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2010

Volume 48, Issue 8, pp. 4821-5653


The Method of Controlled Lagrangians: Energy plus Force Shaping

Dong Eui Chang

SIAM J. Control Optim. 48, pp. 4821-4845 (25 pages) | Cited 1 time

Online Publication Date: September 14, 2010

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We study the method of controlled Lagrangians to stabilize mechanical systems connected with external forces. The basic idea is that we transform by feedback a given controlled Lagrangian system to another controlled Lagrangian system with positive definite energy and a dissipative external force such that a dissipative feedback force stabilizes the closed-loop system. We derive matching conditions for energy plus force shaping that are more general and stronger than those in the literature. We provide various easily verifiable criteria for stabilizability by the method of controlled Lagrangians, including a necessary and sufficient condition for stabilizability by energy shaping for all linear mechanical systems, a necessary and sufficient condition for stabilizability by energy shaping for the class of all mechanical systems with one degree of underactuation, and a sufficient condition for stabilizability by energy plus force shaping for the class of all mechanical systems with one degree of underactuation connected with external forces.

Robust Regulation of Distributed Parameter Systems with Infinite-Dimensional Exosystems

Timo Hämäläinen and Seppo Pohjolainen

SIAM J. Control Optim. 48, pp. 4846-4873 (28 pages)

Online Publication Date: September 14, 2010

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In this paper a robust regulation problem for infinite-dimensional systems with infinite-dimensional exosystems is discussed. The input and output spaces are also allowed to be infinite-dimensional. A new definition of internal model in terms of the controller parameters is given. It is shown that there exists a feedback controller containing an internal model of the exosystem, which robustly regulates the class of signals generated by the exosystem and strongly or weakly stabilizes the closed-loop system. As far as the authors know, the results are new even for finite-dimensional systems with infinite-dimensional exosystems.

Finite Horizon Optimal Stopping of Time-Discontinuous Functionals with Applications to Impulse Control with Delay

Jan Palczewski and Łukasz Stettner

SIAM J. Control Optim. 48, pp. 4874-4909 (36 pages) | Cited 1 time

Online Publication Date: September 16, 2010

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We study finite horizon optimal stopping problems for continuous-time Feller–Markov processes. The functional depends on time, state, and external parameters and may exhibit discontinuities with respect to the time variable. Both left- and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, on the initial state of the process, and on the stopping horizon. We construct $\varepsilon$-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag, and execution delay.

HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, I: Regularity of Viscosity Solutions

Salvatore Federico, Ben Goldys, and Fausto Gozzi

SIAM J. Control Optim. 48, pp. 4910-4937 (28 pages) | Cited 1 time

Online Publication Date: September 29, 2010

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We study a class of optimal control problems with state constraints, where the state equation is a differential equation with delays. This class includes some problems arising in economics, in particular, the so-called models with time to build; see [P. K. Asea and P. J. Zak, J. Econom. Dynam. Control, 23 (1999), pp. 1155–1175; M. Bambi, J. Econom. Dynam. Control, 32 (2008), pp. 1015–1040; F. E. Kydland and E. C. Prescott, Econometrica, 50 (1982), pp. 1345–1370]. We embed the problem in a suitable Hilbert space $H$ and consider the associated Hamilton–Jacobi–Bellman (HJB) equation. This kind of infinite dimensional HJB equation has not been previously studied and is difficult due to the presence of state constraints and the lack of smoothing properties of the state equation. Our main result on the regularity of solutions to such an HJB equation seems to be entirely new. More precisely, we prove that the value function is continuous in a sufficiently big open set of $H$, that it solves in the viscosity sense the associated HJB equation, and that it has continuous classical derivative in the direction of the “present.” This regularity result is the starting point to define a feedback map in the classical sense, which gives rise to a candidate optimal feedback strategy.

Analysis of Unconstrained Nonlinear MPC Schemes with Time Varying Control Horizon

Lars Grüne, Jürgen Pannek, Martin Seehafer, and Karl Worthmann

SIAM J. Control Optim. 48, pp. 4938-4962 (25 pages)

Online Publication Date: October 07, 2010

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For discrete time nonlinear systems satisfying an exponential or finite time controllability assumption, we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints. Based on our formula, we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop.

Periodic Optimization Suffices for Infinite Horizon Planar Optimal Control

Zvi Artstein and Ido Bright

SIAM J. Control Optim. 48, pp. 4963-4986 (24 pages)

Online Publication Date: October 14, 2010

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Under quite general conditions we show that an infinite horizon optimal control problem with a state variable in the plane has a periodic solution. The latter may employ relaxed controls. We then examine the possibility of constructing an ordinary control approximation. An application to singularly perturbed control systems is displayed.

Optimal Dividend Policies with Transaction Costs for a Class of Diffusion Processes

Lihua Bai and Jostein Paulsen

SIAM J. Control Optim. 48, pp. 4987-5008 (22 pages)

Online Publication Date: October 14, 2010

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Finding optimal dividend strategies is a classical problem in the financial and actuarial literature. The idea is that the company wants to pay some of its surplus as dividends, and the problem is to find a dividend strategy that maximizes the expected total discounted dividends received by the shareholders until ruin. Here we generalize results in [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669–689] in that the rate of growth of the surplus process is assumed to exceed the discounting factor whenever the surplus process is smaller than a fixed number $x_{\lambda}$. In [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669–689] it was assumed that this rate of growth is always less than or equal to the discounting factor. It turns out that this generalization makes the problem much more complicated, and a simple barrier strategy is no longer always optimal.

Linear Stochastic State Space Theory in the White Noise Space Setting

Daniel Alpay, David Levanony, and Ariel Pinhas

SIAM J. Control Optim. 48, pp. 5009-5027 (19 pages)

Online Publication Date: October 14, 2010

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We study linear stochastic state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely, its average behavior.

Stabilization of Second Order Evolution Equations with Unbounded Feedback with Time-Dependent Delay

Emilia Fridman, Serge Nicaise, and Julie Valein

SIAM J. Control Optim. 48, pp. 5028-5052 (25 pages)

Online Publication Date: October 19, 2010

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We consider abstract second order evolution equations with unbounded feedback with time-varying delay. Existence results are obtained under some realistic assumptions. We prove the exponential decay under some conditions by introducing an abstract Lyapunov functional. Our abstract framework is applied to the wave, to the beam, and to the plate equations with boundary delays.

Optimal Control for an Elliptic System with Polygonal State Constraints

Karl Kunisch, Kewei Liang, and Xiliang Lu

SIAM J. Control Optim. 48, pp. 5053-5072 (20 pages)

Online Publication Date: October 19, 2010

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This work is devoted to stationary optimal control problems with polygonal constraints on the components of the state. Existence of Lagrange multipliers, of different regularities, is verified for the cases with and without Slater condition holding. For the numerical realization a semismooth Newton method is proposed for an appropriately chosen family of regularized problems. The asymptotic behavior of the regularized problem class is studied, and numerical feasibility of the method is shown.

On Ruckle's Conjecture on Accumulation Games

Steve Alpern, Robbert Fokkink, and Ken Kikuta

SIAM J. Control Optim. 48, pp. 5073-5083 (11 pages)

Online Publication Date: October 19, 2010

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In an accumulation game, the Hider secretly distributes his given total wealth $h$ among $n$ locations, while the Searcher picks $r$ locations and confiscates the material placed there. The Hider wins if what is left at the remaining $n-r$ locations is at least 1; otherwise the Searcher wins. Ruckle's conjecture says that an optimal Hider strategy is to put an equal amount $h/k$ at $k$ randomly chosen locations for some $k$. We extend the work of Kikuta and Ruckle by proving the conjecture for several cases, e.g., $r=2$ or $n-2$; $n\leq7$; $n=2r-1$; $h\leq2+1/\,(n-r)$ and $n\leq2r$. The last result uses the Erdős–Ko–Rado theorem. We establish a connection between Ruckle's conjecture and the Hoeffding problem of bounding tail probabilities of sums of random variables.

Optimal Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems by Output Injection

Mark R. Opmeer and Olof J. Staffans

SIAM J. Control Optim. 48, pp. 5084-5107 (24 pages) | Cited 1 time

Online Publication Date: October 19, 2010

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We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by output injection. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over H-infinity. Another equivalent condition is that the filter Riccati equation (of an arbitrary realization) has a solution (in general, unbounded and even nondensely defined). We further show that after renorming the state space in terms of the inverse of the smallest solution of the filter Riccati equation, the closed-loop system is not only input-output stable but also strongly internally $*$-stable.

A Priori Error Analysis for Linear Quadratic Elliptic Neumann Boundary Control Problems with Control and State Constraints

K. Krumbiegel, C. Meyer, and A. Rösch

SIAM J. Control Optim. 48, pp. 5108-5142 (35 pages)

Online Publication Date: October 26, 2010

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In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations.

Efficient Preconditioners for Optimality Systems Arising in Connection with Inverse Problems

Bjørn Fredrik Nielsen and Kent-Andre Mardal

SIAM J. Control Optim. 48, pp. 5143-5177 (35 pages)

Online Publication Date: October 26, 2010

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This paper is devoted to the numerical treatment of linear optimality systems (OS) that arise in connection with inverse problems for partial differential equations. If such inverse problems are regularized by Tikhonov regularization, then it follows from standard theory that the associated OS is well-posed, provided that the regularization parameter $\alpha$ is positive and that the involved state equation satisfies suitable assumptions. We explain and analyze how certain mapping properties of the operators appearing in the OS can be employed to define efficient preconditioners for finite element (FE) approximations of such systems. The key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter $h$, used in the FE discretization, and increases only moderately as $\alpha\rightarrow0$. More specifically, if the stopping criterion for the iteration process is defined in terms of the associated energy norm, then the number of iterations required (in the severely ill-posed case) cannot grow faster than $O((\ln(\alpha))^2)$. Our analysis is based on a careful study of the operators involved, which yields the distribution of the eigenvalues of the preconditioned OS. Finally, the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by Borzi, Kunisch, and Kwak [SIAM J. Control Optim., 41 (2003), pp. 1477–1497] and an inverse problem arising in connection with electrocardiography [Nielsen, Cai, and Lysaker, Math. Biosci., 210 (2007), pp. 523–553].

On the Convergence, Lock-In Probability, and Sample Complexity of Stochastic Approximation

Sameer Kamal

SIAM J. Control Optim. 48, pp. 5178-5192 (15 pages)

Online Publication Date: October 26, 2010

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It is shown that under standard hypotheses, if stochastic approximation iterates remain tight, they converge with probability one to what their o.d.e. limit suggests. A simple test for tightness (and therefore a.s. convergence) is provided. Further, estimates on lock-in probability, i.e., the probability of convergence to a specific attractor of the o.d.e. limit given that the iterates visit its domain of attraction, and sample complexity, i.e., the number of steps needed to be within a prescribed neighborhood of the desired limit set with a prescribed probability, are also provided. The latter improve significantly upon existing results in that they require a much weaker condition on the martingale difference noise.

A Finite Time Horizon Optimal Stopping Problem with Regime Switching

H. Le and C. Wang

SIAM J. Control Optim. 48, pp. 5193-5213 (21 pages) | Cited 1 time

Online Publication Date: October 26, 2010

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We extend the technique developed in [E. Bayraktar, A Proof of the Smoothness of the Finite Time Horizon American Put Option for Jump Diffusion, http://arxiv.org/abs/math/0703782, 2007] to a class of finite time horizonal optimal stopping problems under regime switching models which includes the pricing of American put options. The construction involved also leads to a computational procedure for the solutions of such optimal stopping problems.

Continuous-Time Average-Preserving Opinion Dynamics with Opinion-Dependent Communications

Vincent D. Blondel, Julien M. Hendrickx, and John N. Tsitsiklis

SIAM J. Control Optim. 48, pp. 5214-5240 (27 pages) | Cited 1 time

Online Publication Date: October 28, 2010

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We study a simple continuous-time multiagent system related to Krause's model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multiagent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a nontrivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.

On Some Systems Controlled by the Structure of Their Memory

G. Buttazzo, G. Carlier, and R. Tahraoui

SIAM J. Control Optim. 48, pp. 5241-5253 (13 pages)

Online Publication Date: November 04, 2010

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We consider an optimal control problem governed by an ODE with memory playing the role of a control. We show the existence of an optimal solution and derive some necessary optimality conditions. Some examples are then discussed.

Local Energy Decay for the Elastic System with Nonlinear Damping in an Exterior Domain

M. Daoulatli, B. Dehman, and M. Khenissi

SIAM J. Control Optim. 48, pp. 5254-5275 (22 pages)

Online Publication Date: November 04, 2010

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In three dimension space, under a microlocal geometric condition, we give the rate of decay of the local energy for solutions of the Lamé system on exterior domain, with localized nonlinear damping.

Impulse Control of Multidimensional Jump Diffusions

Mark H. A. Davis, Xin Guo, and Guoliang Wu

SIAM J. Control Optim. 48, pp. 5276-5293 (18 pages)

Online Publication Date: November 04, 2010

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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.

Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

Bin Zhou, Zongli Lin, and Guang-Ren Duan

SIAM J. Control Optim. 48, pp. 5294-5332 (39 pages)

Online Publication Date: November 04, 2010

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This paper studies stabilization problems for linear systems with multiple delays in the input. Two types of delays are considered. The first type of delays is constant delays, which can be arbitrarily large, while the second type is time-varying with an arbitrarily large bound. With the first type of delays, under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis, (globally) stabilizing state and output feedback laws are constructed based on the solution to a family of parametric Riccati equations, which can be obtained explicitly through the solution of a parametric linear matrix equation. With the second type of delays, under the condition that the open-loop system is absolutely controllable with all its eigenvalues on the imaginary axis being zero, (global) state and output feedback laws are explicitly constructed based on the solution to a similar family of parametric Riccati equations. When the input is also subject to magnitude saturation, it is shown that semiglobal stabilization, instead of global stabilization, can still be achieved. Numerical examples illustrate the effectiveness of the proposed approach.

Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

Grégoire Allaire, Arnaud Münch, and Francisco Periago

SIAM J. Control Optim. 48, pp. 5333-5356 (24 pages)

Online Publication Date: November 09, 2010

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We consider a two-phase isotropic optimal design problem within the context of the transient heat equation. The objective is to minimize the average of the dissipated thermal energy during a fixed time interval $[0,T]$. The time-independent material properties are taken as design variables. A full relaxation for this problem was established in [A. Münch, P. Pedregal, and F. Periago, J. Math. Pures Appl. (9), 89 (2008), pp. 225–247] by using the homogenization method. In this paper, we study the asymptotic behavior as $T$ goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. Next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs, which appears in the form of a sequential laminate of rank at most $N$, the spatial dimension. An asymptotic analysis of the optimality conditions lets us prove that, for $T$ large enough, the order of lamination is, in fact, of at most $N-1$. Several numerical experiments in two dimensions complete our study.

Discrete Carleman Estimates for Elliptic Operators in Arbitrary Dimension and Applications

Franck Boyer, Florence Hubert, and Jérôme Le Rousseau

SIAM J. Control Optim. 48, pp. 5357-5397 (41 pages)

Online Publication Date: November 09, 2010

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In arbitrary dimension, we consider the semidiscrete elliptic operator $-\partial_t^2+\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$, where $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ is a finite-difference approximation of the operator $-\nabla_x(\Gamma(x)\nabla_x)$. For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate, such as a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano for $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ and a null-controllability result for the parabolic operator $\partial_t+\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ for the lower part of the spectrum of $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.

Feedback Stabilization of a Fluid-Structure Model

Jean-Pierre Raymond

SIAM J. Control Optim. 48, pp. 5398-5443 (46 pages) | Cited 1 time

Online Publication Date: November 11, 2010

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We study a system coupling the incompressible Navier–Stokes equations in a 2D rectangular-type domain with a damped Euler–Bernoulli beam equation, where the beam is a part of the upper boundary of the domain occupied by the fluid. Due to the deformation of the beam, the fluid domain depends on time. We prove that this system is exponentially stabilizable, locally about the null solution, with any prescribed decay rate, by a feedback control corresponding to a force term in the beam equation. The feedback is applied on the whole structure, and it is determined, via a Riccati equation, by solving an infinite time horizon control problem for the linearized model. A crucial step in this analysis consists of showing that this linearized system can be rewritten thanks to an analytic semigroup of which the infinitesimal generator has a compact resolvent.

Real-Time Nonlinear Optimization as a Generalized Equation

Victor M. Zavala and Mihai Anitescu

SIAM J. Control Optim. 48, pp. 5444-5467 (24 pages)

Online Publication Date: November 11, 2010

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We establish results for the problem of tracking a time-dependent manifold arising in real-time optimization by casting this as a parametric generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a single linear complementarity problem (LCP) at each time step. We derive sufficient conditions guaranteeing that the tracking error remains bounded to second order with the size of the time step even if the LCP is solved only approximately. We use these results to derive a fast, augmented Lagrangian tracking algorithm and demonstrate the developments through a numerical case study.

Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations

Michael Hintermüller and Ronald H. W. Hoppe

SIAM J. Control Optim. 48, pp. 5468-5487 (20 pages) | Cited 1 time

Online Publication Date: November 16, 2010

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We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular $L^2$-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.

Symmetry of Solutions to the Optimal Exit Time Control Problem

Jianghai Hu and Wei Zhang

SIAM J. Control Optim. 48, pp. 5488-5509 (22 pages)

Online Publication Date: November 16, 2010

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In this paper, we study the solutions to the optimal exit time control problem. Such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space, called the safe set, for as long as possible on average. By formulating the problem as an optimization problem with PDE constraints and using symmetrization techniques, we show that, when the safe set is a ball, the optimal feedback control (if it exists) must be radially symmetric. Furthermore, we show that, among all safe sets with a fixed volume, the ball is the best in that it yields the most efficient optimal exit time control. The proofs make essential use of the general isoperimetric inequality.

Morphological Control Problems with State Constraints

Thomas Lorenz

SIAM J. Control Optim. 48, pp. 5510-5546 (37 pages)

Online Publication Date: November 17, 2010

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In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space as states without linear structure. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space $\mathbb{R}^N$. Their family $\mathcal{K}(\mathbb{R}^N)$, however, does not have any obvious linear structure, but in combination with the popular Pompeiu–Hausdorff distance $d\!l$ it is a metric space. Here Aubin's framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space $(\mathcal{K}(\mathbb{R}^N),d\!l)$. Now various control problems, such as open-loop, relaxed, and closed-loop control problems, are formulated for compact sets depending on time, each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions. Finally, this framework is applied to image segmentation and provides a region growing method without regularity restrictions.

Search Games on Trees with Asymmetric Travel Times

Steve Alpern

SIAM J. Control Optim. 48, pp. 5547-5563 (17 pages)

Online Publication Date: December 08, 2010

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A point $H$ is hidden in a rooted tree $Q$ which is endowed with asymmetric distances (travel times) between nodes. We determine the randomized search strategy, starting from the root, which minimizes the expected time to reach $H$, in the worst case. This is equivalent to a zero-sum search game $\Gamma\left(Q\right)$, with minimizing Searcher, maximizing Hider, and payoff equal to the capture time. The worst Hiding distribution (over the leaves) from the Searcher's viewpoint is one where at every node $i$ the probability of each branch is proportional to the minimum time required to tour it from $i$. The optimal randomized search is a mixture over depth-first searches. We also consider briefly some other networks and the possibility of a mobile Hider. Our formulation with asymmetric travel times generalizes that of Gal [SIAM J. Control Optim., 17 (1979), pp. 99–122] for symmetric travel times and also the search games of Kikuta [J. Oper. Res., 38 (1995), pp. 70–88] and Kikuta and Ruckle [Naval Res. Logist., 41 (1994), pp. 821–831], who posited search costs $c_i$ at each node $i$ which were added to the travel time to obtain the payoff. We also briefly consider what happens if we allow the Searcher (Hider) to start (hide) at any leaf node. We determine when properties found by Dagan and Gal [Networks, 52 (2008), pp. 156–161] for the symmetric version of such games hold in our asymmetric context.

An Eigenvalue Perturbation Approach to Stability Analysis, Part I: Eigenvalue Series of Matrix Operators

Jie Chen, Peilin Fu, Silviu-Iulian Niculescu, and Zhihong Guan

SIAM J. Control Optim. 48, pp. 5564-5582 (19 pages) | Cited 1 time

Online Publication Date: December 08, 2010

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This two-part paper is concerned with stability analysis of linear systems subject to parameter variations, of which linear time-invariant delay systems are of particular interest. We seek to characterize the asymptotic behavior of the characteristic zeros of such systems. This behavior determines, for example, whether the imaginary zeros cross from one half plane into another, and hence plays a critical role in determining the stability of a system. In Part I of the paper we develop necessary mathematical tools for this study, which focuses on the eigenvalue series of holomorphic matrix operators. While of independent interest, the eigenvalue perturbation analysis has a particular bearing on stability analysis and, indeed, has the promise to provide efficient computational solutions to a class of problems relevant to control systems analysis and design, of which time-delay systems are a notable example.

An Eigenvalue Perturbation Approach to Stability Analysis, Part II: When Will Zeros of Time-Delay Systems Cross Imaginary Axis?

Jie Chen, Peilin Fu, Silviu-Iulian Niculescu, and Zhihong Guan

SIAM J. Control Optim. 48, pp. 5583-5605 (23 pages) | Cited 1 time

Online Publication Date: December 08, 2010

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This paper presents an application of the eigenvalue series developed in Part I [J. Chen et al., SIAM J. Control Optim., 48 (2010), pp. 5564–5582] to the study of linear time-invariant delay systems, focusing on the asymptotic behavior of critical characteristic zeros on the imaginary axis. We consider systems given in state-space form and as quasi-polynomials, and we develop an eigenvalue perturbation analysis approach which appears to be both conceptually appealing and computationally efficient. Our results reveal that the zero asymptotic behavior of time-delay systems can in general be characterized by solving a simple eigenvalue problem, and, additionally, when described by a quasi-polynomial, by computing the derivatives of the quasipolynomial.

Duality Between Invariant Spaces for Max-Plus Linear Discrete Event Systems

Michael Di Loreto, Stéphane Gaubert, Ricardo D. Katz, and Jean-Jacques Loiseau

SIAM J. Control Optim. 48, pp. 5606-5628 (23 pages)

Online Publication Date: December 16, 2010

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We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system.

Null Controllability of a Parabolic System with a Cubic Coupling Term

Jean-Michel Coron, Sergio Guerrero, and Lionel Rosier

SIAM J. Control Optim. 48, pp. 5629-5653 (25 pages)

Online Publication Date: December 16, 2010

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We consider a system of two parabolic equations with a forcing term present in one equation and a cubic coupling term in the other one. We prove that the system is locally null controllable.
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