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SIAM J. on Control and Optimization

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2001

Volume 39, Issue 6, pp. 1651-1973


Variational Analysis of the Abscissa Mapping for Polynomials

James V. Burke and Michael L. Overton

SIAM J. Control Optim. 39, pp. 1651-1676 (26 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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The abscissa mapping on the affine variety $\cMn$ of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping is continuous on $\cMn$, but not Lipschitz continuous. Furthermore, its natural extension to the linear space $\cPn$ of polynomials of degree n or less is not continuous. In our analysis of the abscissa mapping, we use techniques of modern nonsmooth analysis described extensively in Variational Analysis (R. T. Rockafellar and R. J.-B. Wets, Springer-Verlag, Berlin, 1998). Using these tools, we completely characterize the subderivative and the subgradients of the abscissa mapping, and establish that the abscissa mapping is everywhere subdifferentially regular. This regularity permits the application of our results in a broad context through the use of standard chain rules for nonsmooth functions. Our approach is epigraphical, and our key result is that the epigraph of the abscissa map is everywhere Clarke regular.

On the Controllability of the Linearized Benjamin--Bona--Mahony Equation

Sorin Micu

SIAM J. Control Optim. 39, pp. 1677-1696 (20 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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We study the boundary controllability properties of the linearized Benjamin--Bona--Mahony equation $$\left\{ \begin{array}{ll} u_t-u_{xxt}+u_x=0,& x\in(0,1),\,t > 0,\\ u(t,0)=0,\,\, u(1,t)=f(t),&t>0. \end{array} \right.
We show that the equation is approximately controllable but not spectrally controllable (no finite linear combination of eigenfunctions, other than zero, is controllable). Next, we prove a finite controllability result and we estimate the norms of the controls needed in this case.

Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

Luis H. R. Alvarez

SIAM J. Control Optim. 39, pp. 1697-1710 (14 pages) | Cited 17 times

Online Publication Date: July 26, 2006

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We consider a class of singular stochastic control problems arising frequently in applications of stochastic control. We state a set of conditions under which the optimal policy and its value can be derived in terms of the minimal r-excessive functions of the controlled diffusion, and demonstrate that the optimal policy is of the standard local time type. We then state a set of weak smoothness conditions under which the value function is increasing and concave, and demonstrate that given these conditions increased stochastic fluctuations decrease the value and increase the optimal threshold, thus postponing the exercise of the irreversible policy. In line with previous studies of singular stochastic control, we also establish a connection between singular control and optimal stopping, and show that the marginal value of the singular control problem coincides with the value of the associated stopping problem whenever 0 is not a regular boundary for the controlled diffusion.

An LMI-Based Algorithm for Designing Suboptimal Static $\cal H_2/\cal H_\infty$ Output Feedback Controllers

F. Leibfritz

SIAM J. Control Optim. 39, pp. 1711-1735 (25 pages) | Cited 65 times

Online Publication Date: July 26, 2006

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We consider the problem of designing a suboptimal ${\cal H}_2/{\cal H}_{\infty}$ feedback control law for a linear time-invariant control system when a complete set of state variables is not available. This problem can be necessarily restated as a nonconvex optimization problem with a bilinear, multiobjective functional under suitably chosen linear matrix inequality (LMI) constraints. To solve such a problem, we propose an LMI-based procedure which is a sequential linearization programming approach. The properties and the convergence of the algorithm are discussed in detail. Finally, several numerical examples for static ${\cal H}_2/{\cal H}_{\infty}$ output feedback problems demonstrate the applicability of the considered algorithm and also verify the theoretical results numerically.

Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass

Bao-Zhu Guo

SIAM J. Control Optim. 39, pp. 1736-1747 (12 pages) | Cited 21 times

Online Publication Date: July 26, 2006

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Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler--Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability are readily obtained. The main results of [ SIAM J. Control Optim., 36 (1998), pp. 1962--1986] are concluded as a special case, and the additional conditions imposed there are removed.

Achieving Arbitrarily Large Decay in the Damped Wave Equation

Carlos Castro and Steven J. Cox

SIAM J. Control Optim. 39, pp. 1748-1755 (8 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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We exhibit a sequence of viscous dampings for the fixed string that yields arbitrarily fast attenuation of any and all initial disturbances. The limit case produces extinction of all solutions in finite time.

The Topological Asymptotic for PDE Systems: The Elasticity Case

Stéphane Garreau, Philippe Guillaume, and Mohamed Masmoudi

SIAM J. Control Optim. 39, pp. 1756-1778 (23 pages) | Cited 96 times

Online Publication Date: July 26, 2006

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The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole. In this paper, such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique. The method is general and can be easily adapted to other linear PDEs and other types of boundary conditions.

Optimal Control Problems for Stochastic Reaction-Diffusion Systems with Non-Lipschitz Coefficients

Sandra Cerrai

SIAM J. Control Optim. 39, pp. 1779-1816 (38 pages) | Cited 7 times

Online Publication Date: July 26, 2006

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By using the dynamic programming approach, we study a control problem for a class of stochastic reaction-diffusion systems with coefficients having polynomial growth. In the cost functional a non-Lipschitz term appears, and this allows us to treat the quadratic case, which is of interest in the applications. The corresponding Hamilton--Jacobi--Bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates. The main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system.

The Lattice Structure of Behaviors

Shiva Shankar

SIAM J. Control Optim. 39, pp. 1817-1832 (16 pages) | Cited 6 times

Online Publication Date: July 26, 2006

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If a linear, continuous, shift invariant distributed system is considered as a (dynamical) system converting input signals to output signals, then this information is encapsulated in the impulse response or the transfer function of the system. The set of all transfer functions has the structure of a ring, corresponding to the operations of parallel and cascade connections of two systems. However, in the behavioral theory of Willems, a system is not described in terms of its input-output transformation property. Indeed, the concept of a behavior does not even need the notions of inputs and outputs and is therefore more fundamental than the classical concept of a system given by its transfer function. The question then arises as to what is the structure of the set of all behaviors. This paper argues that the relevant structure here is that of a modular lattice.

Stabilizability of Systems of One-Dimensional Wave Equations by One Internal or Boundary Control Force

Farid Ammar Khodja and Ahmed Bader

SIAM J. Control Optim. 39, pp. 1833-1851 (19 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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We study the internal and boundary stabilizability of a system of wave equations by one control force. We prove that the "classical" internal damping applied to only one of the equations never gives exponential stability if the wave speeds are different and, if the wave speeds are the same, we give explicit necessary and sufficient conditions for the stability to occur. We also study the simultaneous boundary stabilization of the same system.

Variational Inequality Problems with a Continuum of Solutions: Existence and Computation

P. Jean-Jacques Herings, Dolf Talman, and Zaifu Yang

SIAM J. Control Optim. 39, pp. 1852-1873 (22 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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In this paper three sufficient conditions are provided under each of which an upper semicontinuous point-to-set mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes. These results follow in a constructive way by designing a new simplicial algorithm. The algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a consequence, our resultsgeneralize Browder's fixed point theorem [Summa Brasiliensis Mathematicae, 4 (1960), pp. 183--191] and an earlier result by the authors [Math. Oper. Res., 21 (1996), pp. 675--696] (1996) on the n-dimensional unit cube. An application in economics and some numerical examples are also discussed.

Input-Output-to-State Stability

Mikhail Krichman, Eduardo D. Sontag, and Yuan Wang

SIAM J. Control Optim. 39, pp. 1874-1928 (55 pages) | Cited 25 times

Online Publication Date: July 26, 2006

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This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators," and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.

Relative Flatness and Flatness of Implicit Systems

Paulo Sérgio Pereira da Silva and Carlos Corrêa Filho

SIAM J. Control Optim. 39, pp. 1929-1951 (23 pages) | Cited 7 times

Online Publication Date: July 26, 2006

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In this work we define the concept of relative flatness of a system with respect to a subsystem. The subsystem associated to a set of outputs of a system is constructed, and called here output subsystem. It is shown that the relative flatness of a system with respect to the output subsystem implies the flatness of the corresponding implicit system obtained by setting these outputs to zero. A sufficient condition of relative flatness based on a relative derived flag is presented. Based on these results, a sufficient condition for the flatness of a class of nonlinear implicit systems is obtained.

Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

Kazuyoshi Mori and Kenichi Abe

SIAM J. Control Optim. 39, pp. 1952-1973 (22 pages) | Cited 9 times

Online Publication Date: July 26, 2006

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This paper is concerned with the coordinate-free approach to control systems. The coordinate-free approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for multi-input multi-output (MIMO) systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32 (1994), pp. 1675--1695]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered.
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