a diffusion model for optimal portfolio selection in the presence of brokerage fees we consider a financial market model with two assets one has deterministic rate of growth while the rate of growth of the second asset is governed by a brownian motion with drift we can shift money from one asset to another however there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset we want to maximize the expected rate of growth of funds it is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort

explicit solution of a stochastic irreversible investment problem and its moving threshold we consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity the firm aims to maximize its expected total discounted real profit net of investment on a finite horizont the capacity is modeled as a controlled ita process where the control is the real investment which is not necessarily a rate but more generally a monotone process the result is a singular stochastic control problem we introduce the associated optimal stopping problem that is the optimal cost of not investing its variational formulation turns out to be a parabolic obstacle problem which we explicitly solve in the case of constant relative risk aversion crra production functions the moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital s replacement cost then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits

linear quadratic differential games with cheap control abstract this paper considers a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero it is shown that if a certain minimum phase condition is satisfied then the value of the game will approach zero as the weighting on the minimizing control approaches zero

fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost in this paper a guaranteed cost fuzzy impulsive control gcfic problem is addressed for uncertain continuous time nonlinear systems which can be represented by the takagi sugeno t s fuzzy model with parametric uncertainties based on the t s fuzzy model a novel time varying lyapunov function is initially constructed to derive the existence condition of guaranteed cost fuzzy impulsive controllers which cannot only exponentially stabilize the uncertain fuzzy system but also provide an upper bound on the quadratic cost function then two procedures for designing suboptimal guaranteed cost fuzzy impulsive controllers are given in the sense of minimizing an upper bound of the cost function one casts the controller design into a parameter dependent linear matrix inequality lmi optimization problem and the other casts the controller design into a sequential minimization problem subject to lmi constraints by using the cone complementary linearization ccl algorithm finally an example is presented to illustrate the effectiveness of the proposed method

singular stochastic control linear diffusions and optimal stopping a class of solvable problems we consider a class of singular stochastic control problems arising frequently in applications of stochastic control we state a set of conditions under which the optimal policy and its value can be derived in terms of the minimal r excessive functions of the controlled diffusion and demonstrate that the optimal policy is of the standard local time type we then state a set of weak smoothness conditions under which the value function is increasing and concave and demonstrate that given these conditions increased stochastic fluctuations decrease the value and increase the optimal threshold thus postponing the exercise of the irreversible policy in line with previous studies of singular stochastic control we also establish a connection between singular control and optimal stopping and show that the marginal value of the singular control problem coincides with the value of the associated stopping problem whenever 0 is not a regular boundary for the controlled diffusion

a stabilization algorithm for a class of uncertain linear systems abstract this paper presents an algorithm for the stabilization of a class of uncertain linear systems the uncertain systems under consideration are described by state equations which depend on time varying unknown but bounded uncertain parameters the construction of the stabilizing controller involves solving a certain algebraic riccati equation furthermore the solution to this riccati equation defines a quadratic lyapunov function which is used to establish the stability of the closed loop system this leads to a notion of quadratic stabilizability it is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable the paper also deals with a notion of overbounding for uncertain linear systems this procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process

a multidimensional stochastic singular control problem via dynkin game and dirichlet form the traditional difficulty with stochastic singular control is characterizing the regularities of the value function and the optimal control policy in this paper a multidimensional singular control problem is considered we found the optimal value function and the optimal control policy of this problem via a dynkin game whose solution is given by the saddle point of the cost function the existence and uniqueness of the solution to this dynkin game are proved through an associated variational inequality problem involving dirichlet form as a consequence the properties of the value function of this dynkin game imply the smoothness of the value function of the stochastic singular control problem in this way we are able to show the existence of a classical solution to this multidimensional singular control problem which was traditionally solved in the sense of viscosity solutions and this enables the application of the verification theorem to prove optimality

instantaneous control of brownian motion a controller continuously monitors a storage system such as an inventory or bank account whose content z zt t 0 fluctuates as a 2 brownian motion in the absence of control holding costs are incurred continuously at rate h zt at any time the controller may instantaneously increase the content of the system incurring a proportional cost of r times the size of the increase or decrease the content at a cost of l times the size of the decrease we consider the case where h is convex on a finite interval and h outside this interval the objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon it is shown that there exists an optimal control limit policy characterized by two parameters a and b a b roughly speaking this policy exerts the minimum amounts of control sufficient to keep zt a b for all t 0 put another way the optimal control limit policy imposes on z a lower reflecting barrier at a an

minimizing infinite time horizon discounted cost with mean variance and bounded variation controls we consider an infinite time horizon discounted cost minimization problem for a class of ito processes the available controls are drift and diffusion coefficients and the added bounded variation process we show that the value function is twice continuously differentiable and derive an optimal policy which has feedback type drift and diffusion coefficients when the absolute value of the optimal drift grows faster than the running cost function the optimal bounded variation process is identically zero when it grows weaker than the running cost function optimal bounded variation process is a local time type process in this case we relate the control problem with an optimal stopping problem we also establish the abelian limit relations between the value functions of the discounted cost problem and the stationary problem

characterization of the optimal boundaries in reversible investment problems this paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good our model allows for general diffusion dynamics on the demand as well as general cost functional the resulting optimization problem leads to a degenerate two dimensional bounded variation singular stochastic control problem for which explicit solution is not available in general and the standard verification approach cannot be applied a priori we use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution in the quadratic cost case we are able to prove a smooth fit c 2 property which gives rise to a full characterization of the optimal boundaries and value function

a bounded variation control problem for diffusion processes we consider an infinite horizon discounted cost minimization problem for a one dimensional stochastic differential equation model the available control is an added bounded variation process the cost structure involves a running cost function and a proportional cost for the use of the control process the running cost function is not necessarily convex we obtain sufficient conditions to guarantee the optimality of the zero control also for unbounded cost functions we provide sufficient conditions which make our optimal state process a reflecting diffusion on a compact interval in both cases the value function is a c2 function for bounded cost functions under additional assumptions we obtain a complex optimal strategy which turned out to be a mixture of jumps and local time type processes in this case we show that the value function is only a c1 function and that it fails to be a c2 function we also discuss a related variance control problem

technical communique delay dependent output feedback guaranteed cost control for uncertain time delay systems this paper considers the problem of output feedback guaranteed cost controller design for uncertain time delay systems by representing the time delay system in the descriptor form and using a recent result on bounding of cross products of vectors we obtain new delay dependent sufficient conditions for the existence of the guaranteed cost output feedback controller in terms of matrix inequalities a numerical algorithm is proposed to construct a full order output feedback controller achieving a suboptimal guaranteed cost such that the system can be stabilized for all admissible uncertainties an example is presented which shows that the proposed method can produce a lower guaranteed cost than the delay independent method

connections between optimal stopping and singular stochastic control ii reflected follower problems the stochastic control problem of following a brownian path by a process of bounded variation and subject to a reflecting barrier at the origin reflected follower is reduced to a question of optimal stopping with absorption direct probabilistic arguments are used to establish the equivalence of the two problems under suitable conditions on the cost functions

parabolic variational inequality with parameter and gradient constraints abstract this paper concerns a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints the method is to transform a two dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies moreover we analyze the behaviors of the free boundary surfaces the main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter

an abelian limit approach to a singular ergodic control problem we consider an ergodic stochastic control problem for a class of one dimensional ito processes where the available control is an added bounded variation process the corresponding infinite horizon discounted control problem was solved in a weerasinghe siam j control optim 44 2005 pp 389 417 here we show that as the discount factor approaches zero the optimal strategies derived in a weerasinghe siam j control optim 44 2005 pp 389 417 converge to an optimal strategy for the ergodic control problem under different assumptions two types of optimal strategies were derived also the abelian limit relationships among the ergodic control problem the infinite horizon discounted control problem and the finite time horizon control problem are established here a solution to a constrained optimization problem is obtained as an application

optimal switching in an economic activity under uncertainty this paper considers the problem of finding the optimal sequence of opening starting and closing stopping times of a multi activity production process given the costs of opening running and closing the activities and assuming that the state of the economic system is a stochastic process the problem is formulated as an extended impulse control problem and solved using stochastic calculus as an application the optimal starting and stopping strategy are explicitly found for a resource extraction when the price of the resource is following a geometric brownian motion

long term optimal investment strategies in the presence of adjustment costs we consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment in particular we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project s output commodity we model this economic indicator by means of a general one dimensional ergodic diffusion at any time the project s capacity level can be increased or decreased at given proportional costs the aim is to maximize an ergodic performance criterion that reflects the long term average payoff resulting from the project s management we solve this genuinely two dimensional stochastic control problem by constructing an explicit solution to an appropriate hamilton jacobi bellman equation and by fully characterizing an optimal investment strategy

singular optimal controls of stochastic recursive systems and hamilton jacobi bellman inequality abstract in this paper we study the optimal singular controls for stochastic recursive systems in which the control has two components the regular control and the singular control under certain assumptions we establish the dynamic programming principle for this kind of optimal singular controls problem and prove that the value function is a unique viscosity solution of the corresponding hamilton jacobi bellman inequality in a given class of bounded and continuous functions at last an example is given for illustration

asymptotically optimal admission control of a queue with impatient customers we consider a gi gi 1 queue with impatient customers in heavy traffic we use the solution of an approximating singular diffusion control problem to construct an admission control policy for the queue the approximating control problem does not admit a so called pathwise solution hence the resulting admission control policy depends on second moment data we prove asymptotic optimality of the constructed policy using weak convergence methods rh

on the principle of smooth fit for a class of singular stochastic control problems for diffusions this paper considers the principle of smooth fit for a class of one dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type the existence and the uniqueness of a convex c 2 solution to the corresponding variational inequality are obtained it is proved that this solution gives the value function of the control problem and the optimal control process is constructed as an example of the degenerate case it is proved that the conclusion is also true for linear systems and the explicit formula for the smooth fit points is derived

a model for reversible investment capacity expansion we consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and or demand fluctuations we model market uncertainty by means of a geometric brownian motion and we consider general running payoff functions our model allows for capacity expansion as well as for capacity reduction with each of these actions being associated with proportional costs the resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly we illustrate our results by means of the so called cobb douglas production function the problem that we study presents a model in which the associated hamilton jacobi bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function which may be identically equal to infty thus our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics as well as in other fields

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

optimal control under a dynamic fuel constraint we present a new approach to solve optimal control problems of the monotone follower type the key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint instead of dynamic programming we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the snell envelope of the objective functional s gradient at the optimum the optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in bank and el karoui 2004 ann probab 32 pp 1030 1067 as an illustration we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics

delay dependent admissible consensualization for singular time delayed swarm systems abstract admissible consensus analysis and design problems for high order linear time invariant singular swarm systems with time delays are investigated firstly by state decomposition the admissible consensus problem is transformed into admissible problems of multiple singular subsystems with lower dimensions then linear matrix inequality lmi criteria for admissible consensualization are presented which only involve eight lmi constraints independent of the number of agents moreover an explicit expression of the consensus function which is independent of time delays is presented and the impacts of protocol states and interaction topologies on the consensus function are revealed finally a numerical example is given to illustrate the effectiveness of theoretical results

delay dependent robust stabilization of uncertain state delayed systems this paper concerns a problem of robust stabilization of uncertain state delayed systems a new delay dependent stabilization condition using a memoryless controller is formulated in terms of matrix inequalities an algorithm involving convex optimization is proposed to design a controller guaranteeing a suboptimal maximal delay such that the system can be stabilized for all admissible uncertainties

output feedback stabilisation for uncertain nonlinear time delay systems subject to input constraints robust stabilisation of a class of imperfectly known systems with time varying time delays via output feedback is investigated the systems addressed are composed of a nonlinear nominal system influenced by nonlinear perturbations which may be time state delayed state and or input dependent the output of the system is modelled by a nonlinear function which may depend on the delayed states and inputs together with a feed through term using bounding information on the perturbations in terms of specified growth conditions classes of unconstrained and constrained output feedback controllers are designed in order to guarantee a prescribed stability property for the closed loop systems provided appropriate stability criteria hold two stability criteria are given one in terms of a linear matrix inequality lmi and the other is algebraic in nature obtained using a gersgorin theorem

connections between singular control and optimal switching this paper builds a new theoretical connection between singular control of finite variation and optimal switching problems this correspondence provides a novel method for solving high dimensional singular control problems and enables us to extend the theory of reversible investment sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions consequently our regularity result links singular controls and dynkin games through sequential optimal stopping problems

technical communique an lmi approach to guaranteed cost control of linear uncertain time delay systems the guaranteed cost control problem via memoryless state feedback controllers is studied in this paper for a class of linear time delay systems with norm bounded time varying parametric uncertainty and a given quadratic cost function sufficient conditions for the existence of guaranteed cost controllers are given in terms of linear matrix inequalities lmis furthermore a convex optimization problem with lmi constraints is formulated to design the optimal guaranteed cost controller which minimizes the guaranteed cost of the closed loop uncertain system

a spectral radius formula for the fourier transform on compact groups and applications to random walks abstract let g be a compact group not necessarily abelian let be its unitary dual and for f l1 g let f n f f denote n fold convolution of f with itself and f the fourier transform of f in this paper we derive the following spectral radius formula f n 1 1 n max r g f r where f r is the spectral radius of f r thereby extending the well known beurling gelfand spectral radius formula for the fourier transform on a locally compact abelian group we also establish a partial result in this direction for arbitrary regular borel measures on g as applications we give conceptual and rather short proofs of existing results concerning convergence in the total variation norm of random walks on compact groups and uniform convergence of the corresponding densities we also describe how these results may be transferred to homogeneous spaces

singular control of stochastic linear systems with recursive utility we formulate a class of singular stochastic control problem with recursive utility where the cost function is determined by a backward stochastic differential equation some characteristics of the value function of the control problem are obtained by the method of approximation via penalization and the optimal control process is constructed

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

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

impulse control of multidimensional jump diffusions this paper studies regularity properties of the value function for an infinite horizon discounted cost impulse control problem where the underlying controlled process is a multidimensional jump diffusion with possibly infinite activity jumps surprisingly despite these jumps we obtain the same degree of regularity as for the diffusion case at least when the jump satisfies certain integrability conditions

explicit solution to an optimal switching problem in the two regime case this paper considers the problem of determining the optimal sequence of stopping times for a diffusion process subject to regime switching decisions this is motivated in the economics literature by the investment problem under uncertainty for a multi activity firm involving opening and closing decisions we use a viscosity solutions approach combined with the smooth fit property and explicitly solve the problem in the two regime case when the state process is of geometric brownian nature the results of our analysis take several qualitatively different forms depending on model parameter values