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SIAM J. on Control and Optimization

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2011

Volume 49, Issue 6, pp. 2239-2626


Distributed Consensus for Multiagent Systems with Communication Delays and Limited Data Rate

Shuai Liu, Tao Li, and Lihua Xie

SIAM J. Control Optim. 49, pp. 2239-2262 (24 pages)

Online Publication Date: November 01, 2011

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This paper considers the average consensus problem for multiagent networks with communication delays and limited data rate. On one hand, communication delays often exist in information acquisition and transmission; on the other hand, only limited state information of agents can be transmitted to their neighbors at each time step due to bandwidth constraints. The average consensus problem becomes much more complicated when both delays and data-rate constraints are to be considered. In this paper, a distributed consensus protocol is proposed based on dynamic encoding and decoding. It is shown that for a connected network, as long as the time delays are bounded, the average consensus can be achieved with a finite communication data rate. In particular, it is shown that merely a one-bit information exchange between each pair of adjacent agents at each time step suffices to guarantee the average consensus.

Backstepping for Nonlinear Systems with Delay in the Input Revisited

Frédéric Mazenc, Silviu-Iulian Niculescu, and Mounir Bekaik

SIAM J. Control Optim. 49, pp. 2263-2278 (16 pages)

Online Publication Date: November 01, 2011

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In this paper, a new solution to the problem of globally asymptotically stabilizing a nonlinear system in feedback form with a known pointwise delay in the input is proposed. The result covers a family of systems wider than those studied in the literature and endows with control laws with a single delay, in contrast to those given in previous works which include two distinct pointwise delays or distributed delays. The strategy of design is based on the construction of an appropriate Lyapunov–Krasovskii functional. An illustrative example ends the paper.

Optimal Switching of One-Dimensional Reflected BSDEs and Associated Multidimensional BSDEs with Oblique Reflection

Shanjian Tang, Wei Zhong, and Hyeng Keun Koo

SIAM J. Control Optim. 49, pp. 2279-2317 (39 pages)

Online Publication Date: November 01, 2011

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In this paper, an optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (RBSDEs) where the generators, the terminal values, and the barriers are all switched with positive costs. The value process is characterized by a system of multidimensional RBSDEs with oblique reflection, whose existence and uniqueness are by no means trivial and are therefore carefully examined. Existence is shown using both methods of the Picard iteration and penalization, but under some different conditions. Uniqueness is proved by representation either as the value process to our optimal switching problem for one-dimensional RBSDEs or as the equilibrium value process to a stochastic differential game of switching and stopping. Finally, the switched RBSDE is interpreted as a real option.

$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier–Stokes Equations

Sheetal Dharmatti, Jean-Pierre Raymond, and Laetitia Thevenet

SIAM J. Control Optim. 49, pp. 2318-2348 (31 pages)

Online Publication Date: November 01, 2011

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We study the robust or $H^\infty$ exponential stabilization of the linearized Navier–Stokes equations around an unstable stationary solution in a two-dimensional domain $\Omega$. The disturbance is an unknown perturbation in the boundary condition of the fluid flow. We determine a feedback boundary control law, robust with respect to boundary perturbations, by solving a max-min linear quadratic control problem. Next we show that this feedback law locally stabilizes the Navier–Stokes system. Similar problems have been studied in the literature in the case of distributed controls and disturbances. To the authors' knowledge, it is the first time that the robust stabilization of the Navier–Stokes equations is studied for boundary controls and boundary disturbances.

Differential Games and Zubov's Method

Lars Grüne and Oana Silvia Serea

SIAM J. Control Optim. 49, pp. 2349-2377 (29 pages)

Online Publication Date: November 10, 2011

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In this paper we provide generalizations of Zubov's equation to differential games without the Isaacs condition. We show that both generalizations of Zubov's equation (which we call the min-max and max-min Zubov equation, respectively) possess unique viscosity solutions which characterize the respective controllability domains. As a consequence, we show that under the usual Isaacs condition the respective controllability domains as well as the local controllability assumptions coincide.

HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, II: Verification and Optimal Feedbacks

Salvatore Federico, Ben Goldys, and Fausto Gozzi

SIAM J. Control Optim. 49, pp. 2378-2414 (37 pages)

Online Publication Date: November 15, 2011

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This paper, which is the natural continuation of part I [S. Federico, B. Goldys, and F. Gozzi, SIAM J. Control Optim., 48 (2010), pp. 4910–4937], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In part I the problem is embedded in a suitable Hilbert space $H$ and the regularity of the associated Hamilton–Jacobi–Bellman equation is studied. The goal of the present paper is to exploit the regularity result of part I to prove a verification theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of $V$ and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-to-build models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

State Maps from Integration by Parts

Arjan van der Schaft and Paolo Rapisarda

SIAM J. Control Optim. 49, pp. 2415-2439 (25 pages)

Online Publication Date: November 17, 2011

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We develop a new approach to the construction of state vectors for linear time-invariant systems described by higher-order differential equations. The basic observation is that the concatenation of two solutions of higher-order differential equations results in another (weak) solution once their remainder terms resulting from (repeated) integration by parts match. These remainder terms can be computed in a simple and efficient manner by making use of the calculus of bilinear differential forms and two-variable polynomial matrices. Factorization of the resulting two-variable polynomial matrix defines a state map, as well as a state map for the adjoint system. Minimality of these state maps is characterized. The theory is applied to three classes of systems with additional structure, namely self-adjoint Hamiltonian, conservative port-Hamiltonian, and time-reversible systems. For the first two classes it is shown how the factorization leading to a (minimal) state map is equivalent to the factorization of another two-variable polynomial matrix, which is immediately derived from the external system characterization, and defines a symplectic, respectively, symmetric, bilinear form on the minimal state space.

Minimum-Time Frictionless Atom Cooling in Harmonic Traps

Dionisis Stefanatos, Heinz Schaettler, and Jr-Shin Li

SIAM J. Control Optim. 49, pp. 2440-2462 (23 pages)

Online Publication Date: November 29, 2011

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Frictionless atom cooling in harmonic traps is formulated as a time-optimal control problem, and a synthesis of optimal controlled trajectories is obtained.

Computational Tools for the Safety Control of a Class of Piecewise Continuous Systems with Imperfect Information on a Partial Order

Michael R. Hafner and Domitilla Del Vecchio

SIAM J. Control Optim. 49, pp. 2463-2493 (31 pages)

Online Publication Date: December 01, 2011

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This paper addresses the two-agent safety control problem for piecewise continuous systems with disturbances and imperfect state information. In particular, we focus on a class of systems that evolve on a partial order and whose dynamics preserve the ordering. While the safety control problem with imperfect state information is prohibitive for general classes of nonlinear and hybrid systems, the class of systems considered in this paper admits an explicit solution. We compute this solution with linear complexity discrete-time algorithms that are guaranteed to terminate. The proposed algorithms are illustrated on a two-vehicle collision avoidance problem and implemented on a hardware roundabout test-bed.

Asymptotic Expansion for the Solutions of Control Constrained Semilinear Elliptic Problems with Interior Penalties

J. Frédéric Bonnans and Francisco J. Silva

SIAM J. Control Optim. 49, pp. 2494-2517 (24 pages)

Online Publication Date: December 06, 2011

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In this work we consider the optimal control problem of a semilinear elliptic partial differential equation (PDE) with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated family of penalized problems, parametrized by $\varepsilon>0$, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to recover some known error bounds for the logarithmic barrier approximation and to obtain some new ones for a rather general class of barrier functions. In this manner, we generalize the results of [F. Alvarez et al., Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Math. Program. Ser. A, to appear], which were obtained in the ordinary differential equation (ODE) framework.

Fattening and Comparison Principle for Level-Set Equations of Mean Curvature Type

Qing Liu

SIAM J. Control Optim. 49, pp. 2518-2541 (24 pages)

Online Publication Date: December 08, 2011

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In this paper, we give several applications of the discrete game approach to partial differential equations (PDEs). We first present a rigorous game-theoretic proof of fattening phenomenon for motion by curvature with figure-eight–shaped initial curves without using parabolic PDE theory. The proof is based on a comparison between the game value and its inverse. Accompanied by the example of figure eight, our second result shows, for the stationary equation of mean curvature type in an arbitrary region $\Omega$, that fattening of positive curvature flow with initial surface $\partial\Omega$ causes loss of the weak comparison principle, which partially answers an open question posed by Kohn and Serfaty in 2006. In addition, we prove the existence of solutions of the stationary problem and its game approximation in the absence of comparison principles but under regularity conditions of the flow. The main difference between our games and those in other papers is that we take the domain perturbation into consideration.

Riccati Equations on Noncommutative Banach Algebras

Ruth Curtain

SIAM J. Control Optim. 49, pp. 2542-2557 (16 pages)

Online Publication Date: December 08, 2011

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Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentially stabilizing solution on this larger space. Applications to spatially distributed systems are given.

Exterior Sphere Condition and Time Optimal Control for Differential Inclusions

Piermarco Cannarsa and Khai T. Nguyen

SIAM J. Control Optim. 49, pp. 2558-2576 (19 pages)

Online Publication Date: December 13, 2011

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The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened setup, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.

Stochastic Target Problems with Controlled Loss in Jump Diffusion Models

Ludovic Moreau

SIAM J. Control Optim. 49, pp. 2577-2607 (31 pages)

Online Publication Date: December 15, 2011

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In this paper, we consider a mixed-diffusion version of the stochastic target problem introduced in [B. Bouchard, R. Elie, and N. Touzi, SIAM J. Control Optim., 48 (2009), pp. 3123–3150]. This consists in finding the minimum initial value of a controlled process which guarantees to reach a controlled stochastic target with a given level of expected loss. It can be converted into a standard stochastic target problem by increasing both the state space and the dimension of the control. In our mixed-diffusion setting, the main difficulty comes from the presence of jumps, which leads to the introduction of a new kind of control that takes values in an unbounded set of measurable maps. This has a nontrivial technical impact on the formulation and derivation of the associated partial differential equations.

An Iterative Procedure for Constructing Subsolutions of Discrete-Time Optimal Control Problems

Markus Fischer

SIAM J. Control Optim. 49, pp. 2608-2626 (19 pages)

Online Publication Date: December 20, 2011

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An iterative procedure for constructing subsolutions of deterministic or stochastic optimal control problems in discrete time with continuous state space is introduced. The procedure generates a nondecreasing sequence of subsolutions, giving true lower bounds on the minimal costs. Convergence of the values at any fixed initial state is shown.
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