ϵ). We also prove the following corrector's result: This allows us to pass to the limit in (0ϵ) and to obtain This problem is of the same type as the initial one. When (0ϵ) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then (00) is also a Hamilton-Jacobi-Bellman equation but one corresponding to a modified set of controls.

CONTENTSIntroduction § 1. Optimal time in a stochastic control § 2. Optimal time and optimal design § 3. Stationary quasi-variational inequalities References

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 soluti...

On presente dans cet article une methodologie generale de traitement des problemes de controle impulsionnel, fondee sur l'etude systematique des inequations quasi variationnelles. Celles-ci jouent pour le controle impulsionnel un role analogue a celui des equations de Hamilton-Jacobi-Bellman pour le controle continu.

We prove the existence of a solution of the nonlinear elliptic equation: A(u) + g(x, u, Du) = h(x), where A is a Leray-Lions operator from W 0 1 , p ( Ω ) into W−1, p′(Ω) and g is a nonlinear term with “natural” growth with respect to Du [i.e. such that |g(x, u, ξ)| ≦ b(|u|) (|ξ|p + c(x))], satisfying the sign condition g(x, u, ξ)u ≧ 0 but no growth condition with respect to u. Here h belongs to W−1, p′(Ω), thus the solution u of the problem does not in general be more smooth than W 0 1 , p ( Ω ) . The existence of a solution is also proved for the corresponding obstacle problem.

We study in this paper a problem of regular perturbations in Optimal control. This problem has already been considered by J. Cruz [1]. However the treatment of Cruz is limited to the case without constraints on the control. The main objective of this paper is to extend the theory to get rid of this restriction. Our method of proof is also different.

Inventory models with deterministic and stochastic deterioration rates are given. The first model, assuming deterministic deterioration rates, is a hyperbolic equation while the second one assuming a stochastic deterioration rate is reduced to a parabolic equation of the second order. Optimization of these in inventory models when costs are quadratic yields linear feedback replenishment policies. This short paper provides an analytic solution to both problems. The importance of such inventory models is to be found when the demand and value of stored products is also dependent on their quality.

The existence and uniqueness of the solution to the Bellman equation for ergodic control of one-dimensional diffusions is established under a ‘near-monotonicity’ condition on the cost. Necessary and sufficient conditions for the optimality of a stable Markov control are given in terms of the same.

In this article we give necessary and sufficient conditions of optimality for linear stochastic distributed parameter systems, with convex differentiable payoffs and partial observation. They are obtained through variational methods, which can be applied only in the case of fixed information (i.e., not dependent on the control or the state). However, by a density argument it is proven that an optimal control adapted to the observation is also optimal for a space of controls adapted to some fixed information. Therefore we can get the necessary and sufficient conditions also in the case of feedback controls. We then prove, as a consequence, the separation principle for distributed parameter systems in the case of a quadratic payoff.

A method for constructing observers for dynamical systems as asymptotic limits of filters is described. The program is carried out in detail for linear systems, and in addition an observer is obtai...

An outline of impulsive control and its applications in management is reviewed. Conditions for optimal impulsive controls when the dynamic process is given by a stochastic differential equation with Wiener and jump processes are given. Applications to inventory control, capacity expansion, vehicle dispatching, maintenance-replacement-inspection, and pricing problems are formulated, and optimality conditions found. Although there are many other applications that can be outlined, these provide some motivation for further study in this emerging field.

We consider a continuous-time stochastic control problem with partial observations. Given some assumptions, we reduce the problem in successive approximation steps to a discrete-time, complete-observation, stochastic control problem with a finite number of possible states and controls. For the latter problem an optimal control can always be explicitly computed. Convergence of the approximations is shown, which in turn implies that an optimal control for the last-stage approximating problem is ∈-optimal for the original problem.

We shall consider in this article a general class of stochastic PDE which in particular covers the Zakai equation of nonlinear filtering and natural formulations of distributed systems involving control variables.

The problem of nonlinear filtering is studied for class of diffusions whose statistics depend periodically on the process state and a small parameter e. The problem exhibits a homogenization property as e↓0.

Composite control originally proposed in a deterministic context is generalized to the problem with white-noise inputs. However, the approach used here is radically different from the deterministic approach. Presence of noise smoothes the system behavior and allows a more complete solution than in the deterministic case.