Class Of Singular Control Problems Research Articles

On Singular Control Problems, the Time-Stretching Method, and the Weak-M1 Topology

We consider a general class of singular control problems with state constraints. Budhiraja and Ross [Ann. Appl. Probab., 16 (2006), pp. 2235--2255] established the existence of optimal controls for...

On a Class of Impulsive Control Problems for Continuity Equations

Abstract The talk presents a class of singular control problems for the continuity equation driven by a control-affine vector fields subject to a constraint on the Li-norm of control inputs, ranged in the whole space. Solutions of such distributed systems may occur to be arbitrary close (in a certain natural sense) to discontinuous measure-valued functions, and — as a consequence — related extremal problems are generically ill-posed. In connection with the addressed model, we discuss the following control-theoretical issues: i) relaxation of the tube of solutions in an appropriate coarse topology; ii) representation of generalized (discontinuous in time) solutions in terms of continuous arcs through a discontinuous time reparameterization of the characteristic ordinary differential equation, and iii) a constructive formula for generalized solutions. For the relaxed model, we state an optimal impulsive ensemble control problem and ensure the existence of a minimizer. Finally, we elaborate a conceptual numeric technique for optimal control and exhibit a case study.

A solvable singular control problem driven by a jump diffusion process with applications

ABSTRACTWe consider a class of singular control problems driven by a double exponential jump diffusion process, which come from the reversible investment problem. In some interesting cases (e.g., the running cost function is given by the so-called Cobb-Douglas production function), we give the explicit solutions to the singular control problem by using the connection between singular control and optimal switching. We solve a collection of consistent optimal switching problems and yield the explicit solution for the singular control problem. We then give an application to a particular inventory control problem in a single random period.

Optimal multi-dimensional stochastic harvesting with density-dependent prices

We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a “chattering policy”, i.e. the limit as \(\Delta x\) and \(\Delta t\) go to \(0\) of taking out a sequence of small quantities of size \(\Delta x\) within small time periods of size \(\Delta t\).

Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems

This work develops optimal harvest strategies for Lotka–Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. To better reflect reality, a continuous-time Markov chain is used to model the random environment. The underlying systems are thus controlled regime-switching diffusions that belong to the class of singular control problems. Starting with a model having multiple species, we construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. Moreover, in some cases, the lower bounds provide a good approximation of the value functions.

Optimal stopping and free boundary characterizations for some Brownian control problems

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is $C^1$ and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133--2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is $C^2$ in the interior of the no-action region. We then use a verification argument applied to a suitable $C^2$ approximation of the value function to establish optimality of the conjectured control.

Solving Singular Control from Optimal Switching

This report summarizes some of our recent work (Guo and Tomecek, SIAM J Control Optim 47(1):421–443, 2008; A class of singular control problems and smooth fit principle, 2008) on a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence not only provides a novel method for analyzing multi-dimensional singular control problems, but also builds links among singular controls, Dynkin games, and sequential optimal stopping problems.

A Class of Singular Control Problems and the Smooth Fit Principle

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and continuation regions are smooth and strictly monotonic, as postulated and exploited in the existing literature (see [A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994]; [M. H. A. Davis et al., Adv. in Appl. Probab., 19 (1987), pp. 156-176]; [T. O. Kobila, Stochastics Stochastics Rep., 43 (1993), pp. 29-63]; [A. B. Abel and J. C. Eberly, J. Econom. Dynam. Control, 21 (1997), pp. 831-852]; [A. Oksendal, Finance Stoch., 4 (2000), pp. 223-250]; [J. A. Scheinkman and T. Zariphopoulou, J. Econom. Theory, 96 (2001), pp. 180-207]; [A. Merhi and M. Zervos, SIAM J. Control Optim., 46 (2007), pp. 839-876]; and [L. H. Alvarez, A General Theory of Optimal Capacity Accumulation under Price Uncertainty and Costly Reversibility, Working Paper, Helsinki Center of Economic Research, Helsinski, Finland, 2006]). Illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions.

Existence of optimal controls for singular control problems with state constraints

1. Introduction. This paper is concerned with a class of singular control problems with state constraints. The presence of state constraints, a key feature of the problem, refers to the requirement that the controlled diffusion process take values in a closed convex cone at all times [see (3)]. We consider an infinite horizon discounted cost of the form (4). The main objective of the paper is to establish the existence of an optimal control. Singular control is a well-studied but rather challenging class of stochastic control problems. We refer the reader to [7], especially the sections at the end of each chapter, for a thorough survey of the literature. Classical compactness arguments that are used for establishing the existence of optimal controls for problems with absolutely continuous control terms (cf. [8]) do not naturally extend to singular control problems. For one-dimensional models, one can typically establish existence constructively, by characterizing an optimally controlled process as a reflected diffusion (cf. [2, 3, 15]). In higher dimensions, one approach is to study the regularity of solutions of variational inequalities associated with singular control problems and the smoothness of the corresponding free boundary. Such smoothness results are the starting points in the characterization of the optimally controlled process as a constrained diffusion with reflection at the free boundary. Excepting specific models (cf. [30, 31]), this approach encounters substantial difficulties, even for linear dynamics (cf. [32]); a key difficulty is that little is known about the regularity of the free boundary in higher dimensions. Alternative approaches for establishing the existence of optimal controls based on compactness arguments are developed in [12, 17, 25]. The first of these papers considers linear dynamics, while the last two consider models with nonlinear coefficients. In all

Observation Control Problem for Discrete-Continuous Systems as a Singular Control Problem

Abstract The problem of the estimation and observation control is considered for the stochastic system with a state-estimation dependent noise in observations. The optimal estimation in the class of linear filters was obtained and the separation principle for the linear-quadratic optimization problem was also proved. It gives the opportunity to solve simultaneously the control problem for the process and observations. This new problem belongs to a class of singular control problems and can be solved by standard methods of impulsive control theory.

Approximate solution of a class of singular control problems

A class of singular control problems involving amplitude constraints on the controls is examined. IfL ∞ is the space of control functionsU, the control constraint setS can be identified with the unit ball inL ∞. Now, for anyn ∈ (1, ∞), an analogous problem may be set up withL n forU and the unit ball inL n forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn → ∞ is investigated.

Computation of optimal singular controls

A class of singular control problems is made nonsingular by the addition of an integral quadratic functional of the control to the cost functional; a parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon &gt; 0</tex> multiplies this added functional. The resulting nonsingular problem is solved for a monotonically decreasing sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{\epsilon; \epsilon_{1} &gt; \epsilon_{2} &gt; ... &gt; \epsilon_{k} &gt; 0\}</tex> . As <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k \rightarrow \infty</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon_{k} \rightarrow 0</tex> the solution of the modified problem tends to the solution of the original singular problem. A variant of the method which does not require that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon \rightarrow 0</tex> is also presented. Four illustrative numerical examples are described.