Showing papers in "Siam Journal on Control and Optimization in 1982"
TL;DR: It is shown that Dk can be calculated simply on the basis of second derivatives of f so that the resulting Newton-like algorithm has a typically superlinear rate of convergence.
Abstract: We consider the problem $\min \{ f(x)|x \geqq 0\} $, and propose algorithms of the form $x_{k + 1} = [x_k - \alpha _k D_k
abla f(x_k )]^ + $, where $[ \cdot ]^ + $ denotes projection on the positive orthant, $\alpha _k $ is a stepsize chosen by an Armijo-like rule and $D_k $ is a positive definite symmetric matrix which is partly diagonal. We show that $D_k $ can be calculated simply on the basis of second derivatives of f so that the resulting Newton-like algorithm has a typically superlinear rate of convergence. With other choices of $D_k $ convergence at a typically linear rate is obtained. The algorithms are almost as simple as their unconstrained counterparts. They are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low-dimensional problems. The effectiveness of the Newton-like algorithm is demonstrated via computational examples involving as many as 10,000 variables. Extensions to general linearly constrained pr...
631 citations
TL;DR: In this paper, the controllability of systems of the form {dw} / {dt} = \mathcal {A}w + p(t) w + √ √ {B}w$ where W is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators on a Banach space X and W is a control.
Abstract: This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t} $ on a Banach space X, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of X to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in X of $w_0$ is impossible for $\dim X = \infty $. Examples of hyperbolic partial differential equations are provided.
335 citations
TL;DR: In this paper, a pathwise version of Lambda-t is introduced, which depends continuously on observation and control trajectories Y, U. Under a suitable non-degeneracy condition, the existence of an optimal control is obtained in a suitable class, larger than the usual class of controls admissible in the strict sense that $U_t$ is measurable on the ϵ-algebra ϵ t (Y) generated by observations $Y_s, $s \leqq t ).
Abstract: Stochastic control problems are considered in which a state process $X_t$ and an observation process $Y_t$ are governed by Ito-sense stochastic differential equations driven by independent Brownian motions. The control $U_t$ enters linearly in the dynamics of $X_t$. A “separated”control problem is introduced, in which the state at any time t is a measure $\Lambda_t$ representing an unnormalized conditional distribution for $X_t$ given $Y_s$, $U_s$ for $s \leqq t$. The method depends on introducing a pathwise version of $\Lambda_t$ which depends continuously on observation and control trajectories Y, U. Existence of an optimal control is obtained in a suitable class, larger than the usual class of controls admissible in the strict sense that $U_t$ is measurable on the $\sigma $-algebra $\mathcal{F}_t (Y)$ generated by observations $Y_s$, $s \leqq t$. The dynamics of $\Lambda_t$ are studied using a method of forward and backward partial differential equations. Under a suitable nondegeneracy condition, the m...
157 citations
TL;DR: In this paper, the authors consider the application of a general class of quasi-Newton methods to the solution of the classical equality constrained nonlinear optimization problem and develop necessary and sufficient conditions for the Q-superlinear convergence of such methods and present a companion linear convergence theorem.
Abstract: We consider the application of a general class of quasi-Newton methods to the solution of the classical equality constrained nonlinear optimization problem. Specifically, we develop necessary and sufficient conditions for the Q-superlinear convergence of such methods and present a companion linear convergence theorem. The essential conditions relate to the manner in which the Hessian of the Lagrangian function is approximated.
156 citations
TL;DR: In this article, a new min-max controller for uncertain dynamical systems is presented and a new model for switching action is presented, and the authors re-examine asymptotic stability.
Abstract: We discuss properties of min-max controllers (previously obtained) for uncertain dynamical systems. In particular, we present a new model for switching action and re-examine asymptotic stability. We demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty.
156 citations
TL;DR: In this article, the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered, where at each stage a bang-bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem.
Abstract: The numerical approximation of a parabolic time optimal control problem via piecewise linear splines, is considered. At each stage of the approximation a bang-bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem. The well-posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated. Numerical results for some one- and two-dimensional parabolic control problems are given.
127 citations
TL;DR: In this paper, the authors investigate properties of existence, unicity, representation, of the (causal) solutions of implicit linear systems (or generalized systems) when the underlying matrix pencil is singular.
Abstract: We investigate properties of existence, unicity, representation, of the (causal) solutions of implicit linear systems (or “generalized systems”) when the underlying matrix pencil is singular. We relate the geometric and the algebraic approaches. The main conclusion is that if the underlying matrix pencil is “column singular” (i.e., has a nonempty set of column minimal indices) the causal solutions, when they exist, can exactly be represented as the output of a classical two-player dynamical system, where the second player accounts for the nonunicity. Properties of the equivalent system are related to those of the singular matrix pencils made with the given matrices.
117 citations
TL;DR: In this article, a semigroup model with no explicit delays in control, but with an unbounded control operator is introduced, and it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation.
Abstract: The quadratic cost problem of evolution equations with delays in control is considered. A semigroup model which involves no explicit delays in control, but contains an unbounded control operator is introduced. With the aid of a family of approximating systems, it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Three examples are given to illustrate the theory. The filtering problem of evolution equations with observation delays is also solved through the duality relation.
101 citations
TL;DR: A mathematical model of the network with slow-learning algorithms distributed at various nodes is presented and two linear updating algorithms, under certain conditions, are shown to have desirable equilibrium behavior like load equalization and minimum blocking probability for the entire network.
Abstract: The aim of this paper is to develop a theory of adaptive routing in telephone networks using learning methods. A mathematical model of the network with slow-learning algorithms distributed at various nodes is presented. The algorithms update the routing probabilities on the basis of network feedback information (like call blocking or completion) only. Convergence of the routing strategies is established. Two linear updating algorithms, under certain conditions, are shown to have desirable equilibrium behavior like load equalization and minimum blocking probability for the entire network.
82 citations
TL;DR: The definition of a smooth nonlinear system as proposed recently by Willems, is elaborated as a natural generalization of the more common definitions of a Smooth nonlinear input-output system.
Abstract: The definition of a smooth nonlinear system as proposed recently by Willems, is elaborated as a natural generalization of the more common definitions of a smooth nonlinear input–output system. Mini...
82 citations
TL;DR: In this article, the authors consider the solution of a stochastic integral control problem and study its regularity, and characterize the optimal cost as the maximum solution of \[\begin{gathered} \forall...
Abstract: We consider the solution of a stochastic integral control problem and we study its regularity. In particular, we characterize the optimal cost as the maximum solution of \[\begin{gathered} \forall ...
TL;DR: In this article, a new method for quickly getting the ODE associated with the asymptotic properties of the stochastic approximation of the projected algorithm for the constrained problem is presented.
Abstract: A new method is presented for quickly getting the ODE (ordinary differential equation) associated with the asymptotic properties of the stochastic approximation $X_{n + 1} = X_n + a_n f(X_n ,\zeta _n )$ (or the projected algorithm for the constrained problem). Either $a_n \to 0$, or $a_n $ can be constant, in which case the analysis is on the sequence obtained when $a \to 0$. The method requires that $\{ X_n ,\zeta _{n - 1} \} $ be Markov with a “Feller” transition function, but little else. The simplest result requires that if $X_n \equiv x$, the corresponding noise process $\{ \zeta _n (x),n \geqq 0\} $ have a unique invariant measure; but the “nonunique” case can also be treated. No mixing condition is required, nor the construction of averaged test functions, and $f(\cdot ,\cdot )$ need not be continuous. A detailed analysis of the way that $\{ \zeta _n \} $ varies with $\{ X_n \} $ is not required. For the class of sequences treated, the conditions seem easier to verify than for other methods. There ...
TL;DR: If the players use a learning algorithm of the reward-penalty type, with proper choice of certain parameters in the algorithm, the expected value of the mixed strategies for both players can be made arbitrarily close to optimal strategies.
Abstract: This paper extends recent results [Lakshmivarahan and Narendra, Math. Oper. Res., 6 (1981), pp. 379–386] in two-person zero-sum sequential games in which the players use learning algorithms to update their strategies. It is assumed that neither player knows (i) the set of strategies available to the other player or (ii) the mixed strategy used by the other player or its pure realization at any stage. The outcome of the game depends on chance and the game is played sequentially. The distribution of the random outcome as a function of the pair of pure strategies chosen by the players is also, unknown to them. It is shown that if the players use a learning algorithm of, the reward-penalty type, with proper choice of certain parameters in the algorithm, the expected value of the mixed strategies for both players can be made arbitrarily close to optimal strategies.
TL;DR: In this article, a countable state-controlled Markov chain whose transition probability is specified up to an unknown parameter is considered, and the asymptotic behavior of this control scheme is investigated for the cases when the true parameter value does or does not belong to A, and for the case when $\zeta $ is chosen to minimize an average cost criterion.
Abstract: Consider a countable state controlled Markov chain whose transition probability is specified up to an unknown parameter $\alpha $ taking values in a compact metric space A. To each $\alpha $ is associated a prespecified stationary control law $\zeta (\alpha )$. The adaptive control law selects at each time t the control action $\zeta (\alpha _t ,x_t )$ where $x_t$ is the state and $\alpha_t$ is the maximum likelihood estimate of $\alpha $. The asymptotic behavior of this control scheme is investigated for the cases when the true parameter value $\alpha_0 $ does or does not belong to A, and for the case when $\zeta $ is chosen to minimize an average cost criterion. The analysis uses an appropriate extension of the notions of recurrence to nonstationary Markov chains.
TL;DR: In this article, the authors studied state variable representations of systems with a bounded-input, bounded-output stability property which are uniformly stabilizable and detectable are shown to have their associated homogeneous state-variable systems exponentially stable.
Abstract: Linear, finite-dimensional, time-varying systems are studied State variable representations of systems with a bounded-input, bounded-output stability property which are uniformly stabilizable and detectable are shown to have their associated homogeneous state-variable systems exponentially stable
TL;DR: In this paper, an analogous approach is proposed for systems over a principal ideal domain, where a correspondence between fractional representations of the transfer function matrix of a given i/o map and its reachable or observable realizations is established.
Abstract: The polynomial model approach to linear dynamical systems over a field was developed principally by P. A. Fuhrmann, starting in 1976 [J. Franklin Inst., 305 (1976), pp. 521–540]. In this paper an analogous approach is proposed for systems over a principal ideal domain.When the concept of extended linear i/o map is introduced, fractional representations of transfer function matrices arise naturally in this theoretical framework. A correspondence between fractional representations of the transfer function matrix of a given i/o map and its reachable or observable realizations is established. The McMillan degree of a linear i/o map is proved to be equal to the degree of the determinant of the matrix appearing as “denominator” in a coprime fractional representation of the associated transfer function matrix.
TL;DR: In this paper, the tangent sets to the inverse image of a set under a function were deduced and used to derive necessary conditions for extremality in the form of a multiplier rule.
Abstract: In this paper we deduce some relations concerning the tangent sets to the inverse image of a set under a function and we use these relations to derive necessary conditions for extremality in the form of a multiplier rule.
TL;DR: In this article, the authors considered a control system on a real analytic n-dimensional manifold M with dynamics described by real analytic vector fields and showed the relationship between the map (X, Y) being one-one on a neighborhood of $0 \in \mathbb{R}^n $ and other known necessary conditions for local controllability.
Abstract: Let X, Y be real analytic vector fields on a real analytic n-dimensional manifold M. Consider a control system on M with dynamics described by \[(1)\qquad {{dx} / {dt}} = X(x(t)) + u(t)Y(x(t)),\quad x(0) = p,\]where an admissible control is Lebesgue measurable with values $| {u(t)} | \leqq 1$. The relationship between the map \[(2)\qquad (s_1 , \cdots ,s_n )\to (\exp s_1 Y) \circ \cdots \circ (\exp s_n ({\text{ad}}^{n - 1} X,Y)) \circ (\exp tX)(p)\] being one-one on a neighborhood of $0 \in \mathbb{R}^n $ for each $0 < t < \varepsilon $ and other known necessary conditions for local controllability are studied. In dimension $ n =2 $, many necessary conditions are equivalent, and also sufficient. For $n \geqq 3$, the map (2) being locally one-one implies many necessary conditions are satisfied, but these need not be sufficient. Examples which illustrate what occurs geometrically are given.
TL;DR: In this paper, necessary and sufficient conditions are given for regulation of linear systems over rings using observers and causal dynamic state feedback systems with the polynomial fractional representation property, and the results are then used to obtain stabilizability conditions for systems over integers, delay-differential systems, and to obtain conditions to make a 2-D system nonrecursive.
Abstract: Necessary and sufficient conditions are given for regulation of linear systems over rings using observers and causal dynamic state feedback systems with the polynomial fractional representation property. The results are then used to obtain stabilizability conditions for systems over integers, delay-differential systems, systems over polynomial rings, and to obtain conditions to make a 2-D system nonrecursive.
TL;DR: This paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory and uses these representations in establishing new results for various classes of systems including split systems.
Abstract: This paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory. These representations are then used in establishing new results for various classes of systems including split systems. The relevance of these results to the regulation of various “nonclassical” classes of linear systems (for example, delay-differential systems, two-dimensional systems, etc.) is also discussed.
TL;DR: In this paper, a complementary result to the Jurdjevic-Kupka criterion for controllability of bilinear systems is obtained, which is based on the well-known Jurdjeevic Kupka (JK) criterion.
Abstract: This paper deals with bilinear systems. A complementary result to the well-known Jurdjevic–Kupka criterion for controllability is obtained.
TL;DR: In this article, the authors apply Grenander's method of sieves to the problem of estimation of the infinite dimensional parameter in a nonstationary linear diffusion model, and show that if the dimension of the sieves tends to infinity with the sample size with a rate not too fast, then the sequence of restricted maximum likelihood estimators for the parameter is consistent and asymptotically normal.
Abstract: In this paper, we apply Grenander’s method of sieves to the problem of estimation of the infinite dimensional parameter in a nonstationary linear diffusion model. We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximize the likelihood function. We show that if the dimension of the sieves tends to infinity with the sample size with a rate not too fast then the sequence of restricted maximum likelihood estimators for the parameter is consistent and asymptotically normal.
TL;DR: A (theoretical) solution involving Riccati integral equations and a general approximation scheme are proposed and axiomatically discussed for spline and averaging approximations.
Abstract: The linear regulator problem for delay equations is discussed. We propose a (theoretical) solution involving Riccati integral equations and then axiomatically discuss a general approximation scheme. The details are given for spline and averaging approximations.
TL;DR: In this article, the problem of optimally controlling a partially observed diffusion process is shown to have a solution in two cases: when the set of admissible controls is compact, or when the subset of controlled variables that can be controlled by randomized controls.
Abstract: The problem of optimally controlling a partially observed diffusion process is shown to have a solution in two cases: when the set of admissible controls is compact, or when the set of admissible controls is the set of randomized controls.
TL;DR: In this article, an extension of a previous article of Fleming and Pardoux [SIAM J. Control Optim, 20 (1982), pp. 261-265] on the control of partially observed diffusions, when the control enters linearly into the controlled stochastic differential equation was presented.
Abstract: This paper is an extension of a previous article of Fleming and Pardoux [SIAM J. Control Optim, 20 (1982), pp. 261–265] on the control of partially observed diffusions, when the control enters linearly into the controlled stochastic differential equation.
TL;DR: For a lower-semicontinuous convex function f, the approximate second-order directional derivative is defined through the directional derivative f'_\varepsilon (x;d) as discussed by the authors.
Abstract: For a lower-semicontinuous convex function f, the approximate second-order directional derivative $(d,\delta ) \mapsto f''_\varepsilon (x_0 ;d,\delta )$ is defined through the $\varepsilon $-directional derivative $f'_\varepsilon (x;d)$. The function $v_d :x \mapsto v_d (x) = f'_\varepsilon (x;d)$ is, for all $\varepsilon > 0$, locally Lipschitz on inf (domf) and, at those points where it is not differentiable, $v_d$ admits a directional derivative $v'_d (x_0 ;\delta )$ for all $\delta $, which we precisely denote by $f''_\varepsilon (x_0 ;d,\delta )$. The objective of the present work is two-fold: to classify all the possible differentiability properties of $v_d$ according to the behavior of the function $\lambda \mapsto f(x_0 + \lambda d)$ on $R_ + $, and to study the existence or nonexistence of the limit of $f''_\varepsilon (x_0 ;d,\delta )$ when $\varepsilon \to 0^ + $.
TL;DR: In this article, the authors considered the bilinear system with the property that any solution of the problem x(t) with x(0) \geqq 0 (i.e., all components of x nonnegative) will remain in the positive orthant, and detailed results were presented for the case $n = 2.
Abstract: Consider the bilinear system $\dot x = (A + uB)x$, $x \in R^n $, and u unrestricted. The system has the property that any solution $x(t)$ with $x(0) \geqq 0$ (i.e., all components of x nonnegative) will remain in the positive orthant, $R_ + ^n = \{ x \in R^n |x \geqq 0\} $ for $0 \leqq t 0$ and detailed results are presented for the case $n = 2$.
TL;DR: In this paper, the first order necessary conditions of optimality were obtained for boundary control problems governed by parabolic equations with nonlinear boundary value conditions, and the first-order optimality was obtained for the problem of boundary control with non-linear boundary values.
Abstract: First order necessary conditions of optimality are obtained for boundary control problems governed by parabolic equations with nonlinear boundary value conditions.
TL;DR: In this article, the authors considered the quasilinear parabolic equation and showed that one has existence of an optimal control under suitable hypotheses, and that this satisfies the appropriate optimality system.
Abstract: We consider the quasilinear parabolic equation $\dot y + {\bf A}y + {\bf F}(y) = \varphi $ on $Q: = (0,T) \times \Omega $. Viewing $\varphi $ as a control, we seek to minimize $J(\varphi ): = \| {\varphi - \hat \varphi } \|^2 + \lambda \left\| {y - \hat y} \|^2 + \mu \| {y(T) - \hat \eta } \|^2 $. Under suitable hypotheses it is shown that one has existence of an optimal control $\varphi _ * $ , and that this satisfies the appropriate optimality system. Further, for small data $J_ * : = \min J$ is small enough) one has global uniqueness and continuous dependence on the data.
TL;DR: In this paper, second-order conditions are given which are sufficient for a point to be a local minimizer for a finite-dimensional nonlinear programming problem with a finite number of constraints.
Abstract: Second-order conditions are given which are sufficient for a point to be a local minimizes for a finite-dimensional nonlinear programming problem with a finite number of constraints. In the most general theorem, the functions which comprise the problem are required only to be locally Lipschitz. The sufficiency conditions are given in terms of Clarke generalized gradients. These conditions assume a somewhat more familiar form when the functions in the problem are assumed to be both semismooth and subdifferentiably regular.