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SIAM J. on Control and Optimization

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1991

Volume 29, Issue 6, pp. 1251-1508


Balanced Parametrization of Classes of Linear Systems

Raimund Ober

SIAM J. Control Optim. 29, pp. 1251-1287 (37 pages) | Cited 21 times

Online Publication Date: July 14, 2006

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Canonical forms and parametrizations are presented for several sets of minimal systems of given dimension: asymptotically stable systems, allpass systems, bounded real systems, positive real systems, minimum-phase systems, and the class of all minimal systems. The approach is based on balancing techniques for these classes of systems. Applications of these results to Hankel operators and model reduction are discussed.

Stochastic Regulator Theory for a Class of Abstract Wave Equations

A. V. Balakrishnan

SIAM J. Control Optim. 29, pp. 1288-1299 (12 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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A class of steady-state stochastic regulator problems for abstract wave equations in a Hilbert space—of relevance to the problem of feedback control of large space structures using co-located controls/sensors—is studied. Both the control operator, as well as the observation operator, are finite-dimensional. As a result, the usual condition of exponential stabilizability invoked for existence of solutions to the steady-state Riccati equations is not valid. Fortunately, for the problems considered it turns out that strong stabilizability suffices. In particular, a closed form expression is obtained for the minimal (asymptotic) performance criterion as the control effort is allowed to grow without bound.

Feedback Equivalence for Nonlinear Systems and the Time Optimal Control Problem

B. Bonnard

SIAM J. Control Optim. 29, pp. 1300-1321 (22 pages) | Cited 12 times

Online Publication Date: July 14, 2006

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This article relates the classification of affine control systems under the action of the feedback group, with a differential classification of a set of constrained Hamiltonian vector fields, arising from Pontryagin’s Maximum Principle, for the time minimal control problem. They represent the singularities of the input-state mapping. This relation provides a method to compute feedback invariants.

Some Characterizations of Optimal Trajectories in Control Theory

Piermarco Cannarsa and Halina Frankowska

SIAM J. Control Optim. 29, pp. 1322-1347 (26 pages) | Cited 31 times

Online Publication Date: July 14, 2006

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Several characterizations of optimal trajectories for the classical Mayer problem in optimal control are provided. For this purpose the regularity of directional derivatives of the value function is studied: for instance, it is shown that for smooth control systems the value function $V$ is continuously differentiable along an optimal trajectory $x:[t_0 ,1] \to {\bf R}^n $ provided $V$ is differentiable at the initial point $(t_0 ,x(t_0 ))$.
Then the upper semicontinuity of the optimal feedback map is deduced. The problem of optimal design is addressed, obtaining sufficient conditions for optimality. Finally, it is shown that the optimal control problem may be reduced to a viability one.

New Size $ \times $ Curvature Conditions for Strict Quasiconvexity of Sets

Guy Chavent

SIAM J. Control Optim. 29, pp. 1348-1372 (25 pages) | Cited 10 times

Online Publication Date: July 14, 2006

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Given a closed, not necessarily convex set $D$ of a Hilbert space, the problem of the existence of a neighborhood $\mathcal{V}$ on which the projection on $D$ is uniquely defined and Lipschitz continuous is considered, and such that the corresponding minimization problem has no local minima. After having equipped the set $D$ with a family $\mathcal{P}$ of paths playing for $D$ the role the segments play for a convex set, the notion of strict quasiconvexity of $(D,\mathcal{P})$ is defined, which will ensure the existence of such a neighborhood $\mathcal{V}$. Two constructive sufficient conditions for the strict-quasiconvexity of $D$ are given, the $R_G $-size $ \times $ curvature condition and the $\Theta $-size $ \times $ curvature condition, which both amount to checking for the strict positivity of quantities defined by simple formulas in terms of arc length, tangent vectors, and radii of curvature along all paths of $\mathcal{P}$. An application to the study of wellposedness and local minima of a nonlinear least squares problem is given.

$H_\infty $ Control with Transients

Pramod P. Khargonekar, Krishan M. Nagpal, and Kameshwar R. Poolla

SIAM J. Control Optim. 29, pp. 1373-1393 (21 pages) | Cited 34 times

Online Publication Date: July 14, 2006

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In $H_\infty $ (or uniformly optimal) control problems, it is usually assumed that the system initial conditions are zero. In this paper, an $H_\infty $-like control problem that incorporates uncertainty in initial conditions is formulated. This is done by defining a worst-case performance measure. Both finite and infinite horizon problems are considered. Necessary and sufficient conditions are derived for the existence of controllers that yield a closed-loop system for which the above-mentioned performance measure is less than a prespecified value. State-space formulae for the controllers are also presented.

$H^\infty $ Control of Linear Time-Varying Systems: A State-Space Approach

R. Ravi, K. M. Nagpal, and P. P. Khargonekar

SIAM J. Control Optim. 29, pp. 1394-1413 (20 pages) | Cited 47 times

Online Publication Date: July 14, 2006

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In this paper the standard problem of $H^\infty $ control theory for finite-dimensional linear time-varying continuous-time plants is considered. The problem is: given a real number $\gamma > 0$, find (if one exists) an internally stabilizing controller such that the closed-loop operator norm is less than $\gamma $. Under rather weak assumptions on the plant model, it is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space formulae for one solution to the problem are also given.

Velocity Method and Lagrangian Formulation for the Computation of the Shape Hessian

Michel C. Delfour and Jean-Paul Zolésio

SIAM J. Control Optim. 29, pp. 1414-1442 (29 pages) | Cited 15 times

Online Publication Date: July 14, 2006

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The object of this paper is to study the shape Hessian of a shape functional by the velocity (speed) method. It contains a review and an extension of the velocity method and its connections with methods using first- or second-order perturbations of the identity. The key point is that all these methods yield the same shape gradient but different and unequal shape Hessian since each method depends on a choice of “connection.” However, for autonomous velocity fields the velocity method yields a canonical bilinear Hessian. Expressions obtained by other methods can be recovered by adding to that canonical term the shape gradient acting on the acceleration of the velocity field associated with the choice of perturbation of the identity. The second part of the paper is an application of the Lagrangian method with function space embedding to compute the shape gradient and Hessian of a simple cost function associated with the nonhomogeneous Dirichlet problem.

Numerical Methods for Stochastic Singular Control Problems

Harold J. Kushner and Luiz Felipe Martins

SIAM J. Control Optim. 29, pp. 1443-1475 (33 pages) | Cited 14 times

Online Publication Date: July 14, 2006

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The paper develops a powerful class of numerical methods for stochastic singular control problems. The basic models used are diffusion or reflected diffusions, but the method is of general applicability. The central idea is that of the Markov chain approximation method, where an approximation to the control problem is found for which an optimal solution is computable, and which is an arbitrarily good approximation to the original problem and its optimal value function. The methods are convenient to program and use (and they have been used with success), and they cover a wide variety of problems. In fact, for the singular problem, they seem to be the only ones currently available. Owing to problems in proving tightness of certain processes that occur in the convergence proofs, the methods of proof used for the nonsingular problems need modifications. Examples of useful approximations, the algorithms, and the convergence proofs are given. To illustrate the power of the methods, two classes of problems are dealt with: the first is a class of discounted problems, and the second is an average-cost-per-unit time problem subject to some constraints, which arises in the study of multicustomer class queueing networks under conditions of heavy traffic. The method is applicable to the more standard singular control and ergodic problems with greater ease.

A Framework for Two-Dimensional Hyperstability Theory Based Provably Convergent Adaptive Two-Dimensional IIR Filtering

Sankar Basu

SIAM J. Control Optim. 29, pp. 1476-1508 (33 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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A set of results for two-dimensional quarter plane causal systems reminiscent of one-dimensional hyperstability theory have been reported. The key to this development is a little known result of Landau [Math. Ann., 62 (1906), p. 272], which asserts that a positive polynomial in two variables can be expressed as the sum of squares of polynomials in one variable whose coefficients are real rational functions of the other variable. The tools used are largely based on notions of passivity and the results obtained can be interpreted as a two-dimensional quarter plane causal generalization of the fact that if the total flow of energy into a dissipative system is upper bounded then both input and output asymptotically die out to zero. An adaptive two-dimensional recursive filtering scheme potentially useful in propagating wave type two-dimensional problems is considered next. It is then shown via our two-dimensional hyperstability results that the adaptive scheme converges in an appropriate sense.
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