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SIAM J. on Control and Optimization

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1996

Volume 34, Issue 6, pp. 1831-2179


Infinite-Dimensional Hamilton–Jacobi Equations and Dirichlet Boundary Control Problems of Parabolic Type

Piermarco Cannarsa and Maria Elisabetta Tessitore

SIAM J. Control Optim. 34, pp. 1831-1847 (17 pages) | Cited 9 times

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The paper is concerned with an infinite-dimensional Hamilton–Jacobi equation. This equation is related to boundary control problems of Dirichlet type for semilinear parabolic systems.
The viscosity solution approach is adapted to the equation under consideration, using the properties of fractional powers of generators of analytic semigroups. An existence-and-uniqueness result for such problem is obtained.

Value Iteration in a Class of Communicating Markov Decision Chains with the Average Cost Criterion

Rolando Cavazos-Cadena

SIAM J. Control Optim. 34, pp. 1848-1873 (26 pages) | Cited 2 times

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Markov decision processes with denumerable state space and discrete time parameter are considered. The performance index of a control policy is the (lim sup expected) average cost criterion, and the the main structural restrictions on the model are the following: (i) under the action of any stationary policy, the state space is a communicating class; (ii) the cost function has an almost monotone—or penalized—structure [V S. Borkar, SIAM J. Control Optim., 21(1983), pp. 652–666; 22 (1984), pp. 965–978]; and (iii) some stationary policy induces an ergodic chain with finite average cost. In this context it is shown that the value iteration scheme can be used to construct convergent approximations of a solution to the optimality equation, as well as a sequence of stationary policies whose limit points are optimal.

Causal Feedback Optimal Control for Volterra Integral Equations

A. J. Pritchard and Yuncheng You

SIAM J. Control Optim. 34, pp. 1874-1890 (17 pages) | Cited 1 time

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The optimal control problem for Volterra integral equations with respect to quadratic criteria is studied by a projection causality approach. The work features a synthesis result where the optimal control is implemented via a linear causal feedback in which the feedback operator is determined by solving an independent Fredholm integral operator equation.

On the Use of Consistent Approximations for the Optimal Design of Beams

C. Kirjner Neto and E. Polak

SIAM J. Control Optim. 34, pp. 1891-1913 (23 pages) | Cited 2 times

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This paper presents a discretization strategy, based on the concept of consistent approximations, for certain optimal beam design problems, where the beam is modeled using Euler–Bernoulli beam theory. It is shown that any accumulation point of the sequence of the stationary points of the family of resulting approximating problems is a stationary point of the original, infinite-dimensional problem. The construction of approximating problems requires the development of a relaxation of constraints to ensure existence of solutions. The numerical solution of the approximating problems, by means of nonlinear programming algorithms that are not scale invariant, must be preceded by a change of variables to guard against deterioration of performance. The use of such approximating problems, in conjunction with a diagonalization strategy, is illustrated by a numerical example.

Classification of Generic Singularities for the Planar Time-Optimal Synthesis

B. Piccoli

SIAM J. Control Optim. 34, pp. 1914-1946 (33 pages) | Cited 16 times

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This paper is concerned with control systems on the plane with control appearing linearly. It is known that under generic conditions the problem of reaching points from the origin in minimum time admits a regular synthesis. The minimum time function is piecewise smooth, possibly nondifferentiable on a set that is a finite union of embedded submanifolds of dimension 1 or 0, called singularities. The purpose of the present paper is to provide a classification of all types of singularities arising under generic conditions.

Bifurcation Problems for Some Parametric Nonlinear Programs in Banach Spaces

Aubrey B. Poore

SIAM J. Control Optim. 34, pp. 1947-1971 (25 pages) | Cited 1 time

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Singularities in a class of parametric nonlinear programming problems in Banach spaces are investigated using bifurcation theory. Motivated by the Fritz John first-order necessary conditions and a nonstandard normalization of the multipliers, this problem is first formulated as a system of nonlinear equations. Conditions for this system to be Fredholm are then derived, and singularities are shown to arise from a violation of one or more of the following conditions: strict complementarily, surjectivity of the Fréchet derivative of the active constraints, and a second-order condition. A branching analysis is developed for each of these singularities under a second-order nondegeneracy assumption. Examples from the calculus of variations are then used to illustrate these singularities.

Stability Radii of Systems with Stochastic Uncertainty and Their Optimization by Output Feedback

D. Hinrichsen and A. J. Pritchard

SIAM J. Control Optim. 34, pp. 1972-1998 (27 pages) | Cited 11 times

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We consider linear plants controlled by dynamic output feedback which are subjected to blockdiagonal stochastic parameter perturbations. The stability radii of these systems are characterized, and it is shown that, for real data, the real and the complex stability radii coincide. A corresponding result does not hold in the deterministic case, even for perturbations of single-output feedback type. In a second part of the paper we study the problem of optimizing the stability radius by dynamic linear output feedback. Necessary and sufficient conditions are derived for the existence of a compensator which achieves a suboptimal stability radius. These conditions consist of a parametrized Riccati equation, a parametrized Liapunov inequality, a coupling inequality, and a number of linear matrix inequalities (one for each disturbance term). The corresponding problem in the deterministic case, the optimal $\mu $-synthesis problem, is still unsolved.

Linearization of Discrete-Time Systems

E. Aranda-Bricaire, Ü. Kotta, and C. H. Moog

SIAM J. Control Optim. 34, pp. 1999-2023 (25 pages) | Cited 30 times

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The algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete-time nonlinear systems. The accessibility problem for nonlinear discrete-time systems is shown to be easy to tackle by means of standard linear algebraic tools, whereas this is not the case for nonlinear continuous-time systems, in which case the most suitable approach is provided by differential geometry. The feedback linearization problem for discrete-time systems is recasted through the language of differential forms. In the event that a system is not feedback linearizable, the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a Pfaffian system. A discrete-time system may be linearizable by dynamic state feedback, though it is not linearizable by static state feedback. Necessary and sufficient conditions are given for the existence of a so-called linearizing output, which in turn is a sufficient condition for dynamic state feedback linearizability.

Numerical Stabilization of Bilinear Control Systems

Lars Grüne

SIAM J. Control Optim. 34, pp. 2024-2050 (27 pages) | Cited 9 times

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Extremal Lyapunov exponents for bilinear control systems with constrained control values are computed numerically by solving discounted optimal control problems. Based on this computation a numerical algorithm to calculate stabilizing control functions is developed.

Convergence of the BFGS Method for $LC^1 $ Convex Constrained Optimization

Xiaojun Chen

SIAM J. Control Optim. 34, pp. 2051-2063 (13 pages) | Cited 10 times

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This paper proposes a BFGS-SQP method for linearly constrained optimization where the objective function $f$ is required only to have a Lipschitz gradient. The Karush–Kuhn–Tucker system of the problem is equivalent to a system of nonsmooth equations $F(v) = 0$. At every step a quasi-Newton matrix is updated if $\| {F(v_k )} \|$ satisfies a rule. This method converges globally, and the rate of convergence is superlinear when $f$ is twice strongly differentiable at a solution of the optimization problem. No assumptions on the constraints are required. Thisgeneralizes the classical convergence theory of the BFGS method, which requires a twice continuous differentiability assumption on the objective function. Applications to stochastic programs with recourse on a CM5 parallel computer are discussed.

Existence Results for Noncoercive Variational Problems

Graziano Crasta and Annalisa Malusa

SIAM J. Control Optim. 34, pp. 2064-2076 (13 pages) | Cited 6 times

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The aim of this paper is to give an existence result for a class of one-dimensional, nonconvex, noncoercive problems in the calculus of variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.

Relaxation of Constrained Control Problems

E. N. Barron and R. Jensen

SIAM J. Control Optim. 34, pp. 2077-2091 (15 pages) | Cited 2 times

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The problem of relaxation of optimal control problems with state and control constraints is formulated in this paper. We determine that if the original problem consists of minimizing, over control functions $\zeta ( \cdot )$, $g(\xi (T))$ subject to ${{d\xi } / {ds}} = f(s,\xi ,\zeta )$, $t < s \leq T$, and $h(s,\xi (s),\zeta (s)) \leq 0$ for a.e. $t \leq s \leq T$, then the relaxed problem consists of minimizing, over measure-valued control functions $\mu ( \cdot )$, $g(\widehat\xi (T))$, subject to ${{d\widehat\xi } / {ds}} = \int_Z f (s,\widehat\xi (s),z)\mu (s,dz)$ and $\mu (s) - {\operatorname{ess}}\,\sup _z h(s,\widehat\xi (s),z) \leq 0$ for a.e. $t \leq s \leq T$. For each $s$ this is the essential supremum of $h$ in $z$ with respect to the measure $\mu (s)$.

Solvability and Right-Inversion of Implicit Nonlinear Discrete-Time Systems

T. Fliegner, Ü. Kotta, and H. Nijmeijer

SIAM J. Control Optim. 34, pp. 2092-2115 (24 pages) | Cited 4 times

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In this paper the problems of solvability and right-invertibility for implicit nonlinear discrete-time control systems are investigated. The concept “solvability” is defined in such a way that consistency of the implicit system equations is locally guaranteed for all input sequences, and an algorithm is introduced to verify the solvability of an implicit system in that sense. It is demonstrated how this mechanism may be used to decide on the right-invertibility or functional reproducibility of a given system. In contrast to previous work on right-invertibility for special classes of implicit nonlinear systems, the approach is not restricted to the characterization of right-invertibility, but it is shown in addition how an inverse system can actually be obtained. The theory is illustrated by a realistic economic example in which the inversion procedure is applied using formula manipulation.

A Target Recognition Problem: Sequential Analysis and Optimal Control

Mark H. A. Davis and Mohammad Farid

SIAM J. Control Optim. 34, pp. 2116-2132 (17 pages)

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An iterative computational method for determining the value function of an optimal control problem, related to target tracking, is presented. The target is assumed to be located in a fixed known position in space, but its identity (hostile or friendly) is known only with a prior probability. An observation of the target can be made at any location, and its error has position-dependent probability. The objective is finding the optimal navigation and observation strategy which leads to a final decision (i.e., the target is friendly or hostile). The value function is shown to be the unique viscosity solution of a variational inequality. Furthermore it is the unique fixed point of a nondecreasing concave operator.

Heavy Traffic Convergence of a Controlled, Multiclass Queueing System

L. F. Martins, S. E. Shreve, and H. M. Soner

SIAM J. Control Optim. 34, pp. 2133-2171 (39 pages) | Cited 6 times

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This paper provides a rigorous proof of the connection between the optimal sequencing problem for a two-station, two-customer-class queueing network and the problem of control of a multidimensional diffusion process, obtained as a heavy traffic limit of the queueing problem. In particular, the diffusion problem, which is one of “singular control” of a Brownian motion, is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network. The results are proved by a viscosity solution analysis of the related Hamilton–Jacobi–Bellman equations.

On the Lavrentiev Phenomenon for Optimal Control Problems with Second-Order Dynamics

Chih-Wen Cheng and Victor J. Mizel

SIAM J. Control Optim. 34, pp. 2172-2179 (8 pages) | Cited 2 times

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The present article examines control problems in one dimension for which there is an autonomous running cost and a specified terminal state. In this case, when the running cost involves only the control and the state, it is known that the infimal cost corresponding to any initial state is unaffected by the precise choice of $L^p $ space $(1 \leq p < \infty )$ which is specified for controls to be admissible. Here we show that the situation is different in the case of an autonomous running cost involving, in addition to the control, the state and its derivative. That is, despite the density of each space with higher exponent in those with lower exponent, the infimal cost will generally depend on the choice of $p$ if sign constraints are present.
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