Showing papers in "Siam Journal on Control and Optimization in 1987"
TL;DR: In this paper, the control of a class of discrete event processes, i.e., processes that are discrete, asynchronous and possibly non-deterministic, is studied. And the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language, where the control process is described as the generator of a formal language, while the supervisor is constructed from the grammar of a specified target language that incorporates the desired closed-loop system behavior.
Abstract: This paper studies the control of a class of discrete event processes, i.e. processes that are discrete, asynchronous and possibly nondeter-ministic. The controlled process is described as the generator of a formal language, while the controller, or supervisor, is constructed from the grammar of a specified target language that incorporates the desired closed-loop system behavior. The existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language. Two examples are provided.
3,432 citations
TL;DR: In this paper, a general consumption/investment problem is considered for an agent whose actions cannot affect the market prices, and who strives to maximize total expected discounted utility of both consumption and investment.
Abstract: A general consumption/investment problem is considered for an agent whose actions cannot affect the market prices, and who strives to maximize total expected discounted utility of both consumption ...
1,107 citations
TL;DR: In this paper, the supremal controllable sublanguage S of a given language L is characterized as the largest fixpoint of a monotone operator, and the fixpoint S can be computed as the limit of the (finite) sequence given by the language.
Abstract: The concept of controllable language has been shown to play a basic role in the existence theory of supervisory controls for discrete event processes. In this paper the supremal controllable sublanguage S of a given language L is characterized as the largest fixpoint of a monotone operator $\Omega $. In the case where the languages involved are regular it is shown that the fixpoint S can be computed as the limit of the (finite) sequence $\{ {K_j } \}$ given by $K_{j + 1} = \Omega (K_j )$, $K_0 = L$. An effective computational algorithm is developed, and three examples are provided for illustration.
749 citations
TL;DR: In this article, a general sufficient condition for local controllability of a nonlinear system at an equilibrium point is given. And a number of new necessary and sufficient conditions are obtained.
Abstract: We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and Grossmann, are shown to be particular cases of this result Also, a number of new sufficient conditions are obtained All these results follow from one simple general principle, namely, that local controllability follows whenever brackets with certain symmetries can be “neutralized,” in a suitable way, by writing them as linear combinations of brackets of a lower degree Both the class of symmetries and the definition of “degree” can be chosen to suit the problem
628 citations
TL;DR: A modular approach to the synthesis of state feedback controls for the problem of maintaining a predicate on the state set of a discrete dynamic system invariant by determining conditions under which it is possible to synthesize the appropriate control in a modular fashion.
Abstract: We examine a modular approach to the synthesis of state feedback controls for the problem of maintaining a predicate on the state set of a discrete dynamic system invariant. Dynamical systems are modeled by automata together with a mechanism for enabling and disabling a subset of state transitions. The basic problem of interest is to ensure by appropriate control action that a given predicate on the state set of the process remains invariantly true whenever it is initially satisfied. Assuming the predicate can be decomposed into the conjunction or disjunction of component predicates, we determine conditions under which it is possible to synthesize the appropriate control in a modular fashion.
403 citations
TL;DR: In this paper, it is shown that the Timoshenko beam can be uniformly stabilized by means of a boundary control, and a numerical study on the spectrum is also presented, showing that the beam is uniformly stabilized with respect to the boundary control.
Abstract: It is shown that the Timoshenko beam can be uniformly stabilized by means of a boundary control. A numerical study on the spectrum is also presented.
379 citations
TL;DR: In this article, the authors considered the simplest type of such structures which is formed by N seri..., a large number of components coupled end to end in the form of a chain.
Abstract: Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this paper, we consider the simplest type of such structures which is formed by N seri...
378 citations
TL;DR: In this paper, the principal results in control theory in the context of continuous-time linear systems are discussed, but the focus is on the mathematical theory rather than computational methods, and not on computational methods.
Abstract: This expository paper sets out the principal results in ${\bf H}_\infty $ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods.
358 citations
TL;DR: In this paper, the controllability of an abstract semilinear control system under the assumption of a simple form and can be easily checked in many examples, and the same assumption is used for controllable linear control systems.
Abstract: While various equivalent conditions for controllability have been obtained in the case of linear control systems, controllability problems of semilinear control systems usually require some complicated and limited assumptions. In this paper we show the approximate controllability of an abstract semilinear control system under the assumption, which has a simple form and can be easily checked in many examples.
222 citations
TL;DR: In this article, the authors consider the problem of finding local minima in a global minimum of $U:\mathbb{R}^n \to \mathbb {R}.
Abstract: We seek a global minimum of $U:\mathbb{R}^n \to \mathbb{R}$. The solution to $( * )({d / {dt}})X(t) = -
abla U(X(t))$ will find local minima. Using the idea of simulated annealing, we consider th...
219 citations
TL;DR: In this article, a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators is established, which is similar to our framework.
Abstract: This paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators.
TL;DR: In this paper, it was shown that solutions of linear inequalities, linear programs and certain linear complementarity problems are Lipschitz continuous with respect to changes in the right-hand side data of the problem.
Abstract: It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g. those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the problem. Solutions of linear programs are not Lipschitz continuous with respect to the coefficients of the objective function. The Lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right-hand side perturbation.
TL;DR: The present value of the rewards associated with a discrete-time Markov process has a probability distribution which depends on the initial state as mentioned in this paper, and the authors in this paper apply fixed point theory to a system of equations for the distribution functions of the present value.
Abstract: The present value of the rewards associated with a discrete-time Markov process has a probability distribution which depends on the initial state. The first part of the paper applies fixed point theory to a system of equations for the distribution functions of the present value. The second part of the paper expands the model to a Markov decision process (MDP) and considers the maximization of the expected utility of the present value when the utility function is exponential.
TL;DR: In this paper, the authors prove that the time-optimal trajectories are finite concatenations of "bang-bang" and singular arcs, with local bounds on the number of switchings.
Abstract: For single-input $C^\infty $ systems in the plane, in which the control enters linearly, we prove, if the system is suitably nondegenerate, that the time-optimal trajectories are finite concatenations of “bang-bang” and singular arcs, with local bounds on the number of switchings
TL;DR: In this article, the authors examined the validity of such claims and found that this relationship, interpreted as a differential inclusion involving the generalized gradient, is indeed true, almost everywhere and at the endpoints, for a very large class of nonsmooth optimal control problems.
Abstract: Let $V(t,x)$ be the infimum cost of an optimal control problem, viewed as a function of the initial time and state $(t,x)$. Dynamic Programming is concerned with the properties of $V( \cdot , \cdot )$ and in particular with its characterization as a solution to the Hamilton–Jacobi–Bellman equation. Heuristic arguments have long been advanced relating the Maximum Principle to Dynamic Programming according to \[p(t) = - V_x \left( {t,x_0 (t)} \right).\] Here $x_0 ( \cdot )$ is the minimizing state function under consideration and $p( \cdot )$ is the costate function of the Maximum Principle. In this paper we examine the validity of such claims and find that this relationship, interpreted as a differential inclusion involving the generalized gradient, is indeed true, almost everywhere and at the endpoints, for a very large class of nonsmooth optimal control problems.
TL;DR: This paper proposes an algorithm of the successive quadratic programming (SQP) type which, unlike other algorithms of this type, does enjoy such properties, and is shown to converge from any initial point, locally superlinearly.
Abstract: When iteratively solving optimization problems arising from engineering design applications, it is sometimes crucial that all iterates satisfy a given set of “hard” inequality constraints, and generally desirable that the objective function value improve at each iteration. In this paper, we propose an algorithm of the successive quadratic programming (SQP) type which, unlike other algorithms of this type, does enjoy such properties. Under mild assumptions, the new algorithm is shown to converge from any initial point, locally superlinearly. Numerically tested, it has proven to be competitive with the most successful currently available nonlinear programming algorithms, while the latter do not exhibit the desired properties.
TL;DR: The asymptotic properties of extensions of the type of distributed or decentralized stochastic approximation proposed in [1] are developed and have numerous potential applications in decentralized estimation, detection and adaptive control, or in decentralized Monte Carlo simulation for system optimization.
Abstract: The asymptotic properties of extensions of the type of distributed or decentralized stochastic approximation proposed in [1] are developed. Such algorithms have numerous potential applications in decentralized estimation, detection and adaptive control, or in decentralized Monte Carlo simulation for system optimization (where they can exploit the possibilities of parallel processing). The structure involves several isolated processors (recursive algorithms) that communicate to each other asynchronously and at random intervals. The asymptotic (small gain) properties are derived. The communication intervals need not be strictly bounded, and they and the system noise can depend on the (communicating) system state. State space constraints are also handled. In many applications, the dynamical terms are merely indicator functions, or have other types of discontinuities. The “typical” such case is also treated, as is the case where there is noise in the communication. The linear stochastic differential equation ...
TL;DR: In this paper, a generalized approach is taken to linear and quadratic programming in which dual as well as primal variables may be subjected to bounds, and constraints may be represented through penalties.
Abstract: A generalized approach is taken to linear and quadratic programming in which dual as well as primal variables may be subjected to bounds, and constraints may be represented through penalties. Corresponding problem models in optimal control related to continuous-time programming are then set up and theorems on duality and the existence of solutions are derived. Optimality conditions are obtained in the form of a global saddle point property which decomposes into an instantaneous saddle point condition on the primal and dual control vectors at each time, along with an endpoint condition.
TL;DR: The structure of the dual allows the successful application of a distributed asynchronous method whereby relaxation iterations are carried out in parallel by several processors in arbitrary order and with arbitrarily large interprocessor communication delays.
Abstract: We consider the solution of the single commodity strictly convex network flow problem in a distributed asynchronous computation environment. The dual of this problem is unconstrained, differentiable, and well suited for solution via Gauss-Seidel relaxation. We show that the structure of the dual allows the successful application of a distributed asynchronous method whereby relaxation iterations are carried out in parallel by several processors in arbitrary order and with arbitrarily large interprocessor communication delays.
TL;DR: A state-of-the-art review of many iterative methods for solving large convex quadratic programs is given, within which a unified iterative scheme will be introduced and its convergence analyzed.
Abstract: In this paper, we give a state-of-the-art review of many iterative methods for solving large convex quadratic programs. We attempt to classify several of the more basic methods in two categories, within each of which a unified iterative scheme will be introduced and its convergence analyzed. Hybrid iterative methods (such as the proximal point algorithm and a diagonalization scheme) that make use of the more basic schemes will also be described. The results of an extensive computer experimentation which is aimed at comparing the relative performance of the various methods will be reported and discussed. Finally, several important topics which require future research will be highlighted.
TL;DR: In this paper, the problem of approximating a given stable rational matrix function G(s) of McMillan degree n by a function Ĝ(s)+F(s), where ǫ is antistable, is reduced to a symmetric Wiener-Hopf factorization problem, which can be solved by applying a result of Kaashoek and Ran.
Abstract: We consider the problem of approximating a given stable rational matrix function G(s) of McMillan degree n by a function Ĝ(s)+F(s), where Ĝ has McMillan degree l
TL;DR: In this paper, the authors considered the problem of single commodity network flow with both linear and strictly convex possibly non-differentiable arc costs and studied the convergence of a dual Gauss-Seide! type relaxation method that is well suited for parallel computation.
Abstract: We consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs. For the case where all arc costs are strictly convex we study the convergence of a dual Gauss-Seide! type relaxation method that is well suited for parallel computation. We then extend this method to the case where some of the arc costs are linear. As a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in Bertsekas (1) and Bertsekas and Tseng (2).
TL;DR: In this paper, the authors considered an infinite horizon stochastic production planning problem with demand assumed to be a continuous-time Markov chain and showed that a unique optimal feedback solution exists, after first showing that convex viscosity solutions to the associated dynamic programming equation are continuously differentiable.
Abstract: This paper considers an infinite horizon stochastic production planning problem with demand assumed to be a continuous-time Markov chain. The problems with control (production) and state (inventory) constraints are treated. It is shown that a unique optimal feedback solution exists, after first showing that convex viscosity solutions to the associated dynamic programming equation are continuously differentiable.
TL;DR: In this article, the effect of changes in problem functions and/or distributions in certain two-stage stochastic programming problems with recourse is analyzed under reasonable assumptions that the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters.
Abstract: We analyze the effect of changes in problem functions and/or distributions in certain two-stage stochastic programming problems with recourse. Under reasonable assumptions the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters (including the probability distribution in the second stage).
TL;DR: For arbitrary single-input real analytic systems in the plane, in which the control enters linearly, the authors proved the existence of a regular synthesis for the optimal control problem in which it is desired to minimize the integral of a strictly positive real analytic Lagrangian that does not depend on the control variable.
Abstract: For arbitrary single-input real analytic systems in the plane, in which the control enters linearly, we prove the existence of a regular synthesis for the optimal control problem in which it is desired to minimize the integral of a strictly positive real analytic Lagrangian that does not depend on the control variable. The analysis proceeds by applying our previous results on nondegenerate $\mathcal{C}^\infty $ systems, as well as those on arbitrary real analytic ones, to study the local structure of the time optimal trajectories. The structure of the optimal trajectories for our problem is derived by reparametrization of time. The existence of a synthesis is then proved by using subanalytic set theory.
TL;DR: For arbitrary single-input real analytic systems in the plane, the authors showed that the time-optimal trajectories are finite concatenations of "bang-bang" and singular arcs, with local bounds on the number of switchings.
Abstract: For arbitrary single-input real analytic systems in the plane, in which the control enters linearly, we prove that the time-optimal trajectories are finite concatenations of “bang-bang” and singular arcs, with local bounds on the number of switchings. No “nondegeneracy” assumption is made other than real-analyticity. The analysis proceeds by applying our previous results on nondegenerate $C^\infty $ systems to study the behavior of the trajectories on a neighborhood of every point, except for the points in a discrete set of “branch points.” The branch points are then handled by a combination of control-theoretic arguments and the use of subanalytic set theory.
TL;DR: In this article, the authors present backward, forward, and backward-forward models that weaken previous sufficient conditions and that include, but are not restricted to, optimization problems, including extremization and nonextremization problems.
Abstract: The unifying purpose of the abstract dynamic programming models is to find sufficient conditions on the recursive definition of the objective function that guarantee the validity of the dynamic programming iteration. This paper presents backward, forward, and backward-forward models that weaken previous sufficient conditions and that include, but are not restricted to, optimization problems. The backward-forward model is devoted to the simultaneous solution of a collection of interrelated sequential problems based on the independent computation of a cost-to-arrive function and a cost-to-go function. Several extremization and nonextremization problems illustrate the applicability of the proposed models.
TL;DR: In this article, the pole-zero cancellations which occur in a class of H-optimal control problems which may be embedded in the configuration of Fig. 1 are studied, and a general bound on the McMillan degree of all controllers which are stabilizing and lead.
Abstract: The aim of this paper is to study the pole-zero cancellations which occur in a class of H-optimal control problems which may be embedded in the configuration of Fig. 1. H control problems are said to be of the first kind if both P12(s) and P21(s) are square but not necessarily of the same size. It is primarily this class Of problems which will concern us here. A general bound on the McMillan degree of all controllers which are stabilizing and lead.to a closed loop which satisfies (((s)((o -< p (p need not be optimal in the L-norm sense) is derived. As illustrated in Fig. 1, (s) is the transfer function relating yl(s) to Ul(S). If the McMillan degree of P(s) in Fig. is n, we show that in the single-loop (SISO) case the corresponding (unique) H-optimal controller never requires more than n states. In the multivariable case, there is a continuum of optimal controllers whose McMillan degree satisfies this same bound, although other controllers with higher McMillan degree also exist. The derivation of these bounds require several steps, each of which is of independent system theoretic interest.
TL;DR: For the abstract controlled Volterra equation, the reachable set invariant under affine perturbations was studied in this paper, where conditions were obtained under which a nonlinear perturbation of an otherwise linear control problem can be viewed as a non-linear control problem.
Abstract: For the abstract controlled Volterra equation \[( * )\qquad x(t) = \bar x(t) + \int_0^t {{\boldsymbol{\psi}} (t,s)[\phi (s,x(s)) + {\bf B}(s)v(s)]ds} \] on $[0,T]$, we consider the reachable set $\mathcal{K}_\phi : = \{ {x(T):( * ){\text{ for some }}v \in \mathfrak{B}} \}$. Viewing $\phi ( \cdot , \cdot )$ as a nonlinear perturbation of an otherwise linear control problem, conditions are obtained under which $\mathcal{K}_\phi = \mathcal{K}_0 $ for a suitable class $\mathfrak{F}$ of such nonlinear perturbations $\phi $ whenever the linear problem is known to have a reachable set invariant under affine perturbations: $\phi ( \cdot , \cdot ) = g \in \mathfrak{Z}$. The results generalize those obtained by Naito [7] for control of the heat equation.
TL;DR: In this article, the effect of non-minimum phase zeros on the weighted sensitivity of a general delay system was analyzed. But the results were not extended to more general delay systems.
Abstract: In this paper we extend some of the results of [IEEE Trans. Automat. Control, AC-31 (1986), pp. 763–766] to more general delay systems. In particular, we analyze the effect of the interaction of delays and nonminimum phase zeros on the $H^\infty $-optimal weighted sensitivity.