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SIAM J. on Control and Optimization

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2007

Volume 46, Issue 6, pp. 1923-2302


Existence and Nonexistence Results of an Optimal Control Problem by Using Relaxed Control

Hongwei Lou

SIAM J. Control Optim. 46, pp. 1923-1941 (19 pages)

Online Publication Date: November 28, 2007

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Relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory. Many positive results have been obtained in the literature. However, negative results have also made their rare appearances. The optimal control problem considered in this paper looks quite simple. Yet, by treating such a problem, we can get interesting results, substantiating our idea as to whether an optimal control exists or not. In our opinion, the method used in the paper can be applied to more generalized cases.

Numerical Approximations for Nonzero-Sum Stochastic Differential Games

Harold J. Kushner

SIAM J. Control Optim. 46, pp. 1942-1971 (30 pages) | Cited 1 time

Online Publication Date: November 28, 2007

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The Markov chain approximation method is a widely used and efficient family of methods for the numerical solution of many types of stochastic control problems in continuous time for reflected-jump-diffusion–type models. It converges under broad conditions, and it has been extended to zero-sum stochastic differential games. We apply the method to a class of nonzero stochastic differential games with a diffusion system model where the controls for the two players are separated in the dynamics and cost function. There have been successful applications of the algorithms, but convergence proofs have been lacking. It is shown that equilibrium values for the approximating chain converge to equilibrium values for the original process and that any equilibrium value for the original process can be approximated by an $\epsilon$-equilibrium for the chain for arbitrarily small $\epsilon > 0$. The numerical method solves a stochastic game for a finite-state Markov chain.

On a Model for the Efficient Operation of a Bank or Insurance Company

Joseph G. Conlon and Hyekyung Min

SIAM J. Control Optim. 46, pp. 1972-1994 (23 pages)

Online Publication Date: November 28, 2007

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In this paper the authors study a model for the optimal operation of a bank or insurance company which was recently introduced by Peura and Keppo. The model generalizes a previous one of Milne and Robertson by allowing the bank to raise capital as well as to pay out dividends. Optimal operation of the bank is determined by solving an optimal control problem. In this paper it is shown that the solution of the optimal control problem proposed by Peura and Keppo exists for all values of the parameters and is unique.

Stability of Nonlinear Feedback Systems: A New Small-Gain Theorem

Anna L. Chen, Gui-Qiang Chen, and Randy A. Freeman

SIAM J. Control Optim. 46, pp. 1995-2012 (18 pages) | Cited 1 time

Online Publication Date: November 30, 2007

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For the feedback interconnection of general nonlinear systems, the classical small-gain condition is sufficient but not necessary for robust stability. We introduce a weaker notion of gain which yields a small-gain condition that is both necessary and sufficient for robust stability. We also discuss conditions under which the two notions coincide, and we further provide results for dissipation performance measures that are more general than the classical gain measures.

On Worst-Case Portfolio Optimization

Ralf Korn and Mogens Steffensen

SIAM J. Control Optim. 46, pp. 2013-2030 (18 pages)

Online Publication Date: November 30, 2007

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We formulate a worst-case portfolio optimization problem that technically appears as a game where the investor chooses a portfolio and his opponent, the market, chooses some market crashes. The asymmetry of the opponents' decision processes leads to a new and delicate generalization of the classical Hamilton–Jacobi–Bellman equation in stochastic control. We characterize the optimal controls in general and specify them further in the cases of Hara, logarithmic, and exponential utilities of the investor.

Primal-Dual Symmetric Intrinsic Methods for Finding Antiderivatives of Cyclically Monotone Operators

Heinz H. Bauschke, Yves Lucet, and Xianfu Wang

SIAM J. Control Optim. 46, pp. 2031-2051 (21 pages) | Cited 5 times

Online Publication Date: November 30, 2007

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A fundamental result due to Rockafellar states that every cyclically monotone operator $A$ admits an antiderivative $f$ in the sense that the graph of $A$ is contained in the graph of the subdifferential operator $\partial f$. Given a method $\mathfrak{m}$ that assigns every finite cyclically monotone operator $A$ some antiderivative $\mathfrak{m}_A$, we say that the method is primal-dual symmetric if $\mathfrak{m}$ applied to the inverse of $A$ produces the Fenchel conjugate of $\mathfrak{m}_A$. Rockafellar's antiderivatives do not possess this property. Utilizing Fitzpatrick functions and the proximal average, we present novel primal-dual symmetric intrinsic methods. The antiderivatives produced by these methods provide a solution to a problem posed by Rockafellar in 2005. The results leading to this solution are illustrated by various examples.

Computing the $L_1$-Norm of Continuous-Time Linear Systems

Arno Linnemann

SIAM J. Control Optim. 46, pp. 2052-2070 (19 pages)

Online Publication Date: December 05, 2007

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An algorithm is presented for the computation of the $L_1$-norm of a linear time-invariant continuous-time system. It is based on the numerical integration of the impulse response and is shown to be quadratically convergent.

Exact Controllability for the Time Dependent Transport Equation

Michael V. Klibanov and Masahiro Yamamoto

SIAM J. Control Optim. 46, pp. 2071-2195 (125 pages)

Online Publication Date: December 12, 2007

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The exact controllability theorem for the time dependent transport equation is proved.

The Multi-Agent Rendezvous Problem. Part 1: The Synchronous Case

J. Lin, A. S. Morse, and B. D. O. Anderson

SIAM J. Control Optim. 46, pp. 2096-2119 (24 pages) | Cited 3 times

Online Publication Date: December 12, 2007

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This paper is concerned with the collective behavior of a group of $n>1$ mobile autonomous agents, labelled $1$ through $n$, which can all move in the plane. Each agent is able to continuously track the positions of all other agents currently within its “sensing region,” where by an agent's sensing region we mean a closed disk of positive radius $r$ centered at the agent's current position. The multi-agent rendezvous problem is to devise “local” control strategies, one for each agent, which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location. This paper describes a solution to this problem consisting of individual agent strategies which are mutually synchronized in the sense that all depend on a common clock.

The Multi-Agent Rendezvous Problem. Part 2: The Asynchronous Case

J. Lin, A. S. Morse, and B. D. O. Anderson

SIAM J. Control Optim. 46, pp. 2120-2147 (28 pages) | Cited 3 times

Online Publication Date: December 21, 2007

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This paper is concerned with the collective behavior of a group of $n>1$ mobile autonomous agents, labelled $1$ through $n$, which can all move in the plane. Each agent is able to continuously track the positions of all other agents currently within its “sensing region,” where by an agent's sensing region we mean a closed disk of positive radius $r$ centered at the agent's current position. The multi-agent rendezvous problem is to devise “local” control strategies, one for each agent, which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location. This paper describes a family of unsynchronized strategies for solving the problem. Correctness is established appealing to the concept of “analytic synchronization.”

The Analysis of Exact Controllability of Neutral-Type Systems by the Moment Problem Approach

Rabah Rabah and Grigory M. Sklyar

SIAM J. Control Optim. 46, pp. 2148-2181 (34 pages)

Online Publication Date: December 21, 2007

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The problem of exact null-controllability is considered for a wide class of linear neutral-type systems with distributed delay. The main tool of the analysis is the application of the moment problem approach and the theory of the basis property of exponential families. A complete characterization of this problem is given. The minimal time of controllability is specified. The results are based on the analysis of the Riesz basis property of eigenspaces of the neutral-type systems in Hilbert space.

On the Genericity of the Differential Observability of Controlled Discrete-Time Systems

Sabeur Ammar, Mohamed Mabrouk, and Jean-Claude Vivalda

SIAM J. Control Optim. 46, pp. 2182-2198 (17 pages)

Online Publication Date: December 21, 2007

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In this paper, we prove the genericity of the differential observability for discrete-time systems with more outputs than inputs.

An Introduction to Quantum Filtering

Luc Bouten, Ramon Van Handel, and Matthew R. James

SIAM J. Control Optim. 46, pp. 2199-2241 (43 pages) | Cited 14 times

Online Publication Date: December 21, 2007

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This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

Single Output-Dependent Observability Normal Form

Gang Zheng, Driss Boutat, and Jean-Pierre Barbot

SIAM J. Control Optim. 46, pp. 2242-2255 (14 pages) | Cited 2 times

Online Publication Date: December 21, 2007

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This paper gives the sufficient and necessary conditions which guarantee the existence of a diffeomorphism in order to transform a nonlinear system without inputs into a canonical normal form that is output dependent. Moreover, we extend our results to a class of systems with inputs.

Controllability of a Class of Newtonian Filtration Equations with Control and State Constraints

Xu Liu and Hang Gao

SIAM J. Control Optim. 46, pp. 2256-2279 (24 pages)

Online Publication Date: December 21, 2007

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This paper addresses a study of the controllability of a class of Newtonian filtration equations, with nonnegative constraints on the control and state variables. When the control enters the system through the whole domain where the equation evolves, we characterize the set of nonnegative targets which are approximately controllable at any time $T>0$. The proof combines the Fenchel–Rockafellar duality theory and a fixed point argument. When the control is restricted to be active in a proper open subset of the whole domain, we prove a negative controllability result by means of a localization technique which reflects the underlying obstruction phenomenon in the system.

Robust Control Approach to Option Pricing: A Representation Theorem and Fast Algorithm

Pierre Bernhard, Naïma El Farouq, and Stéphane Thiery

SIAM J. Control Optim. 46, pp. 2280-2302 (23 pages)

Online Publication Date: December 21, 2007

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The so-called interval model for security prices, together with a robust control approach, allows one to construct a consistent theory of option pricing, including discrete time trading and arbitrary transaction costs. In this context, a new representation theorem for the viscosity solution of the relevant Isaacs (differential) quasi-variational inequality leads to simple formulas and fast numerical algorithms to compute a hedging portfolio. We argue that in spite of a less satisfactory market model, the overall theory is not much less realistic than the classical Black and Scholes theory but rather only that it shifts from the portfolio model to the market model the place where the model is violated when sudden large price changes occur on the market. As such, and subject to a more detailed validation, the new theory might be the basis of a possible alternative as a normative theory whenever transaction costs or discrete time trading are the main concerns.
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