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Showing papers in "Siam Journal on Control and Optimization in 1995"


Journal ArticleDOI
TL;DR: In this article, the problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market, and the main result of this work is that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities.
Abstract: The problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market. When the market is incomplete, prices cannot be derived from the absence of arbitrage, since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities. In this situation, there is a price range for the actual market price of the contingent claim. The maximum and minimum prices are studied using stochastic control methods. The main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities. A similar result is obtained for the minimum price (which corresponds to the purchase price).

715 citations


Journal ArticleDOI
TL;DR: These tools for uniform semiglobal stabilization by partial state and output feedback are developed and the usefulness of considering local convergence separate from boundedness of solutions is demonstrated.
Abstract: We develop tools for uniform semiglobal stabilization by partial state and output feedback. We show, by means of examples, that these tools are useful for solving a variety of problems. One application is a general result on semiglobal output feedback stabilizability when global state feedback stabilizability is achievable by a control function that is uniformly completely observable. We provide more general results on semiglobal output feedback stabilization as well. Globally minimum phase input--output linearizable systems are considered as a special case. Throughout our discussion we demonstrate the usefulness of considering local convergence separate from boundedness of solutions. For the former we employ a sufficient small gain condition guaranteeing convergence. For the latter we rely on Lyapunov techniques.

619 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered stochastic control problems on an infinite time horizon with exponential cost criteria, and the Donsker-Varadhan large deviation rate was used as a criterion to optimize the optimum rate.
Abstract: Stochastic control problems on an infinite time horizon with exponential cost criteria are considered. The Donsker--Varadhan large deviation rate is used as a criterion to be optimized. The optimum rate is characterized as the value of an associated stochastic differential game, with an ergodic (expected average cost per unit time) cost criterion. If we take a small-noise limit, a deterministic differential game with average cost per unit time cost criterion is obtained. This differential game is related to robust control of nonlinear systems.

331 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown that if the regressor vector is constructed out of radial basis function approximants, it will be persistently exciting, provided a kind of "ergodic" condition is satisfied.
Abstract: In this paper, identification algorithms whose convergence and rate of convergence hinge on the regressor vector being persistently exciting are discussed. It is then shown that if the regressor vector is constructed out of radial basis function approximants, it will be persistently exciting, provided a kind of "ergodic" condition is satisfied. In addition, bounds on parameters associated with the persistently exciting regressor vector are provided; these parameters are connected with both the convergence and rates of convergence of the algorithms involved.

274 citations


Journal ArticleDOI
TL;DR: In this paper, the stabilizability problem for control stochastic nonlinear systems driven by a Wiener process is studied and sufficient conditions for the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium point of the system, are provided by means of stochiastic Lyapunov-like techniques.
Abstract: The purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a Wiener process. Sufficient conditions for the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium point of the system, are provided by means of stochastic Lyapunov-like techniques. The notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied.

253 citations


Journal ArticleDOI
TL;DR: This paper constructs a class of certainty equivalence control with forcing schemes and derive asymptotic upper bounds on their learning loss that are stronger than the $o(n)$ required for optimality with respect to the average-cost-per-unit-time criterion.
Abstract: In this paper we consider the multiarmed bandit problem where the arms are chosen from a subset of the real line and the mean rewards are assumed to be a continuous function of the arms. The problem with an infinite number of arms is much more difficult than the usual one with a finite number of arms because the built-in learning task is now infinite dimensional. We devise a kernel estimator-based learning scheme for the mean reward as a function of the arms. Using this learning scheme, we construct a class of certainty equivalence control with forcing schemes and derive asymptotic upper bounds on their learning loss. To the best of our knowledge, these bounds are the strongest rates yet available. Moreover, they are stronger than the $o(n)$ required for optimality with respect to the average-cost-per-unit-time criterion.

242 citations


Journal ArticleDOI
TL;DR: In this article, the rendezvous value of the region is defined as the probability that two players can find each other in the least expected time, and it is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise.
Abstract: The author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time. This time is called the rendezvous value of the region. It is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise. A general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players. These concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network. The discrete rendezvous problem on a cycle graph for players restricted to symmetric Markovian strategies is then solved. Finally, the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other.

232 citations


Journal ArticleDOI
TL;DR: In this article, it is proven that, for any positive time, if there exists an open-loop control with continuous time-varying feedback law, the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4.
Abstract: It is proven that, if, for any positive time $T$, there exists an open-loop control $u(a,t)$ depending continuously on the initial data $a$, vanishing for $a=0$, and steering a small neighborhood of $0$ into $0$ in time $T$, then the control system can be locally stabilized in small time by means of continuous time-varying feedback law, provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds.

151 citations


Journal ArticleDOI
TL;DR: In this article, a finite difference method for studying the Bolza problem with general endpoint constraints is developed and necessary optimality conditions for the intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima.
Abstract: This paper deals with the Bolza problem $(P)$ for differential inclusions subject to general endpoint constraints. We pursue a twofold goal. First, we develop a finite difference method for studying $(P)$ and construct a discrete approximation to $(P)$ that ensures a strong convergence of optimal solutions. Second, we use this direct method to obtain necessary optimality conditions in a refined Euler--Lagrange form without standard convexity assumptions. In general, we prove necessary conditions for the so-called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima. In the case of a Mayer cost functional or boundary solutions to differential inclusions, this Euler--Lagrange form holds without any relaxation. The results obtained are expressed in terms of nonconvex-valued generalized differentiation constructions for nonsmooth mappings and sets.

138 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that singularities in waves are "smoothed one order" as they cross a point mass, and the most general reachable space (from 0) that one can expect is $L^2\times H^{-1}$ to the left of the mass and $H^1\times L^2
Abstract: In this article we examine the problems of boundary control and stabilization for a one-dimensional wave equation with interior point masses. We show that singularities in waves are "smoothed one order" as they cross a point mass. Thus in the case of one interior point mass, with, e.g., $L^2$-Dirichlet control at the left end, the most general reachable space (from 0) that one can expect is $L^2\times H^{-1}$ to the left of the mass and $H^1\times L^2$ to the right of the mass. We show that this is in fact the optimal result (modulo certain compatibility conditions). Several related results for both control and stabilization of such systems are also given.

128 citations


Journal ArticleDOI
TL;DR: In this paper, a complete geometric characterization of the solvability conditions of disturbance decoupling problems for injective systems with coefficients in a commutative ring is given, and practical procedures for testing the conditions and for constructing solutions if any exist.
Abstract: Up to now the use of geometric methods in the study of disturbance decoupling problems (DDPs) for systems over a ring has provided only necessary conditions for the existence of solutions. In this paper we study such problems, considering separately the case in which only static feedback solutions are allowed, and the one in which dynamic feedback solutions are admitted. In the first case, we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring. Practical procedures for testing the solvability conditions and for constructing solutions, if any exist, are given in the case of systems with coefficients in a principal ideal domain (PID). In the second case, we give a complete geometric characterization of the solvability conditions for systems with coefficients in a PID.

Journal ArticleDOI
TL;DR: The properties of averaged functions are studied and a new notion of subgradient is introduced based on approximations generated by mollifiers and is exploited in the design of minimization procedures.
Abstract: To minimize discontinuous functions that arise in the context of systems with jumps, for example, we propose a new approach based on approximation via averaged functions (obtained by convolution with mollifiers). The properties of averaged functions are studied, after it is shown that they can be used in an approximation scheme consistent with minimization. A new notion of subgradient is introduced based on approximations generated by mollifiers and is exploited in the design of minimization procedures.

Journal ArticleDOI
TL;DR: In this article, a general formulation of nonholonomic control systems on a Riemannian manifold modeled by second-order differential equations and using the unique RiemANNian connection defined by the metric is given.
Abstract: This paper gives a general formulation of the theory of nonholonomic control systems on a Riemannian manifold modeled by second-order differential equations and using the unique Riemannian connection defined by the metric. The main concern is to introduce a reduction scheme, replacing some of the second-order equations by first-order equations. The authors show how constants of motion together with the nonholonomic constraints may be combined to yield such a reduction. The theory is applied to a particular class of nonholonomic control systems that may be thought of as modeling a generalized rolling ball. This class reduces to the classical example of a ball rolling without slipping on a horizontal plane.

Journal ArticleDOI
TL;DR: In this paper, the authors apply the compactification method to study the control problem where the state is governed by an Ito stochastic differential equation allowing both classical and singular control.
Abstract: We apply the compactification method to study the control problem where the state is governed by an Ito stochastic differential equation allowing both classical and singular control. The problem is reformulated as a martingale problem on an appropriate canonical space after the relaxed form of the classical control is introduced. Under some mild continuity hypotheses on the data, it is shown by purely probabilistic arguments that an optimal control for the problem exists. The value function is shown to be Borel measurable.


Journal ArticleDOI
TL;DR: In this paper, the authors considered a stochastic control problem with linear dynamics, convex cost criterion, and convex state constraint, in which the control entered both the drift and diffusion coefficients.
Abstract: This paper considers a stochastic control problem with linear dynamics, convex cost criterion, and convex state constraint, in which the control enters both the drift and diffusion coefficients. These coefficients are allowed to be random, and no $L^{p}$-bounds are imposed on the control. An explicit solution for the adjoint equation and a global stochastic maximum principle are obtained for this model. This is evidently the first version of the stochastic maximum principle that covers the consumption-investment problem. The mathematical tools are those of stochastic calculus and convex analysis. When it is assumed, as in other versions of the stochastic maximum principle, that the admissible controls are square-integrable, not only a necessary but also a sufficient condition for optimality is obtained. It is then shown that this particular case of the general model may be applied to solve a variety of problems in stochastic control, including the linear-regulator, predicted-miss, and Benes problems.

Journal ArticleDOI
TL;DR: In this article, the authors formalized the notion of optimal supervisory control of discrete event dynamical systems (DEDSs) in the framework of Ramadge and Wonham.
Abstract: The notion of optimal supervisory control of discrete event dynamical systems (DEDSs) is formalized in the framework of Ramadge and Wonham. A DEDS is modeled as a state machine and is controlled by disabling some of its transitions. Two types of cost functions are defined: a cost of control function corresponding to disabling transitions in the state machine, and a penalty of control function corresponding to reaching some undesired states or not reaching some desired states in the controlled system. The control objective is to design an optimal control mechanism, if it exists, so that the net cost is minimized. Since a DEDS is represented as a state machine---a directed graph---network flow techniques are naturally applied for designing optimal supervisors. It is also shown that our techniques can be used to solve supervisory control problems under complete as well as partial observation. In particular, for the first time, techniques for computing the supremal controllable and normal sublanguage and the infimal controllable and normal/observable superlanguage without having to perform alternate computations of controllable and normal/observable languages are obtained.

Journal ArticleDOI
TL;DR: In this paper, the joint placements that lead to exponential stability for coupled Euler-Bernoulli beams with a dissipative joint are characterized and a spectrum analysis of the zero dynamics of the associated controlled, observed system is performed.
Abstract: Two examples of coupled Euler-Bernoulli beams with a dissipative joint are considered. The joint placements that lead to exponential stability for these systems are characterized. The technique used shows input-output stability of a related controlled, observed system, and then shows that in these examples, input-output stability implies exponential stability. In the first example, the energy dissipation arises from a discontinuity in the shear at the joint. In the second example, the energy dissipation arises from a discontinuity in the bending moment at the joint. The analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled, observed system.

Journal ArticleDOI
TL;DR: It is established that any accumulation point of the parallel algorithm is stationary for the nonconvex case and is a global solution for the convex case.
Abstract: A parallel version is proposed for a fundamental theorem of serial unconstrained optimization. The parallel theorem allows each of $k$ parallel processors to use simultaneously a different algorithm, such as a descent, Newton, quasi-Newton, or conjugate gradient algorithm. Each processor can perform one or many steps of a serial algorithm on a portion of the gradient of the objective function assigned to it, independently of the other processors. Eventually a synchronization step is performed which, for differentiable convex functions, consists of taking a strong convex combination of the $k$ points found by the $k$ processors. A more general synchronization step, applicable to convex as well as nonconvex functions, consists of taking the best point found by the $k$ processors or any point that is better. The fundamental result that we establish is that any accumulation point of the parallel algorithm is stationary for the nonconvex case and is a global solution for the convex case. Computational testing on the Thinking Machines CM-5 multiprocessor indicates a speedup of the order of the number of processors employed.

Journal ArticleDOI
TL;DR: In this article, the dynamic programming principle for a multidimensional singular stochastic control problem is established for the case when assuming Lipschitz continuity on the data, and the value function is continuous and is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.
Abstract: The dynamic programming principle for a multidimensional singular stochastic control problem is established in this paper When assuming Lipschitz continuity on the data, it is shown that the value function is continuous and is the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation

Journal ArticleDOI
TL;DR: In this paper, the supervisory control of non-deterministic discrete event dynamical systems (DEDSs) with driven events in the setting of prioritized synchronization and trajectory models introduced by Heymann is studied.
Abstract: The supervisory control of nondeterministic discrete event dynamical systems (DEDSs) with driven events in the setting of prioritized synchronization and trajectory models introduced by Heymann are studied. Prioritized synchronization captures the notions of controllable, uncontrollable, and driven events in a natural way, and the authors use it for constructing supervisory controllers. The trajectory model is used for characterizing the behavior of nondeterministic DEDSs since it is a sufficiently detailed model (in contrast to the less detailed language or failures models), and serves as a language congruence with respect to the operation of prioritized synchronization. Results concerning controllability and observability in this general setting are obtained.

Journal ArticleDOI
TL;DR: In this article, the problem of boundary feedback stabilization of a Euler-Bernoulli beam with an endmass is considered and the lack of uniform stabilization is proved in the case of a clamped beam with the usual boundary feedbacks applied to the end with the mass.
Abstract: The problem of boundary feedback stabilization of a Euler-Bernoulli beam with an endmass is considered. Using a method of compact perturbation, the lack of uniform stabilization is proved in the case of a clamped beam with the usual boundary feedbacks applied to the end with the mass. Next, the uniform stabilization when the usual boundary feedbacks are applied to the end without the mass is proved. Also the uniform decay of energy by means of higher-order feedbacks applied to the end with the mass is established.

Journal ArticleDOI
TL;DR: In this article, the problem of motion and shape estimation of a moving body with the aid of a monocular camera is considered, and it is shown that the estimation problem reduces to a specific parameter estimation of perspective dynamical system.
Abstract: In this paper, we consider the problem of motion and shape estimation of a moving body with the aid of a monocular camera. We show that the estimation problem reduces to a specific parameter estimation of a perspective dynamical system. Surprisingly, the above reduction is independent of whether the data measured is the brightness pattern which the object produces on the image plane or whether the data observed are points, lines, or curves on the image plane produced as a result of discontinuities in the brightness pattern. Many cases of the perspective parameter estimation problem have been analyzed in this paper. These cases include a fairly complete analysis of a planar textured surface undergoing a rigid flow and an affine flow. These two cases have been analyzed for orthographic, pseudo-orthographic, and image-centered projections. The basic procedure introduced for parameter estimation is to subdivide the problem into two modules, one for "spatial averaging" and the other for "time averaging." The estimation procedure is carried out with the aid of a new "realization theory for perspective systems" introduced for systems described in discrete time and in continuous time. Finally, much of our analysis has been substantiated by computer simulation of specific algorithms developed in order to explicitly compute the parameters. Detailed simulation that would answer the perspective realizability question is a subject of future research.

Journal ArticleDOI
TL;DR: In this article, a minimum principle of Pontryagin's type under some stability conditions of the optimal cost with respect to the state constraints is derived by using Ekeland's principle.
Abstract: This paper deals with state-constrained optimal control problems governed by semilinear elliptic equations or variational inequalities. By using Ekeland's principle, a minimum principle of Pontryagin's type under some stability conditions of the optimal cost with respect to the state constraints is derived.

Journal ArticleDOI
TL;DR: In this paper, a unified approach for deriving many old and new error estimates for linear programs, linear complementarity problems, convex quadratic programs, and affine variational inequality problems is presented.
Abstract: In this paper, we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program. These error estimates provide a unified approach for deriving many old and new error estimates for linear programs, linear complementarity problems, convex quadratic programs, and affine variational inequality problems. The approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program. In a sense, even Robinson's result on the upper Lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program. As an application, we derive new (global) error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the case where the distribution of the players is a bounded, point, discrete, or finite mean distribution and obtain upper bounds or exact values for the least expected rendezvous time.
Abstract: Two players are placed on the real line at a distance $d$ with a distribution $F$ known to both. Neither knows the direction of the other, nor do they have a common notion of a positive direction on the line. We seek the least expected rendezvous time $R=R\left(F\right)$ in which they can meet, given maximum speeds of one. We consider the cases where $F$ is a bounded, point, discrete, or finite mean distribution. We obtain upper bounds or exact values for $R$ and in one case an optimality condition for search strategies. A connection with Beck's linear search problem is established.

Journal ArticleDOI
TL;DR: In this article, a generalized normal map is considered, and the associated generalized normal equation is studied, which provides a unified formulation of several generalized variational inequality and complementarity problems using degree theory.
Abstract: The class of normal maps was recently investigated by Robinson and Ralph in connection with the study of a variational inequality defined on a polyhedral set. In this paper a generalization of such a map is considered, and the associated generalized normal equation is studied. The latter provides a unified formulation of several generalized variational inequality and complementarity problems. Using degree theory, some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed. Specializations of the results to various applications are discussed.

Journal ArticleDOI
TL;DR: In this article, the authors discuss the nature of optimal solutions for a class of continuous linear programs called separated continuous linear Programs (SCLP), and show that under various different assumptions on the problem data there exist optimal solutions that are piecewise constant, piecewise polynomial, or, more generally, piece wise analytic.
Abstract: This paper discusses the nature of optimal solutions for a class of continuous linear programs called separated continuous linear Programs. It is shown that under various different assumptions on the problem data there exist optimal solutions that are piecewise constant, piecewise polynomial, or, more generally, piecewise analytic. These results are reminiscent of bang-bang results in linear optimal control.

Journal ArticleDOI
TL;DR: In this article, an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates.
Abstract: In this paper we first present an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates. We then consider a certainty equivalence adaptive linear-quadratic Gaussian feedback control law using the estimates generated by the nonlinear filter and demonstrate the global existence of solutions to the resulting closed-loop system. A stochastic Lyapunov analysis establishes that the certainty equivalence law stabilizes the Markov jump parameter linear system in the mean square average sense. The conditions for this result are that certain products of (i) the parameter process jump rate and (ii) the solution of the control Riccati equation and its second derivatives should be less than certain given bounds. An example is given where the controlled linear system has state dimension 2. Finally, the stabilizing properties of certainty equivalence laws which depend on (i) the maximum likelihood estimate of the parameter value and (ii) a modified version of this estimate are established under certain conditions.

Journal ArticleDOI
TL;DR: In this paper, a mixed strategy with an expected meeting time of 1.78D+\mu /2 was given, where D is the maximum of a bounded distribution and μ is its mean.
Abstract: Alpern introduced a problem in which two players are placed on the real line at a distance drawn from a bounded distribution $F$ known to both. They can move at maximum velocity one and wish to meet as soon as possible. Neither knows the direction of the other, nor do they have a common notion of a positive direction on the line. It is required to find the symmetric rendezvous value $R^{s}(F)$, which is the minimum expected meeting time achievable by players using the same mixed strategy. This corresponds to the case where the players are indistinguishable they both take directions from a controller who does not know their names. In this paper we give a mixed strategy which has an expected meeting time of $1.78D+\mu /2$, where $D$ is the maximum of $F$ and $\mu$ its mean. This leads to an upper bound $R^{s}(F)\le 1.78D+\mu /2$ on the symmetric rendezvous value, which is better than the upper bound $R^s(F)\le 2D+\mu /2$ obtained by Alpern.