Distributed control for a class of non-Newtonian fluids

We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown.     

A stochastic linear–quadratic problem with Lévy processes and its application to finance

We study a Linear–Quadratic Regulation (LQR) problem with Lévy processes and establish the closeness property of the solution of the multi-dimensional Backward Stochastic Riccati Differential Equation (BSRDE) with Lévy processes. In particular, we consider multi-dimensional and one-dimensional BSRDEs with Teugel’s martingales which are more general processes driven by Lévy processes. We show the existence and uniqueness of solutions to the one-dimensional regular and singular BSRDEs with Lévy processes by means of the closeness property of the BSRDE and obtain the optimal control for the non-homogeneous case. An application of the backward stochastic differential equation approach to a financial (portfolio selection) problem with full and partial observation cases is provided.     

Analysis of a multistate control problem related to food technology

This paper is concerned with an optimal control problem related to the determination of an optimal profile for the steam temperature into the autoclave along the processing of canned foods. The problem studies a system coupling the evolution Navier-Stokes equations with the heat transfer equation by natural convection (the so-called Boussinesq equations), and with the microorganisms removal equation. The essential difficulties in the study of this multistate control problem arise from the lack of uniqueness for the solution of the state system. Here we obtain—after a careful analysis of the problem mathematical formulation—the uniqueness of part of the state, and the existence of optimal solutions.     

A diffusion approximation result for two parameter processes

Summary We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.Partially supported by ONR N00014-91-J-1010     

Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

Third-order nilpotency, nice reachability and asymptotic stability

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].     

Dynamic programming for stochastic target problems and geometric flows

Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed. Received October 24, 2000 / final version received July 24, 2001?Published online November 27, 2001     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which is determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with a finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.     

Variational approach and optimal control of a PEM fuel cell

The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions.     

Entrance laws for Feller diffusions on (0, ∞) and Doob's h-path transformation

We consider the problem of constructing entrance laws for Feller diffusions on the state space (0, ∞). Our method, based on Feller-McKean theory of one-dimensional diffusions, gives an analytic expression for the entrance density in terms of transition density. Moreover, the entrance density is the density of the first passage time to the left boundary {0}. Also, the entrance density is related to the transition density via Doob's h-path transformation.     

Well-posedness and spectral properties of heat and wave equations with non-local conditions

We consider the one-dimensional heat and wave equations but – instead of boundary conditions – we impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain eigenvalue asymptotics of so-called Weyl?s type.     

Large time asymptotic problems for optimal stochastic control with superlinear cost

The paper is concerned with stochastic control problems of finite time horizon whose running cost function is of superlinear growth with respect to the control variable. We prove that, as the time horizon tends to infinity, the value function converges to a function of variable separation type which is characterized by an ergodic stochastic control problem. Asymptotic problems of this type arise in utility maximization problems in mathematical finance. From the PDE viewpoint, our results concern the large time behavior of solutions to semilinear parabolic equations with superlinear nonlinearity in gradients.     

Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

State-dependent Foster–Lyapunov criteria for subgeometric convergence of Markov chains

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster–Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in Connor and Kendall (2007) [2] concerning the existence of so-called ‘tame’ Markov chains. Furthermore, we show that our ‘subsampled drift condition’ implies the existence of finite moments for the return time to a small set.     

Optimal dividend payments in the stochastic Ramsey model

We consider the dividend payments of a self-financing firm in the stochastic Ramsey model. The firm invests in capital stock and its production technology is given by the Cobb–Douglas function. Our objective is to maximize the expected present value of future real dividends subject to a positive constraint on the capital stock. We use the penalization method to obtain a solution for the variational inequality associated with the optimal growth problem and give a synthesis of the optimal dividend policy.     

Infinitesimal methods in control theory: Deterministic and stochastic

In Part I, methods of nonstandard analysis are applied to deterministic control theory, extending earlier work of the author. Results established include compactness of relaxed controls, continuity of solution and cost as functions of the controls, and existence of optimal controls. In Part II, the methods are extended to obtain similar results for partially observed stochastic control. Systems considered take the form:where the feedback control u depends on information from a digital read-out of the observation process y. The noise in the state equation is controlled along with the drift. Similar methods are applied to a Markov system in the final section.     

Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.     

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.