Equations In Finite-dimensional Spaces Research Articles

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n n -player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland’s variational principle on the Wasserstein space.

Identifying the heat sink

<p style='text-indent:20px;'>In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.</p>

Heuristic parameter choice rule for solving linear ill-posed integral equations in finite dimensional space

A new heuristic parameter choice rule is proposed, which is an important process in solving the linear ill-posed integral equation. Based on multiscale Galerkin projection, we establish the error upper bound between the approximate solution obtained by this rule and the exact solution. Under certain conditions, we prove that the approximate solution obtained by this rule can reach the optimal convergence rate. Since the computational cost will be very large when the dimension of space increases, we analyze a special m-dimensional integral operator that can be transformed to m one-dimensional integral operator, which can reduce the computational cost greatly. Numerical experiments show that the proposed heuristic rule is promising among the known heuristic parameter choice rules.

On Multivalent Guiding Functions Method in the Periodic Problem for Random Differential Equations

By applying the random topological degree theory we develop the methods of random multivalent guiding functions and use them for the study of periodic solutions for random differential equations in finite dimensional spaces.

Оптимізація функціоналів в умовах невизначеності для стохастичних диференціальних рівнянь Іто-Скорохода у гільбертових просторах

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.

Maximum principle for partially-observed optimal control problems of stochastic delay systems

This paper is concerned with partially-observed optimal control problems for stochastic delay systems. Combining Girsanov’s theorem with a standard variational technique, the authors obtain a maximum principle on the assumption that the system equation contains time delay and the control domain is convex. The related adjoint processes are characterized as solutions to anticipated backward stochastic differential equations in finite-dimensional spaces. Then, the proposed theoretical result is applied to study partially-observed linear-quadratic optimal control problem for stochastic delay system and an explicit observable control variable is given.

Differential optimization in finite-dimensional spaces

In this paper, a class of optimization problems coupled with differential equations in finite dimensional spaces are introduced and studied. An existence theorem of a Caratheodory weak solution of the differential optimization problem is established. Furthermore, when both the mapping and the constraint set in the optimization problem are perturbed by two different parameters, the stability analysis of the differential optimization problem is considered. Finally, an algorithm for solving the differential optimization problem is established.

Partially observed optimal controls of forward-backward doubly stochastic systems

The partially observed optimal control problem is considered for forward-backward doubly stochastic systems with controls entering into the diffusion and the observation. The maximum principle is proven for the partially observable optimal control problems. A probabilistic approach is used, and the adjoint processes are characterized as solutions of related forward-backward doubly stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully coupled forward-backward doubly stochastic system.

A maximum principle for partially observed optimal control of forward-backward stochastic control systems

This paper studies an optimal control problem for partially observed forward-backward stochastic control system with a convex control domain and the forward diffusion term containing control variable. A maximum principle is proved for this kind of partially observable optimal control problems and the corresponding adjoint processes are characterized by the solutions of certain forward-backward stochastic differential equations in finite-dimensional spaces. One partially observed recursive linear-quadratic (LQ) optimal control example is also given to show the application of the obtained maximum principle. An explicit observable optimal control is obtained in this example.

Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems

This paper is concerned with partially-observed optimal control problems for fully-coupled forward-backward stochastic systems. The maximum principle is obtained on the assumption that the forward diffusion coefficient does not contain the control variable and the control domain is not necessarily convex. By a classical spike variational method and a filtering technique, the related adjoint processes are characterized as solutions to forward-backward stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully-coupled forward-backward stochastic system and an explicit observable control variable is given.

The Maximum Principles for Stochastic Recursive Optimal Control Problems Under Partial Information

A maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables. Such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal-agent problems. By virtue of a classical spike variational approach and a filtering technique, the maximum principle is obtained, and the related adjoint processes are characterized by the solutions of forward-backward stochastic differential equations in finite-dimensional spaces. Then our theoretical results are applied to study a partially observed linear-quadratic recursive optimal control problem. In addition, for the case with initial and terminal state constraints, the corresponding maximum principle is also obtained by using Ekeland's variational principle.

Input constrained adaptive tracking for a nonlinear distributed parameter tubular reactor

AbstractIn this paper we address the problem of local λ tracking of a pre‐specified profile temperature for exothermic chemical reaction in a tubular reactor. In contrast to the various models in the literature, which are usually ordinary differential equations in finite‐dimensional spaces, we use a nonlinear distributed parameter model for both the evolution of the temperature and the reactant concentration. We show under nonrestrictive conditions that an adaptive output feedback controller achieves approximate asymptotic tracking where the pre‐specified asymptotic tracking accuracy, quantified by the design parameter λ>0, is guaranteed. Only a feasibility assumption in terms of the reference temperature and the input constraints is considered. Numerical simulations have been performed to illustrate the performance of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.

Antiperiodic Boundary Value Problems for Finite Dimensional Differential Systems

We study antiperiodic boundary value problems for semilinear differential and impulsive differential equations in finite dimensional spaces. Several new existence results are obtained.

A Newton-Kantorovich-SOR type theorem

In this paper we propose a new method for solving nonlinear systems of equations in finite dimensional spaces, combining the Newton-Raphson's method with the SOR idea. For the proof we adapt Kantorovich's demonstration given for the Newton-Raphson's method. As applications we reobtain the classical Newton-Raphson's method and the author's Newton-Kantorovich-Seidel type result.

General necessary conditions for partially observed optimal stochastic controls

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

General necessary conditions for partially observed optimal stochastic controls

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

Limiting Equations of Integrodifferential Equations in Banach Space

The "limiting equation" results for integrodifferential equations in finite dimensional space are extended to equations [formula] and [formula] in reflexive Banach space in a weak sense when A is a closed and densely defined operator and F( t) is a bounded operator t ≥ 0. The results are applied to study the periodic solutions of Eq. (b) on R . An application to a heat equation is also given.

Componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces

In this paper we use interval arithmetic tools for the computation of componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces. We define a mapping for which under certain assumptions we can construct an interval vector which is mapped into itself. Using Brouwer's fixed point theorem we conclude the existence of a solution of the original equation in this interval vector. Under different assumptions we can construct an interval vector such that the range of the mapping has no point in common with this interval vector. This implies that there is no solution in this interval vector. Furthermore we consider an iteration method which improves componentwise errorbounds for a solution of a quadratic. The theoretical results of this paper are demonstrated by some numerical examples using the algebraic eigenvalue problem which is probably the best known example of a quadratic equation.

Properties of monotone mappings

In the theories of ordinary differential equations, I to 's stochastic differential equations, and equations in Banach spaces one frequently uses the concept of monotone mapping, introduced in [ 1 ]. This concept is used not only in studying equations with monotone operators, but also in studying other equations (cf. [2-6]). The concept of monotonici ty turned out to be convenient, in the first place, because it is easy to pass to the limit under the sign of a monotone operator, and the investigation of certain equations or others in infinite-dimensional spaces with the help of the method of Galerkin usually reduces to the investigation of equations in finite-dimensional spaces. To the author 's mind there is interest in the detailed s tudy of propert ies of monotone mappings of a finite-dimensional space.

A finite difference technique for solving optimization problems governed by linear functional differential equations

Aspects of the approximation and optimal control of systems governed by linear retarded nonautonomous functional differential equations (FDE) are considered. First, certain FDE are shown to be equivalent to corresponding abstract ordinary differential equations (ODE). Next, it is demonstrated that these abstract ODE may be approximated by difference equations in finite dimensional spaces. The optimal control problem for systems governed by FDE is then reduced to a sequence of mathematical programming problems. Finally, numerical results for two examples are presented and discussed.