Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.     

具有反馈控制的N种群非自治LOTKA-VOLTERRA竞争系统的全局吸引性

讨论了非自治N种群Lotka-volterra竞争反馈控制模型,主要采用构造适当的Lyapunov泛函的方法,同时应用Barbalat引理得到了系统全局吸引的判别准则,而且给出了周期系统存在全局吸引的正周期解的充分条件,最后利用数值模拟验证了所得结论.     

Optimal Control of the Periodic String Equation with Internal Control

This paper is concerned with the existence and the maximum principle for optimal control problems governed by the periodic vibrating string equation on (0, )×(0, T) with Dirichlet boundary conditions. The case of internal controllers supported on (0, ) is examined.     

Partially Observed Analytic Systems with Fully Unbounded Actuators and Sensors-FEM Algorithms

Partially observed control systems described by analytic semigroup are considered. Finite-dimensional feedback control based on FEM approximations and accounting for incomplete observations is constructed. It is shown that this feedback control provides uniform stability (in time) of the originally unstable system. The main novel feature of the problem is that both—control and observation operators—are modeled by fully unbounded operators as they frequently arise in modeling of smart sensors and actuators. This contributes to technical difficulties at the level of perturbation theory for analytic semigroups. It is shown that a careful and rather special approximation in the area of support of the unbounded control/observation operators allows to obtain the right stability estimates. Theoretical results are illustrated with several examples of control problems governed by heat and plate equations.     

On the approximate calculation of time-minimal distributed controls of vibrations by finite element approximations

This paper is concerned with distributed null-control of vibrations governed by an abstract wave equation. Based on a method for the exact computation of minimumL ²-norm controls for given time intervals and time-minimal controls which are bounded with respect to theL ²-norm, an approximation method is developed which is based on Galerkin's method ana convergence results are derived.This paper is based on U. Lamp's doctoral dissertation and was supported by the Deutsche Forschungsgemeinschaft.     

On extremal solutions to stochastic control problems,II

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.     

Multitime Controllability,Observability and Bang-Bang Principle

Control problems for multitime first-order PDE arise in many different contexts and ways. The obstruction of complete integrability conditions (path independent curvilinear integrals) has determined the mathematicians to study such problems only in the discrete context, though thus they loose the geometrical character which is proper to the continuous approach. In this paper, we study controllability, observability and bang-bang properties of multitime completely integrable autonomous linear PDE systems, overcoming the existent mathematical prejudices regarding the importance of a multitime evolution of m-flow type. Our geometrical arguments show that each basic theorem has a correspondent in the case of a single-time linear controlled ODE system. The main results include controllability criteria, equivalence between controllability of a PDE system and observability of the dual PDE system, geometry of the control set, extremality and multitime bang-bang principle. All of these show that the passing from controlled single-time evolution (1-flow) to the controlled multitime evolution (m-flow) is not trivial. Changing the geometrical language, the case of nonholonomic evolution can be recovered easily from our theory.     

On the Approximate Controllability of Some Semilinear Parabolic Boundary-Value Problems

We prove the approximate controllability of several nonlinear parabolic boundary-value problems by means of two different methods: the first one can be called a Cancellation method and the second one uses the Kakutani fixed-point theorem. Accepted 10 June 1996     

Control Localized on Thin Structures for the Linearized Boussinesq System

We study optimal control problems for the linearized Boussinesq system when the control is supported on a submanifold of the boundary of the domain. This type of problem belongs to the class of optimal control problems with measures as controls, which has been studied recently by several authors. We are mainly interested in the optimality conditions for such problems. It is known that the differentiability properties needed to obtain the optimality conditions are more demanding, in terms of regularity of the data, than what is needed to prove the existence of optimal controls. Here we are able to derive the optimality conditions by taking advantage of the particular structure of the controls.     

On the Geometric Properties of Reachable and Controllable Sets for Linear Discrete Systems

A linear time-invariant discrete system with a single bounded control input is considered. Various types of constraints imposed on the control input are investigated. The set of system states that can be reached from the origin (reachable set) is studied. The set of system states from where the origin can be reached (null-controllable set) is studied as well. Various examples are presented.