Showing papers in "Siam Journal on Control and Optimization in 1980"
TL;DR: In this article, the authors pointed out that a linear oscillator in an infinite dimensional Hilbert space, with no uniform decay rate, cannot be given a uniform decaying rate with compact linear feedback.
Abstract: This note points out the fact that a linear oscillator in an infinite dimensional Hilbert space, with no uniform decay rate, cannot be given a uniform decay rate with compact linear feedback The motivation for the analysis here is the use of a finite number of control elements to stabilize a system with an infinite number of modes of vibration, and the implications of the inability to produce a uniform decay rate are elaborated in regard to optimal regulation For systems with and without inherent damping, the result is based on approximating a compact operator with a sequence of finite dimensional operators The physical interpretation of this technique is discussed
121 citations
TL;DR: Conditional gradient algorithms with implicit line minimization and Goldstein-Armijo step length rules are considered for the problem of minimizing a bounded convex subset of a real Banach space as discussed by the authors.
Abstract: Conditional gradient algorithms with implicit line minimization and Goldstein–Armijo step length rules are considered for the problem $\min _\Omega F$ with $\Omega $ a bounded convex subset of a real Banach space. When the Frechet derivative $F'$ is uniformly continuous on $\Omega $, the iterates $x_n $ generated by any of the algorithms comprise an “extremizing” sequence in the sense that the quantity, $\langle {F'(x_n ),x_n } \rangle - \inf _{y \in \Omega } \langle {F'(x_n ),y} \rangle $, converges to zero as $n \to \infty $. This ensures that every limit point of $\{ x_n \} $ is an extremal, and for compact $\Omega $ it then follows that $\{ x_n \} $ converges to the set of extremals in $\Omega $. Weak counterparts of these results are also established. Convergence rate estimates are derived for convex F and Lipschitz continuous $F'$. These estimates are closely related to results obtained in an earlier investigation of two explicit step length formulas for conditional gradient methods. Once again, the...
116 citations
TL;DR: In this paper, the existence of equilibrium points in non-cooperative stochastic games is investigated in the monotone contraction operator framework of Denardo (1967), and sufficient conditions are determined for their existence.
Abstract: Noncooperative sequential games, including the noncooperative stochastic game of Rogers (1969) and Sobel (1971), are investigated in the monotone contraction operator framework of Denardo (1967). Sufficient conditions are determined for the existence of equilibrium points in this setting. Techniques for comparing and, approximating dynamic programs previously developed by the author are then applied to these sequential games, yielding conditions for the existence of $\varepsilon $-equilibrium points.
98 citations
TL;DR: In this paper, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other, and a stability theorem for solutions of the matrix Riccati equation is obtained along the way.
Abstract: This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.
98 citations
TL;DR: In this paper, the generalized Legendre-Clebsch higher order tests for optimality of singular arcs in optimal control problems depend upon the orders of the arcs involved, and the features of each definition are discussed with special reference to the applicability of the higher-order tests and the conditions at junctions between singular and nonsingular arcs.
Abstract: The generalized Legendre–Clebsch higher order tests for optimality of singular arcs in optimal control problems depend upon the orders of the arcs involved. To date three distinct definitions of order have been given but many authors do not distinguish among them. The features of each definition are discussed with special reference to the applicability of the higher order tests and of the conditions at junctions between singular and nonsingular arcs; only in terms of one of the definitions are the junction conditions generally valid. An illustrative example is presented.
57 citations
TL;DR: In this paper, the authors consider the relationship between the linear complementarity problem and the linear programming problem and show that the linear program requires the minimization of a linear functional which is proportional to the load borne by the bearing.
Abstract: Let X be a vector lattice Hilbert space with dual $X^ * $. Let M be a continuous linear mapping of X onto $X^ * $. Let $p,q \in X^ * $ with $p > 0$. We consider the relationship between the linear complementarity problem: Find $x \in X$ such that $x \geqq 0$, $Mx + q \geqq 0$, $\langle {x,Mx + q} \rangle = 0$, and the linear programming problem: Find $x \in X$ which minimizes $\langle {x,p} \rangle $ subject to $x \geqq 0$, $Mx + q \geqq 0$. For the problem of a cavitating journal bearing, which is used as an example, the linear program requires the minimization of a linear functional which is proportional to the load borne by the bearing.
56 citations
55 citations
TL;DR: In this paper, the authors proved that the time-optimal controls associated with arbitrary reachable target temperature distributions in boundary control for the heat equation (with bounds on the admissible controls) are "bang-bang".
Abstract: Following previous work by H. O. Fattorini, J. Henry and the present author it is proved that the time-optimal controls associated with arbitrary reachable target temperature distributions in boundary control for the heat equation (with bounds on the admissible controls) are “bang-bang”.
53 citations
TL;DR: In this paper, a class of boundary-distributed linear control systems in Banach spaces is studied and a maximum principle for a convex control problem associated with such systems is obtained.
Abstract: A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.
48 citations
TL;DR: In this paper, a characterization of the optimal cost of an impulse control problem as the maximum solution of a quasi-variational inequality without assuming nondegeneracy is given, and an estimate of the velocity of uniform convergence of the sequence of stopping time problems associated with the impulse control problems is given.
Abstract: In this paper, we give a characterization of the optimal cost of an impulse control problem as the maximum solution of a quasi-variational inequality without assuming nondegeneracy. An estimate of the velocity of uniform convergence of the sequence of stopping time problems associated with the impulse control problem is given.
47 citations
TL;DR: In this paper, the optimal cost of a stopping-time problem is characterized as the maximum solution of a variational inequality without coercivity, and some properties of continuity for optimal cost are also given.
Abstract: In this paper we give a characterization of the optimal cost of a stopping time problem as the maximum solution of a variational inequality without coercivity. Some properties of continuity for the optimal cost are also given.
TL;DR: In this article, necessary and sufficient conditions are given to achieve a given nonsingular matrix in the denominator of a matrix fraction description of a dynamic feedback system, and a characterization of the required dynamic feedback laws is presented.
Abstract: Basedd on some recent results in algebraic system theory, necessary and sufficient conditions are given to achieve a given nonsingular matrix in the denominator of a matrix fraction description of a dynamic feedback system. A characterization of the required dynamic feedback laws is presented.To achieve this, first, a complete solution is given to a problem of algebra on polynomial matrix equations.
TL;DR: In this paper, a relaxed form of Newton's method is analyzed for the problem, where feasible directions are obtained by minimizing local quadratic approximations Q to F, and the relaxation parameters, or step lengths, are obtained from Goldstein's rule.
Abstract: A relaxed form of Newton’s method is analyzed for the problem, $\min _\Omega F$, with $\Omega $ a convex subset of a real Banach space X, and $F:X \to \mathbb{R}^1 $ twice differentiable in the sense of Frechet. In this iterative scheme, feasible directions are gotten by minimizing local quadratic approximations Q to F, and the relaxation parameters, or step lengths, are obtained from Goldstein’s rule. The local and global convergence theorems established here yield two significant extensions of an earlier theorem of Goldstein for the special case $\Omega = X = {\text{a}}$ Hilbert space. In one extension, growth rate conditions on the local approximation Q subsume the classical uniform positivity restriction on $F''$; connections are made here with a recently formulated classification scheme for singular and nonsingular extremals. In the second extension, uniform growth rate conditions are replaced by assumptions of the compactness and boundedness type. This development establishes global convergence of t...
TL;DR: In this paper, a control problem associated with controlled, partially observed diffusion processes is considered, where the state in the separated problem is an unnormalized conditional distribution measure, and the corresponding Nisio nonlinear semigroup associated with the separation problem is found.
Abstract: In this paper a “separated” control problem associated with controlled, partially observed diffusion processes is considered. The state in the separated problem is an unnormalized conditional distribution measure. The corresponding Nisio nonlinear semigroup associated with the separated problem is found.
TL;DR: In this article, the controllability canonical form of the Mufti-pair system is extended to the case that the set of points reachable from zero (the controllable set) is a subspace.
Abstract: In this paper discrete-time systems of the form \[x_{k + 1} = C_p x_k + D_p U_k ,\] in which the pair $(C_p ,D_p )$ is selected from a finite set $\{ (C_i ,D_i )\} _{i = 1}^N $, are studied. Such systems, called “mufti-pair systems,” arise naturally in the study of multi-rate sampled-data systems. It is shown that the set of points reachable from zero (the controllable set) is a subspace under certain hypotheses, but not always. When this is the case, an extended version of the controllability canonical form is obtained, and it is applied to the study of state deadbeat response and more general forms of stabilizability.
TL;DR: In this article, the question of whether a set is reachable by a nonlinear control system is answered in terms of the properties of a convex optimization problem, i.e., whether the value of the optimization problem is zero or infinity.
Abstract: The question of whether a set is reachable by a nonlinear control system is answered in terms of the properties of a convex optimization problem. The set is reachable or not according to whether the value of the optimization problem is zero or infinity. Our findings strengthen earlier sufficient conditions for a point not to be reachable, given in terms of Lyapunov-like functions, in that we assure that the functions exist. Our approach is to embed admissible trajectories in a space of measures, and to apply recently obtained results on the properties of measures arising in this way.
TL;DR: In this paper, the stability properties of a fairly general continuous-time adaptive control scheme are analyzed and sufficient conditions for stability in the presence of disturbances are given, without requiring any a priori stability assumptions.
Abstract: The stability properties of a fairly general continuous-time adaptive control scheme are analyzed. Sufficient conditions for $L^\infty $-stability in the presence of disturbances are given. The stability results are used to prove convergence of the process outputs in the disturbance-free case, without requiring any a priori stability assumptions.
TL;DR: Theorems concerning weak convergence of non-Markovian processes to diffusions, together with an averaging and a stability method, are applied to two (learning or adaptive) processes of current interest: an automata model for route selection in telephone traffic routing, and an adaptive quantizer for use in the transmission of random signals in communication theory.
Abstract: Recently proven theorems concerning weak convergence of nonMarkovian processes to diffusions, together with an averaging and a stability method, are applied to two (learning or adaptive) processes of current interest: (1) an automata model for route selection in telephone traffic routing; (2) an adaptive quantizer for use in the transmission of random signals in communication theory The models are chosen because they are prototypes of a large class to which the methods can be applied The technique of application of the basic theorems to such processes is developed Suitably interpolated and normalized “learning or adaptive” processes converge weakly to a diffusion, as the “learning or adaptation” rate goes to zero For small learning rates, the qualitative properties (eg, asymptotic (large-time) variances and parametric dependence) of the processes can be determined from the properties of the limit
TL;DR: In this paper, the convex span of feasible solutions to a system of facial constraints is generalized through the device of first viewing the characterization as a two-person "game" on a polytope, and then enlarging the class of "moves" open to one of the players.
Abstract: Balas’ characterization, of the convex span of feasible solutions to a system of facial constraints, is generalized through the device of first viewing the characterization as a two person “game” on a polytope, and then enlarging the class of “moves” open to one of the “players.” Both primal and dual cutting-plane algorithms are presented for facial constraint systems, and are then proven finitely-convergent by use of our generalization of Balas’ result.
TL;DR: In this article, an extended notion of stabilizability was used to obtain a stable constant stochastic control law for the linear-quadratic-Gaussian (LQG) problem.
Abstract: The estimation and control of linear stochastic systems with delays in the state, control, and observations are studied. First, the deterministic optimal control problem with quadratic cost over an infinite time interval is examined. Using an extended notion of stabilizability, the existence and characterization of the optimal control law is obtained. Using the additional assumption of detectability the optimal closed-loop system is shown to be $L^2 $-stable. Next, the stochastic filtering problem is studied. A new version of the duality relations between optimal control and filtering is developed. This combined with a suitable notion of detectability, is exploited to show convergence of the filter gains. Under the additional assumption of stabilizability, the optimal stationary filter is shown to be $L^2 $-stable. Finally, by putting together the optimal control and filtering results, a stable constant stochastic control law is obtained for the linear-quadratic-Gaussian problem.
TL;DR: In this article, a functional observer of Luenberger type is derived and then utilized in order to stabilize unstable parabolic equations for which observation of the state and control can be carried out only through the boundary.
Abstract: Feedback stabilization of unstable parabolic equations is of great interest. The fact that it is not necessarily possible to stabilize the equations by means of static feedback schemes when both observation and control can be realized only through the boundary is illustratively shown by a simple example. In view of this, a functional observer of Luenberger type is derived and then utilized in order to stabilize unstable parabolic equations for which observation of the state and control can be carried out only through the boundary.
TL;DR: In this article, the authors studied the properties of the system topology that ensure that the overall system with dissipation is asymptotically stable in both linear and nonlinear systems.
Abstract: In this paper we answer the following question for a large class of (linear and nonlinear) dynamical systems. Given is a system with dissipation and given is the associated conservative system. Suppose the associated conservative system is stable. What properties of the system topology (system configuration) will ensure that the overall system with dissipation is asymptotically stable?Both linear and nonlinear (Hamiltonian) systems are treated. For the linear case, necessary and sufficient conditions for asymptotic stability are established, while for the nonlinear case, sufficient conditions and also some necessary and sufficient conditions for asymptotic stability are obtained.It is emphasized that the application of the present results to specific problems will usually not require a search for appropriate Lyapunov functions. Indeed, a stability analysis by the present method involves the following two steps: (a) given a system with dissipation, the stability of its trivial solution (equilibrium) is asc...
TL;DR: In this article, a bilinear realization theory for a Volterra series input-output map is presented, which involves the definition of appropriate shift operators on linear spaces associated with the transforms of the kernals in the VOLTERRA series and leads to a characterization of finite dimensional realizability in terms of rationality properties of the transforms.
Abstract: Using a transform representation, we present a bilinear realization theory for a Volterra series input–output map. The approach involves the definition of appropriate shift operators on linear spaces associated with the transforms of the kernals in the Volterra series. This approach yields in a very simple manner a theory of minimality and connections with the concepts of span reachability and observability. It also leads to a characterization of finite dimensional realizability in terms of rationality properties of the transforms.
TL;DR: It is demonstrated that the special case of equal columns is solvable in $O(m^2 )$ time, and the algorithm is applicable to a production-sales planning model with concave utilities.
Abstract: The following problem is considered. Given positive integers $(n_1 , \cdots ,n_m )$ and an $n_m \times m$ matrix D with the property that the $n_m $ elements in each column form a monotone sequence, find a set A of $n_m $ elements of D whose sum is maximum, and such that for any j, $j = 1, \cdots ,m$, not more than $n_j $ elements are chosen from columns $1,2, \cdots ,j$. An algorithm, solving the above problem in time $O(m^2 \log ^2 n_m )$ is presented. The algorithm is applicable to a production-sales planning model with concave utilities. It is also demonstrated that the special case of equal columns is solvable in $O(m^2 )$ time.
TL;DR: In this article, the authors focus on the determination of optimal interpolators and control laws from a restricted class when broad assumptions are made about the system and derive consistency and asymptotic normality results for estimates of the parameter of an optimal interpolator when the class of interpolators is restricted.
Abstract: It is usual in time series analysis and control theory to assume that there is a close connection between the structure of the system and the structure of the class of interpolators or control laws under study. Moreover, in practice, restrictive assumptions are often made about the system since this leads ab initio to a simple structure for the optimal interpolator or optimal control law. This paper is concerned with an alternative viewpoint in which attention is focused on the determination of optimal interpolators and control laws from a restricted class when broad assumptions are made about the system. In particular, consistency and asymptotic normality results are developed for estimates of the parameter of an optimal interpolator when the class of interpolators is restricted. Results relevant to the choice of interpolator structure are also established.
TL;DR: In this paper, it was shown that finding such factorizations is equivalent to finding $(A,B)$-invariant subspaces in the kernel of C where A, B, C are linear maps determined by a polynomial matrix.
Abstract: Given a polynomial matrix $B(s)$, we consider the class of nonsingular polynomial matrices $L(s)$ such that $B(s) = R(s)L(s)$ for some polynomial matrix $R(s)$. It is shown that finding such factorizations is equivalent to finding $(A,B)$-invariant subspaces in the kernel of C where A, B, C are linear maps determined by $B(s)$. In particular, the results yield, as a corollary, a method to determine simultaneously a row proper greatest right divisor of a left invertible polynomial matrix as well as the resulting polynomial matrix whose greatest right divisors are unimodular.The results also relate, the same way, such subspaces of constant systems $(\bar C,\bar A,\bar B)$ where $(\bar C,\bar A)$ is observable, to the nonsingular right factors of the numerator polynomial matrices in factorizations of the form $D^{ - 1} (s)B(s)$ of their transfer matrices.
TL;DR: In this paper, an abstract model of stochastic optimal control is described, where the influence of the control is modeled by exponentials of martingales, and sufficient and sufficient conditions for optimality from these papers are adapted to the abstract model.
Abstract: The paper gives a fairly general approach to the stochastic optimal control problem, using some recent results on semimartingales. After a short, review on these results, an abstract model of stochastic optimal control is described, where the influence of the control is modeled by exponentials of martingales. This model gives a synthesis of the martingale approaches of Davis and Varaiya on the one hand, and Striebel on the other hand. Necessary and sufficient conditions for optimality from these papers are adapted to the abstract model. Topological existence results, which apply to the complete observation case as well as to the partial observation case, are described when the likelihood ratios describing the dynamics of the system are subsets respectively of an $L_2 $- and an $L_1 $-space.These abstract results are specialized to the problem where the martingales are represented as stochastic integrals. The results can then be written in a more explicit form; in particular it is possible to formulate the...
TL;DR: In this article, sufficient conditions are obtained for the dependence on a parameter of a local minimizes and associated Lagrange multipliers for parametrized families of nonlinear programming problems in which some constraints do not vary with the parameter.
Abstract: Sufficient conditions are obtained for the $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} ^1 $ dependence on a parameter of a local minimizes and associated Lagrange multipliers for parametrized families of nonlinear programming problems in which some constraints do not vary with the parameter. A class of sets, called “cyrtohedra”, with properties suitable for the representation of fixed constraint sets is discussed.
TL;DR: In this paper, the existence of periodic solutions of nonlinear control systems subjected to sinusoidal forcing functions, using the describing function method, is studied, and relative error bounds between the response of the exact problem and the associated linearized describing function problem are provided.
Abstract: The existence of periodic solutions of nonlinear control systems subjected to sinusoidal forcing functions, using the describing function method, is studied. The setting is general enough to allow systems with delays, systems with discontinuous nonlinearities, systems with hysteresis nonlinearities, and so forth. The present results state that if the linearized describing function problem can be solved and if certain bounds (which depend on the exact form of the solution of the describing function problem) can be satisfied, then there is a periodic solution of the exact problem. Furthermore, the present results provide relative error bounds between the response of the exact problem and the associated linearized describing function problem. To demonstrate the applicability of the method advanced, a specific example is considered.