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2011

Volume 49, Issue 3, pp. 927-1382


Risk Averse Shape Optimization

Sergio Conti, Harald Held, Martin Pach, Martin Rumpf, and Rüdiger Schultz

SIAM J. Control Optim. 49, pp. 927-947 (21 pages)

Online Publication Date: May 04, 2011

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Risk averse optimization has attracted much attention in finite dimensional stochastic programming. In this paper, we propose a risk averse approach in the infinite dimensional context of shape optimization. We consider elastic materials under stochastic loading. As measures of risk awareness we investigate the expected excess and the excess probability. The developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets. We incorporate topological derivatives to allow for topological changes in the shape optimization procedure. Numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios.

Weak Dynamic Programming Principle for Viscosity Solutions

Bruno Bouchard and Nizar Touzi

SIAM J. Control Optim. 49, pp. 948-962 (15 pages) | Cited 2 times

Online Publication Date: May 04, 2011

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We prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-made for the derivation of the dynamic programming equation in the sense of viscosity solutions.

Feedback Stabilization of Magnetohydrodynamic Equations

Cătălin-George Lefter

SIAM J. Control Optim. 49, pp. 963-983 (21 pages)

Online Publication Date: May 04, 2011

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We prove the local exponential stabilizability for the magnetohydrodynamic (MHD) system in space dimension 3, with internally distributed feedback controllers. These controllers take values in a finite dimensional space which is the unstable manifold of the elliptic part of the linearized operator. The stabilization of the linear system is derived using a unique continuation property for systems of parabolic and elliptic equations as well as the equivalence between controllability and feedback stabilizability in the case of finite dimensional systems. The feedback that stabilizes the linearized system is also stabilizing the nonlinear system in a certain interpolation space.

Mixed Finite Element Method for Dirichlet Boundary Control Problem Governed by Elliptic PDEs

Wei Gong and Ningning Yan

SIAM J. Control Optim. 49, pp. 984-1014 (31 pages)

Online Publication Date: May 05, 2011

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In this paper we study the finite element approximation of Dirichlet boundary control problems governed by elliptic PDEs. Based on a mixed variational scheme, we establish a mixed finite element approximation to the underlying optimal control problem. We consider the optimal control problems posed on both polygonal and general smooth domains, and we derive a priori error estimates for optimal control, state, and adjoint state. The optimal and quasi-optimal error estimates are obtained for problems on polygonal and smooth domains, respectively. Numerical experiments are provided to confirm our theoretical results.

Optimal Shape Design Subject to Elliptic Variational Inequalities

M. Hintermüller and A. Laurain

SIAM J. Control Optim. 49, pp. 1015-1047 (33 pages)

Online Publication Date: May 05, 2011

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The shape of the free boundary arising from the solution of a variational inequality is controlled by the shape of the domain where the variational inequality is defined. Shape and topological sensitivity analysis is performed for the obstacle problem and for a regularized version of its primal-dual formulation. The shape derivative for the regularized problem can be defined and converges to the solution of a linear problem. These results are applied to an inverse problem and to the electrochemical machining problem.

Geometry of the Limit Sets of Linear Switched Systems

Moussa Balde and Philippe Jouan

SIAM J. Control Optim. 49, pp. 1048-1063 (16 pages)

Online Publication Date: May 05, 2011

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This paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a nonstrict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call nonchaotic inputs, which generalize the different notions of inputs with dwell time. Next we turn our attention to the behavior for possibly chaotic inputs. Finally, we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

Finite Element Approximation for Shape Optimization Problems with Neumann and Mixed Boundary Conditions

Dan Tiba

SIAM J. Control Optim. 49, pp. 1064-1077 (14 pages)

Online Publication Date: May 05, 2011

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For optimal design problems, defined in domains of class $C$ and in arbitrary space dimension, governed by elliptic equations with boundary conditions of Neumann or mixed type, we introduce the corresponding discretized problems and we prove convergence results. The discretization method is of fixed domain type, in the sense that it is given in the domain that contains all the admissible open sets.

Penalty Method for Finite Horizon Stopping Problems

L. Stettner

SIAM J. Control Optim. 49, pp. 1078-1099 (22 pages)

Online Publication Date: May 10, 2011

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In this paper we use a penalty method to approximate a number of stopping problems over a finite horizon. In particular we prove existence and continuity of the value function corresponding to Dynkin games over a finite horizon. Since stopping problems can be studied in the context of Dynkin games, as a by-product we obtain continuity of the optimal stopping value. We then study Dynkin games with delayed stopping and finally impulse control. In each case the value function is approximated by a solution to a suitable penalty equation.

Stabilization by Means of Approximate Predictors for Systems with Delayed Input

Iasson Karafyllis

SIAM J. Control Optim. 49, pp. 1100-1123 (24 pages)

Online Publication Date: May 17, 2011

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Sufficient conditions for global stabilization of nonlinear systems with delayed input by means of approximate predictors are presented. An approximate predictor is a mapping which approximates the exact values of the stabilizing input for the corresponding system with no delay. A systematic procedure for the construction of approximate predictors is provided for globally Lipschitz systems. The resulting stabilizing feedback can be implemented by means of a dynamic distributed delay feedback law. Illustrative examples show the efficiency of the proposed control strategy.

On the Existence of Time Optimal Controls with Constraints of the Rectangular Type for Heat Equations

Qi Lü and Gengsheng Wang

SIAM J. Control Optim. 49, pp. 1124-1149 (26 pages)

Online Publication Date: May 17, 2011

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This paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations. An existence result of time optimal controls for such a problem is established. The rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations. In the finite dimensional case, there is no difference between such problems with control constraints of the rectangular type and those of the ball type, from the perspective of the study on the existence of optimal controls. Interestingly, in the infinite dimensional case, the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type. For infinite dimensional systems, the existence for time optimal controls with constraints of the ball type has already been discussed in the literature, while the study of the rectangular type has not been touched upon as far as we know.

Space-Time Adaptive Wavelet Methods for Optimal Control Problems Constrained by Parabolic Evolution Equations

Max D. Gunzburger and Angela Kunoth

SIAM J. Control Optim. 49, pp. 1150-1170 (21 pages)

Online Publication Date: May 19, 2011

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An adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation. First, the constraints are represented by means of a full weak space-time formulation as a linear system in $\ell_2$ in wavelet coordinates, following a recent approach by Schwab and Stevenson. Second, a quadratic cost functional involving a tracking-type term for the state and a regularization term for the distributed control is also formulated in terms of $\ell_2$ sequence norms of wavelet coordinates. This functional serves as a representer for a functional involving different Sobolev norms with possibly nonintegral smoothness parameter. Standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in $\ell_2$-coordinates. For this purpose, an adaptive method is proposed, which can be interpreted as an inexact gradient method for the control. In each iteration step, the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm. It is shown that the adaptive algorithm converges. Moreover, the algorithm is proved to be asymptotically optimal: the convergence rate achieved for computing each of the components of the solution (state, adjoint state, and control) up to a desired target tolerance is asymptotically the same as the wavelet best $N$-term approximation of each solution component, and the total computational work is proportional to the number of computational unknowns.

On the Robust Stability, Stabilization, and Stability Radii of Continuous-Time Infinite Markov Jump Linear Systems

Marcos G. Todorov and Marcelo D. Fragoso

SIAM J. Control Optim. 49, pp. 1171-1196 (26 pages)

Online Publication Date: May 19, 2011

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This paper addresses the robust stochastic stability and stabilization of continuous-time Markov jump linear systems (MJLS), with the Markov jump parameters taking values in a countably infinite set. It is assumed that the state and input matrices are subjected to norm-bounded uncertainty with a prespecified structure, which encompasses the block-diagonal setting. We introduce new robust analysis and synthesis characterizations such that, unlike previous approaches in the MJLS literature, the scaling parameters are treated as decision variables in linear matrix inequalities. As a by-product, new contributions to the theory of stability radii of MJLS are provided. When restricted to the finite case, we further introduce new adjoint linear matrix inequality (LMI) characterizations for each of the robust analysis and synthesis problems, as well as for stability radii. Besides the interest in its own right, the adjoint approach allows us to verify that, in the general MJLS case, there is a gap between the complex stability radius and what can be assessed with scaled versions of the small-gain theorem. This suggests a fundamental limitation of the robustness against linear perturbations that the H$_\infty$ control of MJLS may provide. Some numerical examples, which include the robust control of two interconnected oscillators, illustrate the main results.

Trigger Strategy Equilibriums in Stochastic Differential Games with Information Time Lags: An Analysis of Cooperative Production Strategies

Tao Yao, Susan H. Xu, and Bin Jiang

SIAM J. Control Optim. 49, pp. 1197-1220 (24 pages)

Online Publication Date: May 24, 2011

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In the current literature of differential games, most studies formulate optimal strategies in feedback (Markovian) equilibriums and ignore repeated interactions among players. In many real-world settings, however, the competitors' action history may have impacts on a firm's decisions. This paper considers the production strategies for several competing firms in an oligopolistic industry. A firm's profit is determined by a continuous-time stochastic demand shock process together with the production strategies of all firms in the industry. A firm's decision is only observable by other firms after an information time lag, induced by the production lead time. We study a history-dependent trigger strategy, whereby firms adopt the cooperative strategy until a firm fails to do so, and thereafter punish the deviating firm by applying the noncooperative equilibrium. We show that, as long as the information time lag is less than a threshold, the trigger strategy is both a Nash equilibrium and a Pareto optimum. We obtain the analytical solutions to the threshold and investigate how the threshold is affected by market growth rate, market volatility, the number of competitors in the industry, and the risk-free rate. Moreover, we investigate the repeated games in a continuous-time setting and provide a tractable approach to derive the trigger-type repeated equilibrium in a Nash–Cournot framework. While the derivation of equilibrium strategies in a stochastic continuous-time setting can be quite challenging, we obtain a solution that is not only analytically simple but also practically applicable.

Some Results on the Controllability of Coupled Semilinear Wave Equations: The Desensitizing Control Case

Louis Tebou

SIAM J. Control Optim. 49, pp. 1221-1238 (18 pages)

Online Publication Date: May 24, 2011

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First, we consider a semilinear hyperbolic equation with partially known initial data in a bounded domain. For this system, we construct a locally distributed control that desensitizes a certain norm of the state. This result is new, and the method for proving it combines a judicious application of the Carleman estimate and a localization technique. Then we discuss possible extensions of our result. Finally, we apply our method to the exact controllability of two hyperbolic equations (one semilinear, and the other one linear) coupled in cascade through the boundary, the control acting in one equation only.

Controller Design for a Class of Uncertain Systems with Guaranteed Performance

Jianming Lian and Stanislaw H. .Zak

SIAM J. Control Optim. 49, pp. 1239-1261 (23 pages)

Online Publication Date: May 26, 2011

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The tracking control problem for a class of multi-input multi-output nonlinear systems with unknown system dynamics and disturbances is considered. An approximation-free output feedback controller based on feedback linearization is proposed. It incorporates a high-gain observer to simultaneously estimate the state tracking error and the system uncertainty. The state tracking error of the closed-loop system is guaranteed to be semiglobally uniformly ultimately bounded with respect to a closed ball of radius controlled by the design parameters. The key feature of the proposed tracking control strategy is the simple controller architecture without any components to approximate unknown system dynamics. The robustness of the proposed controller in the presence of measurement noise is also analyzed and illustrated with simulations.

Output-Feedback Stabilization of Stochastic High-Order Nonlinear Systems under Weaker Conditions

Wuquan Li, Xue-Jun Xie, and Siying Zhang

SIAM J. Control Optim. 49, pp. 1262-1282 (21 pages)

Online Publication Date: May 31, 2011

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Under the weaker conditions on the power order and the nonlinear functions, this paper investigates the output-feedback stabilization problem for a class of stochastic high-order nonlinear systems. Based on the backstepping design method and homogeneous domination technique, the closed-loop system can be proved to be globally asymptotically stable in probability.

An Approximation Method for Exact Controls of Vibrating Systems

Nicolae Cîndea, Sorin Micu, and Marius Tucsnak

SIAM J. Control Optim. 49, pp. 1283-1305 (23 pages)

Online Publication Date: June 02, 2011

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We propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's “stabilizability implies controllability” principle with the Galerkin method. The main new feature of this work consists of giving precise error estimates. In order to test the efficiency of the method, we consider two illustrative examples (with the finite element approximations of the wave and the beam equations) and describe the corresponding simulations.

Pontryagin Maximum Principle on Almost Lie Algebroids

Janusz Grabowski and MichałJóźwikowski

SIAM J. Control Optim. 49, pp. 1306-1357 (52 pages)

Online Publication Date: June 02, 2011

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The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics. On the other hand, in the presented approach the reduced and unreduced PMPs are parts of the same universal formalism. The framework is based on a very general concept of homotopy of measurable paths and the geometry of AL algebroids.

3D Shape Optimization in Viscous Incompressible Fluid under Oseen Approximation

Michael Zabarankin and Anton Molyboha

SIAM J. Control Optim. 49, pp. 1358-1382 (25 pages)

Online Publication Date: June 09, 2011

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The integral equation constrained optimization approach to finding three-dimensional (3D) minimum-drag shapes for bodies translating in viscous incompressible fluid under the Oseen approximation of the Navier–Stokes equations is presented. The approach formulates the Oseen flow problem as a boundary integral equation and finds solutions to this equation and its adjoint in the form of function series. Minimum-drag shapes, being also represented by function series, are then found by the adjoint equation–based method with a gradient-based algorithm, in which the gradient for shape series coefficients is determined analytically. Compared to partial differential equation (PDE) constrained optimization coupled with the finite element method (FEM), the approach reduces dimensionality of the flow problem, solves the issue with region truncation in exterior problems, finds minimum-drag shapes in semianalytical form, and has fast convergence. The approach is demonstrated in three drag minimization problems for different Reynolds numbers for (i) a body of constant volume, (ii) a torpedo with only fore-and-aft noses being optimized, and (iii) a body of constant volume following another body of fixed shape. The minimum-drag shapes in problem (i) are in good agreement with the existing optimality conditions and conform to those obtained by PDE constrained optimization.
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