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2012

Volume 50, Issue 1, pp. 1-599


A Lyapunov Inequality Characterization of and a Riccati Inequality Approach to $L_{\infty}$ and $L_{2}$ Low Gain Feedback

Bin Zhou, Zongli Lin, and Guang-Ren Duan

SIAM J. Control Optim. 50, pp. 1-22 (22 pages)

Online Publication Date: January 03, 2012

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This paper is concerned with a Lyapunov inequality characterization of the eigenstructure assignment–based low gain feedback laws. With this characterization and our earlier characterizations of other low gain feedback design approaches, all existing low gain feedback designs are unified under this Lyapunov inequality framework, which in turn implies that all of these low gain feedback laws are both $L_{\infty}$ and $L_{2}$ low gain feedback. This Lyapunov inequality characterization also leads to a quadratic Lyapunov function for the closed-loop system, which is expected to play an important role in solving other control problems. This characterization also motivates a new Riccati inequality–based low gain feedback design, which not only possesses the appealing features of the existing low gain designs but also is computationally easy to carry out.

Linear Programming and Constrained Average Optimality for General Continuous-Time Markov Decision Processes in History-Dependent Policies

Xianping Guo, Yonghui Huang, and Xinyuan Song

SIAM J. Control Optim. 50, pp. 23-47 (25 pages)

Online Publication Date: January 03, 2012

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This paper attempts to study the constrained average optimality for continuous-time Markov decision processes in the class of randomized history-dependent policies. The states and actions are in general Polish spaces, and the transition rates are allowed to be unbounded. The optimality criterion to be optimized is expected average costs, multiple constraints are imposed on similar expected average costs, and all costs may be unbounded from above and from below. Under suitable conditions, we first show the existence of a constrained optimal policy by improving the concept of a stable policy in the previous literature and using the analogue of the forward Kolmogorov equation. Then, we develop a linear program (LP), which is equivalent to the constrained optimality problem and is used to obtain a constrained optimal policy. By introducing suitable operators and conditions, we further establish the dual program (DP) of the LP, show that the LP and DP are solvable, and show that there is no duality gap between them. Finally, we use a cash flow model and a controlled birth and death system to illustrate the applications of our main results.

Stability and Resolution Analysis for a Topological Derivative Based Imaging Functional

Habib Ammari, Josselin Garnier, Vincent Jugnon, and Hyeonbae Kang

SIAM J. Control Optim. 50, pp. 48-76 (29 pages) | Cited 1 time

Online Publication Date: January 03, 2012

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The aim of this paper is to study a topological derivative based anomaly detection algorithm. We compare its performance with other imaging approaches such as MUltiple SIgnal Classification, backpropagation, and Kirchhoff migration. We also investigate its stability with respect to medium and measurement noises as well as its resolution. A simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional, both in terms of resolution and stability.

Mean Field Games: Numerical Methods for the Planning Problem

Yves Achdou, Fabio Camilli, and Italo Capuzzo-Dolcetta

SIAM J. Control Optim. 50, pp. 77-109 (33 pages)

Online Publication Date: January 03, 2012

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Mean field games describe the asymptotic behavior of differential games in which the number of players tends to $+\infty$. Here we focus on the optimal planning problem, i.e., the problem in which the positions of a very large number of identical rational agents, with a common value function, evolve from a given initial spatial density to a desired target density at the final horizon time. We propose a finite difference semi-implicit scheme for the optimal planning problem, which has an optimal control formulation. The latter leads to existence and uniqueness of the discrete control problem. We also study a penalized version of the semi-implicit scheme. For solving the resulting system of equations, we propose a strategy based on Newton iterations. We describe some numerical experiments.

Lyapunov Methods for Time-Invariant Delay Difference Inclusions

R. H. Gielen, M. Lazar, and I. V. Kolmanovsky

SIAM J. Control Optim. 50, pp. 110-132 (23 pages)

Online Publication Date: January 03, 2012

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Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov–Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is $\mathcal{KL}$-stable if and only if it admits a Lyapunov–Krasovskii function (LKF). Second, the Lyapunov–Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proved that a DDI is $\mathcal{KL}$-stable if it admits a Lyapunov–Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for $\mathcal{KL}$-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed.

Optimal Structural Policies for Ambiguity and Risk Averse Inventory and Pricing Models

Xin Chen and Peng Sun

SIAM J. Control Optim. 50, pp. 133-146 (14 pages)

Online Publication Date: January 05, 2012

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This paper discusses multiperiod stochastic joint inventory and pricing models when the decision maker is risk and ambiguity averse. We study infinite horizon models with discounted and long run average optimization criteria. The main result of this paper is establishing the optimality of stationary $(s,S,p)$ policies for the infinite horizon inventory and pricing models.

Optimal Control Models of Goal-oriented Human Locomotion

Yacine Chitour, Frédéric Jean, and Paolo Mason

SIAM J. Control Optim. 50, pp. 147-170 (24 pages)

Online Publication Date: January 17, 2012

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In recent papers it has been suggested that human locomotion may be modeled as an inverse optimal control problem. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem that has to be determined. We discuss the modeling of both the dynamical system and the cost to be minimized, and we analyze the corresponding optimal synthesis. The main results describe the asymptotic behavior of the optimal trajectories as the target point goes to infinity.

Action Time Sharing Policies for Ergodic Control of Markov Chains

Amarjit Budhiraja, Xin Liu, and Adam Shwartz

SIAM J. Control Optim. 50, pp. 171-195 (25 pages)

Online Publication Date: January 19, 2012

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Ergodic control for discrete time controlled Markov chains with a locally compact state space and a compact action space is considered under suitable stability, irreducibility, and Feller continuity conditions. A flexible family of controls, called action time sharing (ATS) policies, associated with a given continuous stationary Markov control, is introduced. It is shown that the long-term average cost for such a control policy, for a broad range of one-stage cost functions, is the same as that for the associated stationary Markov policy. In addition, ATS policies are well suited for a range of estimation, information collection, and adaptive control goals. To illustrate the possibilities we present two examples. The first demonstrates a construction of an ATS policy that leads to consistent estimators for unknown model parameters while producing the desired long-term average cost value. The second example considers a setting where the target stationary Markov control $q$ is not known but there are sampling schemes available that allow for consistent estimation of $q$. We construct an ATS policy which uses dynamic estimators for $q$ for control decisions and show that the associated cost coincides with that for the unknown Markov control $q$.

Asynchronous Rendezvous Analysis via Set-valued Consensus Theory

Feng Xiao and Long Wang

SIAM J. Control Optim. 50, pp. 196-221 (26 pages)

Online Publication Date: January 19, 2012

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This paper presents the design and analysis result for asynchronous rendezvous control of multiagent systems with continuous-time dynamics and intermittent interactions. The protocol-designing strategies only impose weak restrictions on anticipated-way-point sets (from which the way-points are selected) and path-planning of each agent and can be applied to the networks of arbitrary dimensional subsystems. Explicitly, the anticipated-way-point sets are in a polytope-like form and the path between any two consecutive way-points is required to be included within the minimum convex region covering the two associated anticipated-way-point sets. Under the assumption of directed and switching interaction topology and the assumption of intermittent and asynchronous interactions with time-varying delays, we perform the set-valued consensus analysis on the evolution of anticipated-way-point sets with respect to update times and provide mild sufficient conditions for the solvability of the asynchronous rendezvous problem. The proof techniques rely much on graph theory and nonnegative matrix theory. The obtained result extends greatly the existing work in the literature and several examples demonstrate its broad potential applications. Particularly, additional distributed control rules, different from the circumcenter algorithm, are devised for network connectivity maintenance.

Some Compact Classes of Open Sets under Hausdorff Distance and Application to Shape Optimization

Bao-Zhu Guo and Dong-Hui Yang

SIAM J. Control Optim. 50, pp. 222-242 (21 pages)

Online Publication Date: January 19, 2012

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In this paper, we introduce three new classes of open sets in a general Euclidean space $\mathbb{R}^N$. It is shown that every class of open sets is compact under the Hausdorff distance. The result is then applied to a shape optimization problem of elliptic equation. The existence of the optimal solution is presented.

An Eulerian Approach to the Analysis of Krause's Consensus Models

C. Canuto, F. Fagnani, and P. Tilli

SIAM J. Control Optim. 50, pp. 243-265 (23 pages)

Online Publication Date: January 19, 2012

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In this paper we analyze a class of multiagent consensus dynamical systems inspired by Krause's original model. As in Krause's model, the basic assumption is the so-called bounded confidence: two agents can influence each other only when their state values are below a given distance threshold $R$. We study the system under an Eulerian point of view considering (possibly continuous) probability distributions of agents, and we present original convergence results. The limit distribution is always necessarily a convex combination of delta functions at least $R$ far apart from each other: in other terms these models are locally aggregating. The Eulerian perspective provides the natural framework for designing a numerical algorithm, by which we obtain several simulations in $1$ and $2$ dimensions.

Coprime Factorization and Optimal Control on the Doubly Infinite Discrete Time Axis

Mark R. Opmeer and Olof J. Staffans

SIAM J. Control Optim. 50, pp. 266-285 (20 pages)

Online Publication Date: January 19, 2012

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We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic Riccati equation (of an arbitrary realization) have a solution (in general unbounded and even nondensely defined) and that a coupling condition involving these solutions is satisfied.

Exponentially Stable Interval Observers for Linear Systems with Delay

Frédéric Mazenc, Silviu-Iulian Niculescu, and Olivier Bernard

SIAM J. Control Optim. 50, pp. 286-305 (20 pages)

Online Publication Date: January 19, 2012

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This paper focuses on the analysis and design of families of interval observers for linear systems with a pointwise delay. First, it is proved that classical interval observers for systems without delays are not robust with respect to the presence of delays, no matter how small delays are. Next, it is shown that, in general, for linear systems with delay, the classical interval observers endowed with a pointwise delay are unstable. A new type of design of interval observers enabling circumvention of these obstacles is proposed. It provides framers that incorporate distributed delay terms. The proposed interval observers are assessed through a nonlinear biotechnological model.

Mean Square Performance of Consensus-Based Distributed Estimation over Regular Geometric Graphs

Federica Garin and Sandro Zampieri

SIAM J. Control Optim. 50, pp. 306-333 (28 pages)

Online Publication Date: January 19, 2012

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Average-consensus algorithms allow one to compute the average of some agents' data in a distributed way, and they are used as a basic building block in many algorithms for distributed estimation, load balancing, formation, and distributed control. Traditional analysis of such algorithms studies, for a given communication graph, the convergence rate (second largest eigenvalue of the transition matrix) and predicts that, for many graph families, performance degrades when the number of agents grows, because of the longer time required to spread information. However, in estimation problems, a growing number of sensor nodes improves the quality of the estimate. To understand whether such an improvement is possible also with distributed algorithms, it is important to specify a suitable performance metric, depending on the specific estimation problem in which the consensus algorithm is used, and to study how performance scales when both the number of iterations and the number of agents grow to infinity. Here, we propose a simple example of a distributed estimation problem solved by average-consensus, and a performance index naturally arising in this context (mean square estimation error, MSE). To understand the performance limitations of sensor networks with limited-range communications, we consider graphs describing local interactions. We give analytic results for some families of such graphs whose symmetries allow the use of suitable mathematical tools. However, simulations indicate that a similar behavior occurs also for random geometric graphs. This suggests that the performance limitations of regular lattices are mainly due to the geometrically local interactions and not to the symmetries.

Structural Invariants of Two-dimensional Systems

Lorenzo Ntogramatzidis

SIAM J. Control Optim. 50, pp. 334-356 (23 pages)

Online Publication Date: January 26, 2012

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In this paper, some fundamental structural properties of two-dimensional (2-D) systems which remain invariant under feedback and output-injection transformation groups are identified and investigated for the first time. As is well known, structural invariants that follow from the definition of controlled and conditioned invariance, output-nulling, input-containing, self-bounded and self-hidden subspaces play pivotal roles in many theoretical studies of systems theory and in the solution of several control/estimation problems. These concepts are developed and studied within a 2-D context in this paper.

Controllability as Minimality

Vakhtang Lomadze

SIAM J. Control Optim. 50, pp. 357-367 (11 pages)

Online Publication Date: January 26, 2012

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Among many other things, Oberst proved in his fundamental paper that controllability is equivalent to minimality in a transfer equivalence class. In this paper we directly and explicitly describe minimal systems in terms of transfer functions. A transfer function is a purely algebraic object that is associated to a linear differential system and that determines in an explicit form a certain “trivial” part of the system. Trajectories belonging to this part are called transfer trajectories. It is shown that a linear differential system is controllable if and only if every one of its trajectories can be obtained from a transfer trajectory by using a finite number of differentiations.

Smooth Morse–Lyapunov Functions of Strong Attractors for Differential Inclusions

Desheng Li and Yanling Wang

SIAM J. Control Optim. 50, pp. 368-387 (20 pages)

Online Publication Date: January 26, 2012

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This paper is concerned with a smooth converse Lyapunov theorem for Morse decompositions of strong attractors of differential inclusion $x'(t)\in F(x(t))$, where $F$ is an upper semicontinuous multivalued mapping on $\mathbb{R}^m$ with compact convex values. Roughly speaking, let there be given a strong attractor $\mathscr{A}$ of the system with attraction basin $\Omega$ and Morse decomposition $\mathcal{M}=\{M_1,\ldots,M_l\}$. We will construct a radially unbounded function $V\in C^\infty(\Omega)$ such that (1) $V$ is constant on each Morse set $M_k$ and (2) $V$ is strictly decreasing along any solution of the system in $\Omega$ outside the Morse sets.

Distributed Optimal Control of the Cahn–Hilliard System Including the Case of a Double-Obstacle Homogeneous Free Energy Density

M. Hintermüller and D. Wegner

SIAM J. Control Optim. 50, pp. 388-418 (31 pages)

Online Publication Date: February 02, 2012

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In this paper we study the distributed optimal control for the Cahn–Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau–Yosida approximation. Corresponding first-order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fréchet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first-order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.

Gossip Coverage Control for Robotic Networks: Dynamical Systems on the Space of Partitions

Francesco Bullo, Ruggero Carli, and Paolo Frasca

SIAM J. Control Optim. 50, pp. 419-447 (29 pages)

Online Publication Date: February 09, 2012

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Future applications in environmental monitoring, delivery of services, and transportation of goods motivate the study of deployment and partitioning tasks for groups of autonomous mobile agents. These tasks may be achieved by recent coverage algorithms, based upon the classic methods by Lloyd. These algorithms, however, rely upon critical requirements on the communication network: information is exchanged synchronously among all agents and long-range communication is sometimes required. This work proposes novel coverage algorithms that require only gossip communication, i.e., asynchronous, pairwise, and possibly unreliable communication. Which robot pair communicates at any given time may be selected deterministically or randomly. A key innovative idea is describing coverage algorithms for robot deployment and environment partitioning as dynamical systems on a space of partitions. In other words, we study the evolution of the regions assigned to each agent rather than the evolution of the agents' positions. The proposed gossip algorithms are shown to converge to centroidal Voronoi partitions under mild technical conditions. Our treatment features a broad variety of results in topology, analysis, and geometry. First, we establish the compactness of a suitable space of partitions with respect to the symmetric difference metric. Second, with respect to this metric, we establish the continuity of various geometric maps, including the Voronoi diagram as a function of its generators, the location of a centroid as a function of a set, and the widely known multicenter function studied in facility location problems. Third, we prove two convergence theorems for dynamical systems on metric spaces described by deterministic and stochastic switches.

The Linear Quadratic Regulator Problem for a Class of Controlled Systems Modeled by Singularly Perturbed Itô Differential Equations

Vasile Dragan, Hiroaki Mukaidani, and Peng Shi

SIAM J. Control Optim. 50, pp. 448-470 (23 pages)

Online Publication Date: February 09, 2012

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This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of $O(\sqrt{\varepsilon})$ approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an $O(\varepsilon)$ approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter $\varepsilon$ is not precisely known.

Experimental Design for Biological Systems

Matthias Chung and Eldad Haber

SIAM J. Control Optim. 50, pp. 471-489 (19 pages)

Online Publication Date: February 14, 2012

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Many problems in biology are governed by a dynamical system of differential equations with unknown parameters. To have a meaningful representation of the system, these parameters need to be evaluated from observations. Experimentalists face the dilemma between accuracy of the parameters and costs of an experiment. The choice of the design of an experiment is important if we are to recover precise model parameters. It is important to know when and what kind of observations should be taken. Taking the wrong measurement can lead to inaccurate estimation of parameters, thus resulting in inaccurate representations of the dynamical system. Each experiment has its own specific challenges. However, optimization methods form the basic computational tool to address eminent questions of optimal experimental design. In this paper we present a methodology for the design of such experiments that can optimally recover parameters in a dynamical system, biological systems in particular. We show that the problem can be cast as a stochastic bilevel optimization problem. We then develop an effective algorithm that allows for the solution of the design problem. The advantages of our approach are demonstrated on a few basic biological models as well as a design problem for the energy metabolism to estimate insulin resistance.

A Convex Condition for Robust Stability Analysis via Polyhedral Lyapunov Functions

R. Ambrosino, M. Ariola, and F. Amato

SIAM J. Control Optim. 50, pp. 490-506 (17 pages)

Online Publication Date: February 14, 2012

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In this paper we study the robustness analysis problem for linear continuous-time systems subject to parametric time-varying uncertainties making use of piecewise linear (polyhedral) Lyapunov functions. A given class of Lyapunov functions is said to be “universal” for the uncertain system under consideration if the robust stability of the system is equivalent to the existence of a Lyapunov function belonging to the class. In the literature it has been shown that the class of polyhedral functions is universal, while, for instance, the class of quadratic functions is not. This fact justifies the effort of developing efficient algorithms for the construction of polyhedral Lyapunov functions. In this context, we provide a low computational cost procedure, based on a novel convex condition, for the construction of a polyhedral Lyapunov function. In the section on the numerical examples, we consider some benchmark problems for the robust stability analysis and we show that the proposed low computational cost approach, though only sufficient, is less conservative than all the other approaches presented so far in the literature.

Linear-quadratic Control for Stochastic Equations in a Hilbert Space with Fractional Brownian Motions

T. E. Duncan, B. Maslowski, and B. Pasik-Duncan

SIAM J. Control Optim. 50, pp. 507-531 (25 pages)

Online Publication Date: February 28, 2012

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A linear-quadratic control problem with a finite time horizon for some infinite-dimensional controlled stochastic differential equations driven by a fractional Gaussian noise is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise. The covariance of the noise as well as the control operator in the system equation can in general be unbounded, so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain. Some examples of controlled stochastic partial differential equations are given.

Stochastic Comparison, Boundedness, Weak Convergence, and Ergodicity of a Random Riccati Equation with Markovian Binary Switching

Li Xie

SIAM J. Control Optim. 50, pp. 532-558 (27 pages)

Online Publication Date: February 28, 2012

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This paper considers the dynamical behavior of a discrete-time random Riccati equation with Markovian binary switching arising from a Kalman filter with intermittent observations. A number of properties of the random Riccati equation are studied such as comparability, monotonicity, boundedness, weak convergence, and ergodicity. A stochastic order method is used to compare random Riccati equations and establish the monotonicity property in an expectation sense. By using the idea of stopping times, we present a sufficient condition for an expectation boundedness property. Moreover, the contraction and nonexpansion, the Riemannian metric, and the Markov–Feller operator are applied to the analysis of weak convergence and ergodicity properties. These properties allow us to establish a relation between time average and the limit of the expectation of covariance matrices. We also revisit the independently and identically distributed case. By making use of the comparison and boundedness results in this paper, we present a variant of the existence theorem about a critical value in existing literature.

On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics

Davide Barilari, Ugo Boscain, and Jean-Paul Gauthier

SIAM J. Control Optim. 50, pp. 559-582 (24 pages)

Online Publication Date: February 28, 2012

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In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a by-product of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6-dimensional, 2-step, corank 2 sub-Riemannian metric.

A Martingale Approach to Optimal Portfolios with Jump-diffusions

Daniel Michelbrink and Huiling Le

SIAM J. Control Optim. 50, pp. 583-599 (17 pages)

Online Publication Date: February 28, 2012

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This paper investigates optimal investment-consumption strategies that maximize the expected utility of consumption and/or terminal wealth under jump-diffusion models using a martingale method. We characterize the optimal trading strategy and the optimal martingale measure in terms of a system of equations and obtain explicit solutions for the power and logarithmic utility case.
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