Showing papers in "Siam Journal on Control and Optimization in 1979"
TL;DR: Submodular games as mentioned in this paper are finite non-cooperative games in which the set of feasible joint decisions is a sublattice and the cost function of each player has properties of submodularity and antitone differences.
Abstract: A submodular game is a finite noncooperative game in which the set of feasible joint decisions is a sublattice and the cost function of each player has properties of submodularity and antitone differences. Examples of submodular games include 1) a game version of a system with complementary products; 2) an extension of the minimum cut problem to a situation where players choose from different sets of nodes and perceive different capacities, with special cases being a game with players choosing whether or not to participate in available economic activities and a game version of the selection problem; 3) the pricing problem of competitors producing substitute products; 4) a game version of the facility location problem; and 5) a game with players determining their optimal usage of available products. A fixed point approach establishes the existence of a pure equilibrium point for certain submodular games. Two algorithms which correspond to fictitious play in dynamic games generate sequences of feasible join...
658 citations
TL;DR: In this paper, necessary and sufficient conditions for the invertibility of nonlinear control systems of the form where the state space is a real analytic manifold are given, where the class of real analytic functions which can appear as outputs of a given nonlinear system is described, and a prefilter is constructed to generate the required control.
Abstract: This paper gives necessary and sufficient conditions for the invertibility of nonlinear control systems of the form $\dot x = A(x) + uB(x)$; $y = c(x)$, where the state space is a real analytic manifold. For invertible systems we construct nonlinear inverse systems. These results are used to study the question of functional controllability for nonlinear systems. The class of real analytic functions which can appear as outputs of a given nonlinear system is described, and a prefilter is constructed to generate the required control.
290 citations
TL;DR: It is proved that the optimal control and the dual multipliers for strictly convex control problems with convex constraints on the state and the control are Lipschitz continuous in time.
Abstract: We study Lipschitz continuity properties for “constrained processes”. As applications of our general theory, we consider mathematical programs and optimal control problems. We show that if thegradients of the binding constraints satisfy an independence condition, then the solution and the dual multipliers of a convex mathematical program are a Lipschitz continuous function of the data. Similarly, it is proved that the optimal control and the dual multipliers for strictly convex control problems with convex constraints on the state and the control are Lipschitz continuous in time. In both applications, estimates of the Lipschitz constant are given.
225 citations
TL;DR: In this paper, different notions of observability are compared for systems defined by polynomial difference equations, and the main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination.
Abstract: Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.
183 citations
TL;DR: In this paper, the second order necessary and sufficient conditions for a minimum, exact smooth penalty theorems and augmented Lagrangian duality theorem were given for the case when more than one collection of Lagrange multipliers exist and the second-order quadratic function can be lower estimated by norms weaker than that in which the cost function and constrained mappings are differentiable.
Abstract: The paper contains second order necessary and sufficient conditions for a minimum, exact smooth penalty theorems and augmented Lagrangian duality theorems which cover the case when more than one collection of Lagrange multipliers exist and the second order quadratic function can be lower estimated only by norms weaker than that in which the cost function and constrained mappings are differentiable.
173 citations
TL;DR: In this paper, the Riccati integral equations for linear-quadratic control problems involving evolution operators on Hilbert spaces are derived and shown to have a common solution, which yields the closed-loop structure of the optimal control.
Abstract: The two Riccati integral equations for linear-quadratic control problems involving evolution operators on Hilbert spaces are derived and shown to have a common solution, which yields the closed-loop structure of the optimal control. Riccati integral equations, instead of differential equations, arise because evolution operators are used to represent system dynamics. The operator representing the closed-loop control perturbs the evolution operator representing the uncontrolled system to produce a second evolution operator, representing the optimally controlled system, hence the two Riccati integral equations in terms of these two evolution operators, respectively.Having both Riccati integral equations facilitates the extension of the analysis of optimal control on finite time intervals to the analysis of optimal control on infinite time intervals, and then existence, uniqueness, and stability results for periodic solutions of the Riccati equations are obtained. Finally, sufficient conditions are given for ...
172 citations
TL;DR: In this paper, the objective function of any solvable linear program can be perturbed by a differentiable, convex or Lipschitz continuous function in such a way that a solution of the original linear program is also a Karush-Kuhn-Tucker point, local or global solution of a perturbed program, or (b) each global solution for the perturbed problem is also the linear program.
Abstract: The objective function of any solvable linear program can be perturbed by a differentiable, convex or Lipschitz continuous function in such a way that (a) a solution of the original linear program is also a Karush–Kuhn–Tucker point, local or global solution of the perturbed program, or (b) each global solution of the perturbed problem is also a solution of the linear program.
167 citations
TL;DR: The notion of strong structural controllability is introduced in this paper, which is defined as "a system is strongly structurally controllable if, whatever values (other than zero) the indeterminate parameters of the system may take, the system is controllably fixed".
Abstract: For linear time-invariant control systems, the system parameters values may vary or be never known precisely with the exception of fixed zeros determined by the physical structure of the system. Dividing the system parameters into two categories, indeterminate parameters and fixed zero parameters, the notion of strong structural controllability is introduced with the following meaning: A system is strongly structurally controllable if, whatever values (other than zero) the indeterminate parameters of the system may take, the system is controllable.The two necessary and sufficient graph theoretic conditions for linear time-invariant control systems to be strongly structurally controllable are given. The one is fundamental for strong structural controllability and shows what is the essential set of indeterminate parameters the change of whose values may cause a system to be uncontrollable. The other is useful because of its very simple and intuitive form in graph theoretic aspect. For sparse systems, its co...
165 citations
TL;DR: In this paper, the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of a mean square continuous stochastic vector process y with stationary increments and a rational spectral density y such that y is finite and nonsingular is considered.
Abstract: Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density $\Phi $ such that $\Phi (\infty )$ is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman–Busy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algori...
161 citations
TL;DR: In this article, a class of optimal control problems is considered in which the cost functional is locally Lipschitz (not necessarily convex or differentiable) and the dynamics linear and/or convex.
Abstract: A class of optimal control problems is considered in which the cost functional is locally Lipschitz (not necessarily convex or differentiable) and the dynamics linear and/or convex. By using genera...
138 citations
TL;DR: In this paper, a new approach to the theory of necessary conditions is described, which replaces the initial constrained problem by a problem without constraints, and the core of the approach is a reduction theorem which replaces a constrained problem with a nonconstrained problem.
Abstract: A new approach to the theory of necessary conditions is described. The core of the approach is a reduction theorem which replaces the initial constrained problem by a problem without constraints ha...
TL;DR: In this article, a new stabilization scheme of combined viscous damping and compensation was proposed for the wave equation and a stabilizing control for a regulator problem was also derived, which is based on a distributed parameter control and stabilization for wave equations.
Abstract: The present note makes a further study on the distributed parameter control and stabilization for the wave equation in an earlier article (SIAM J. Control Optim., 17 (1979) pp. 66–81). Decay rates and control time are improved by a new stabilization scheme of combined viscous damping and compensation. A stabilizing control for a regulator problem is also derived.
TL;DR: In this article, the authors consider search games in which the searcher moves along a continuous trajectory in a set Q until he captures the hider, where Q is either a network or a two-dimensional region.
Abstract: We consider search games in which the searcher moves along a continuous trajectory in a set Q until he captures the hider, where Q is either a network or a two (or more) dimensional region. We distinguish between two types of games; in the first type which is considered in the first part of the paper, the hider is immobile while in the second type of games which is considered in the rest of the paper, the hider is mobile. A complete solution is presented for some of the games, while for others only upper and lower bounds are given and some open problems associated with those games are presented for further research.
TL;DR: In this article, it is proved that the complementarity problem can be solved by a homotopy algorithm developed by Chow, Mallet-Paret, Yorke, and Watson.
Abstract: Let F be a $C^2 $ map from n-dimensional Euclidean space into itself. It is proved that, under some mild conditions on F, the complementarily problem $z \geqq 0$, $F(z) \geqq 0$, $zF(z) = 0$ can be solved by a homotopy algorithm developed by Chow, Mallet-Paret, Yorke, and Watson. The algorithm is globally convergent with probability one, and uses Mangasarian’s nonlinear system equivalent to the complementarity problem. Convergence theorems for the algorithm simultaneously prove existence of a solution, although existence is already well known. Some computational results are included.
TL;DR: In this paper, a new class of augmented Lagrangians is introduced for solving equality constrained problems via unconstrained minimization techniques, and it is proved that a solution of the constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single constrained minimization of the augmented LGA.
Abstract: In this paper a new class of augmented Lagrangians is introduced, for solving equality constrained problems via unconstrained minimization techniques. It is proved that a solution of the constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of the augmented Lagrangian. In particular, in the linear quadratic case, the solution is obtained by minimizing a quadratic function. Numerical examples are reported.
TL;DR: In this article, it was shown that a natural continuous parameter interpolation of the tail part of the SA converges weakly to a stationary Gauss-Markov process, from which the asymptotic properties of SA with constant coefficients can be obtained.
Abstract: Asymptotic properties (as $n \to \infty $, and then $a \to 0$) of the Stochastic Approximation (SA) algorithm \[ ( * )\qquad ,X_{n + 1}^a = X_n^a + ah_a \left( {X_n^a ,\xi _n^a } \right) \] are obtained, where $h_a $ is not necessarily additive in $\xi _n^a $. If $Eh_a (x,\xi _n^a ) = g(x) + O(a)$ and $\dot x = g(x)$ is globally asymptotically stable about a solution $x_t \equiv \theta $, then the asymptotic properties of $\{ {{(X_n^a - \theta )} / {\sqrt a }}\} \equiv \{ U_n^a \} $ are developed. In particular, it is shown that (as $a \to 0$) a natural continuous parameter interpolation of the tail part of $\{ U_n^a \} $ converges weakly to a stationary Gauss-Markov process, from which the asymptotic properties of $\{ U_n^a \} $ and $\{ X_n^a \} $ can be obtained for small a. The conditions on $\{ \xi _n^a \} $ are reasonable from the point of view of the usual applications to adaptive systems and identification. These results seem to be the first of their type for SA’s with constant coefficients. Some r...
TL;DR: In this paper, the problem of parameter identification for delay systems motivated by examples from aerody-namics and biochemistry is considered and a class of theoretical approximation schemes are developed and two specific cases (averaging and spline) are shown to be included in this treatment.
Abstract: Parameter identification problems for delay systems motivated by examples from aerody- namics and biochemistry are considered. The problem of estimation of the delays is included. Using approximation results from semigroup theory, a class of theoretical approximation schemes is developed and two specific cases (“averaging” and “spline” methods) are shown to be included in this treatment. Convergence results, error estimates, and a sample of numerical findings are given.
TL;DR: In this paper, the stochastic regulator problems and optimal stationary control as well as stability are studied for infinite dimensional systems with state and control dependent noise, and the model is described by a semigroup and Wiener processes in Hilbert space.
Abstract: In this paper stochastic regulator problems and optimal stationary control as well as stability are studied for infinite dimensional systems with state and control dependent noise. The stochastic model is described by a semigroup and Wiener processes in Hilbert space and Wonham’s approach using differential generators and dynamic programming is extended to infinite dimensions.
TL;DR: In this paper, two conditional gradient algorithms are considered for convex F and Lipschitz continuous F, and it is shown that they converge geometrically or in finitely many steps according to whether $a(\sigma ) > 0$ for ≥ 0, or $a(sigma) \geqq A\sigma ^2 $ with ≥ 0.
Abstract: Two conditional gradient algorithms are considered for the problem $\min _\Omega F$, with $\Omega $ a bounded convex subset of a Banach space. Neither method requires line search; one method needs no Lipschitz constants. Convergence rate estimates are similar in the two cases, and depend critically on the continuity properties of a set valued operator T whose fixed points $\xi $, are the extremals of F in $\Omega $. The continuity properties of T at $\xi $ are determined by the way the function $a(\sigma ) = \inf \{ \rho = \langle {F'(\xi ), {y - \xi } \rangle } \mid y \in \Omega , \| {y - \xi } \| \geqq \sigma \} $ grows with increasing $\sigma $. It is shown that for convex F and Lipschitz continuous $F'$, the algorithms converge like $o({1 / n})$, geometrically, or in finitely many steps, according to whether $a(\sigma ) > 0$ for $\sigma > 0$, or $a(\sigma ) \geqq A\sigma ^2 $ with $A > 0$, or $a(\sigma ) \geqq A\sigma $ with $A > 0$. These three abstract conditiions are closely related to established ...
TL;DR: In this paper, the authors discuss uniqueness questions for identification of coefficients in a second-order, linear, one-dimensional, parabolic partial differential equation, where the unknowns are spatially-varying coefficients appearing in the equation.
Abstract: This paper discusses uniqueness questions for identification of coefficients in a second-order, linear, one-dimensional, parabolic partial differential equation. Here, the unknowns are spatially-varying coefficients appearing in the equation. The solution of an initial-boundary value problem is observed at one point over a finite time interval. Conditions are given under which the eigenvalues associated with the problem are uniquely determined by such an observation. The coefficients are not uniquely determined. If, however, the equation is in normal form, the single coefficient which appears is, in certain cases, uniquely determined. This can be established by obtaining the spectral function or by obtaining the eigenvalues for two different boundary value problems and applying existing results (I. M. Gelfand and B. M. Levitan (1959), N. Levison (1949)).
TL;DR: Collocation at Gauss points is shown to be a high order accurate discretization of certain unconstrained optimal control problems and best possible convergence rates are established along with superconvergence results.
Abstract: Collocation at Gauss points is shown to be a high order accurate discretization of certain unconstrained optimal control problems. Best possible convergence rates are established along with superconvergence results.
TL;DR: In this article, the authors proved a priori bound on the number of switchings in a neighborhood of a point x, provided that the following condition is satisfied: in every neighborhood of x, it is possible to express, for each j, the vector field $[g,({\text{ad }}f)^j (g)]$ as a linear combination of the $(n, n)^i (g)$, $i \leqq j + 1$, in such a way that the coefficient of the coefficient in this expression is bounded in absolute value
Abstract: For systems of the form $\dot x = f(x) + ug(x)$, with f and g analytic, and $ - 1 \leqq u \leqq 1$, we prove a bang-bang theorem with a priori bounds on the number of switchings, provided that the following condition is satisfied: in a neighborhood of every point x, it is possible to express, for each j, the vector field $[g,({\text{ad }}f)^j (g)]$ as a linear combination of the $({\text{ad }}f)^i (g)$, $i \leqq j + 1$, in such a way that the coefficient of $({\text{ad }}f)^{^j + 1} (g)$ in this expression is bounded in absolute value by a constant $c < 1$.
TL;DR: A new class of outer approximations algorithms which incorporate constraint dropping schemes, based on the use of certain types of optimality functions, which are commonly used in minimization algorithms for defining stationary points are presented.
Abstract: This paper presents a new class of outer approximations algorithms which incorporate constraint dropping schemes. The algorithms are based on the use of certain types of optimality functions, which are commonly used in minimization algorithms, for defining stationary points. The algorithms are implementable in that all the inner minimizations and maximizations need to be carried out only approximately. It is shown that any accumulation point constructed by these algorithms is both feasible and stationary.
TL;DR: In this paper, the authors investigated the possibility of developing algorithms involving only rational computations on the matrices, which would determine whether the rank condition is everywhere satisfied, i.e. whether the bilinear system has the accessibility or controllability property.
Abstract: We consider a bilinear control system on $\mathbb{R}_0^n = \mathbb{R}^n - \{ 0\} $\[\frac{{dx}}{{dt}}(A_0 + \sum\limits_{i = 1}^r {u_i (t)A_i } )x,\] where $x \in \mathbb{R}^n ,A_0$, $A_1 , \cdots ,A_r $ are $n \times n$ real matrices, $\mathfrak{g}$ is the Lie algebra they generate, and $u_1 (t), \cdots ,u_r (t)$ are real valued control functions. Although there exists a standard rank condition in terms of the Lie algebra g which Sussman and Jurdjevic have shown to be sufficient to guarantee accessibility, it is primarily of theoretical interest, being essentially impossible to apply to given data. In this paper the authors investigate the possibility of developing algorithms involving only rational computations on the matrices, which would determine whether the rank condition is everywhere satisfied, i.e. whether the bilinear system has the accessibility or, in some instances, controllability property. This is equivalent to determining whether or not the matrix Lie group G generated by $\exp (tA_i )$, $...
TL;DR: In this article, a semigroup formulation of boundary input problems for systems governed by parabolic partial differential equations is presented and a useful bound on the operator kernel of the input map is established under very general conditions.
Abstract: A semigroup formulation of boundary input problems for systems governed by parabolic partial differential equations is presented. A useful bound on the operator kernel of the input map is established under very general conditions. This bound is used to study the input map and the results used to examine the time optimal boundary control problem.
TL;DR: In this article, a new mufti-parameter singular perturbation problem is formulated and sufficient conditions for uniform asymptotic stability are derived, and the behavior of solution is investigated.
Abstract: A new mufti-parameter singular perturbation problem is formulated. Sufficient conditions for uniform asymptotic stability are derived, and asymptotic behavior of solution is investigated.
TL;DR: In this paper, a new derivation of the classification of the minimal Markovian representations of the given process z is presented which is based on a certain backward filter of the innovations.
Abstract: Invariant directions of the Riccati difference equation of Kalman filtering are shown to occur in a large class of prediction problems and to be related to a certain invariant subspace of the transpose of the feedback matrix. The discrete time stochastic realization problem is studied in its deterministic as well as probabilistic aspects. In particular a new derivation of the classification of the minimal Markovian representations of the given process z is presented which is based on a certain backward filter of the innovations. For each Markovian representation which can be determined from z the space of invariant directions is decomposed into two subspaces, one on which it is possible to predict the state process without error forward in time and one on which this can be done backward in time.
TL;DR: In this article, the stability of a sampled-data system with sampling interval lengths selected from a finite set of matrices is studied, and conditions for stabilizability involving pre-contractiveness, contractiveness and positive definiteness are given.
Abstract: A sampled-data system with sampling interval lengths selected from a finite set is considered. Stabilizability of the system via feedbacks associated with sampling interval lengths is studied, and conditions for stabilizability involving “pre-contractiveness”, “contractiveness” and “positive definiteness” of a finite set of matrices are given. Included in these results is a generalization of a theorem by P. Stein stating that for a real square matrix H, $\lim _{n \to \infty } H^n = 0$ if and only if there is a symmetric matrix Q such that $Q - H^T QH$ is positive definite. Finally, some results concerning a choice of feedbacks which will produce stability are presented.
TL;DR: In this paper, a multi-grid method for the solution of boundary control problems with quadratic cost functions was presented, where the state is a solution of a parabolic initial-boundary value problem.
Abstract: We present a multi-grid method for the solution of boundary control problems with quadratic cost functions, where the state is a solution of a parabolic initial-boundary value problem. The computationalwork is proportional to the work needed for the integration of the parabolic equation, The method can be extended also to nonlinear problems.
TL;DR: In this article, an algebraic theory of linear time-varying discrete-time systems is developed in terms of a module structure defined over a noncommutative polynomial ring.
Abstract: An algebraic theory of linear time-varying discrete-time systems is developed in terms of a module structure defined over a noncommutative polynomial ring. The module setup is induced from a semilinear transformation that is derived from the given system. Various structural properties of the module framework are explored including the concepts of cyclicity and n-cyclicity. The module theory is then applied to the study of reachability and state feedback. Results on the construction of feedback controllers are obtained that resemble pole or coefficient assignability in the theory of time-invariant systems.