Showing papers in "Siam Journal on Control and Optimization in 1981"

Detectability and Stabilizability of Time-Varying Discrete-Time Linear Systems

Abstract: The concepts of detectability and stabilizability are explored for time-varying systems. We study duality, invariance under feedback, an extended version of the lemma of Lyapunov, existence of stabilizing feedback laws, linear quadratic filtering and control, and the existence of approximate canonical forms.

Discrete Time Stochastic Adaptive Control

Abstract: This paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems. It is shown that, with probability one, the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control. Thus, asymptotically, the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known.

Disturbance Decoupling by Measurement Feedback with Stability or Pole Placement

Abstract: In this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system. The problem is attacked using the geometric approach through the concepts of $A(\bmod \mathcal{B})$-invariant and controllability subspaces and their duals, $ A \mid K$-invariant and complementary observability subspaces. The solution of this problem has an interesting structure consisting of a feedback processor which decomposes into (i) a disturbance decoupling loop; (ii) a disturbance input stabilization or pole placement loop, and (iii) a controlled output stabilization or pole placement loop.

The Infinite Time Quadratic Control Problem for Linear Systems with State and Control Delays: An Evolution Equation Approach

Abstract: We show how a linear differential delay equation with delays in the control can be reformulated as an evolution equation with bounded input operator. As a simple application, we solve an infinite t...

Generic observability of differentiable systems

Abstract: A dynamical system consists of a smooth vectorfield defined on a differentiable manifold, and a smooth mapping from the manifold to the real numbers. The vectorfield represents the dynamics of a physical system. The mapping stands for a measuring device by which experimental information on the dynamics is made available. The information itself is modeled as a sampled version of the image of the state trajectory under the smooth mapping. In this paper the observability of this set-up is discussed from the viewpoint of genericity. First the observability property is expressed in terms of transversality conditions. Then the theory of transversal intersection is called upon to yield the desired results. It is shown that almost any measuring device will combine with a given physical system to form an observable dynamical system, if $(2n + 1)$ samples are taken and not fewer, where n is the dimension of the manifold. Dually, it is shown that almost any physical system will combine with a given measuring device ...

A Note on the Boundary Stabilization of the Wave Equation

Abstract: We study the energy decay rates of the wave equation in a domain where boundary damping is present. We generalize the geometrical conditions obtained earlier in (J. Math. Purer Appl., 58 (1979), pp. 249–273) by using some more general multipliers of Strauss (Comm. Pure Appl. Math., 28 (1975), pp. 265–278). The interaction between distributed damping and boundary damping is discussed. A regulator problem is also formally discussed by the synthesis method.

$(A,\mathcal{B})$-Invariant Distributions and Disturbance Decoupling of Nonlinear Systems

Abstract: The concept of $(A,B)$-invariant subspaces has resulted in a unified approach to many of the basic structural properties of time-invariant linear systems (W. M. Wonham, Lecture Notes in Economics and Mathematical Systems, vol. 101, Springer-Verlag, New York, 1974). The purpose of this paper is to introduce the more general notion of $(A,\mathcal{B})$-invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds.

Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes

Abstract: Projected gradient processes of the Goldstein–Levitin–Polyak type are considered for constrained minimization problems, $\min _\Omega F$, with $\Omega $ a convex set in a Hilbert space X and $F:X \to \mathbb{R}^1 $ a differentiable functional. Global and local convergence theorems are established for a large class of these processes, including those generated with implicit step length rules proposed by Bertsekas and Goldstein. In this analysis, traditional uniform strong positivity conditions on the Hessian $ abla ^2 F$ are replaced by weaker pseudoconvexity conditions and growth conditions on F. When F has a unique minimizes in $\Omega $, convergence rates are shown to depend on how rapidly the function $\gamma (\sigma ) = \inf \{ r = F(x) - F(\xi )\mid x . \in \Omega \| {x - \xi } \| \geqq \sigma \} $ grows with increasing $\sigma > 0$. If $\gamma (\sigma ) \geqq B\sigma ^ u $ for some $B > 0$, the processes $\{ F_n \} $ in question converge to $\inf F$ like $O(n^{{{ - u } / {( u - 2)}}} )$, linear...

An Analysis of Optimal Modal Regulation: Convergence and Stability

Abstract: This paper treats the linear-quadratic regulator problem for infinite dimensional, second order (in time), linear oscillators. We solve the problem approximately by modeling a finite number of mode...

An approximation theory for nonlinear partial differential equations with applications to identification and control

Abstract: Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

Dynamical Realizations of Finite Volterra Series

Abstract: In this paper, realizations of finite Volterra series are viewed as nonlinear analytic input–output systems, with state space described by an analytic manifold. For a minimal realization guaranteed by H. J. Sussmann, the state space, which is unique up to diffeomorphism, is shown to have the homogeneous space structure of a nilmanifold, the quotient of two nilpotent Lie groups. The structure of nilmanifolds as described by A. Malcev is used to show that for these systems, the state space has a vector space structure. As a consequence of this result, it is shown that a minimal realization of a finite Volterra series can be described as a cascade of linear subsystems with polynomial link maps, in which the dimension o f each linear subsystem is independent of the realization considered.

On Some Impulse Control Problems with Long Run Average Cost

Abstract: Some particular impulse control problems with infinite horizon and long run average cost are considered for Markov processes having “nice” ergodicity properties.The main feature of the method is to start with discounted cost and then let the discount factor go to zero. This also gives an opportunity to study the asymptotic behavior of some optimal stopping time problems when the discount factor goes to zero. Probabilistic and analytical methods are used and examples are given, especially for Markov jump process and diffusion processes with reflection.

Necessary Conditions for Distributed Control Problems Governed by Parabolic Variational Inequalities

Abstract: Necessary conditions for optimality in distributed control problems governed by semilinear and variational parabolic inequalities are given. The optimality conditions are expressed in terms of generalized gradients, and are obtained by means of an abstract approximating control process.

Control and Stabilization for the Wave Equation, Part III: Domain with Moving Boundary

Abstract: We use energy invariants to study the growth and decay estimates for solutions of the wave equation in a domain with moving boundary. Sufficient conditions are formulated which insure the exact (distributed-parameter) controllability of the wave equation.

Existence of Value and Randomized Strategies in Zero-Sum Discrete-Time Stochastic Dynamic Games

Abstract: Two players with conflicting objectives are simultaneously controlling a discrete-time stochastic system. The goal of this paper is to analyze such zero-sum, discrete-time, stochastic systems when the two players are allowed to use randomized strategies.Previous results have been restricted to systems with finite or compact state spaces. Such restrictions are usually untenable from the point of view of applications, since many applications frequently use either the integers or $\mathbb{R}^n $ as a state space. Our results are proved for complete, separable, metric spaces which are very useful for applications.All previously known results emerge as special cases of our results. In addition, a variety of conjectures and open problems are resolved regarding the existence of a value function, its properties such as Borel measurability or continuity, and the existence for either or both players of optimal or $\varepsilon $-optimal stationary strategies.

On Spectrum Distribution of Completely Controllable Linear Systems

Abstract: This paper is concerned with the placement of the spectrum of the closed-loop operator $A + BK$ resulting from use of a linear feedback control law $u = Kx$ in the infinite dimensional linear control system $x' = Ax + Bu$. For a class of systems in Hilbert space with certain assumptions on the spectrum of the operator A, a complete characterization of the achievable spectra is obtained. The proofs are carried out in an operator-theoretic context.

Optimal Play in a Stochastic Differential Game

Abstract: This paper considers play in a two-person zero-sum differential game where the dynamics are given by a differential equation with additive white noise. Feedback strategies are employed. Standard results from control theory show that the maximizing player has an optimal response to any pre-announced strategy of the minimizing player. Here it is shown that the minimizing player can achieve the upper value of the game by playing a strategy which is constructed by performing a pointwise min-max on a certain fixed Hamiltonian function.

Necessary and Sufficient Conditions of Approximate Controllability for General Linear Retarded Systems

Abstract: Necessary and sufficient conditions of approximate controllability, in the space $R^n \times L_2 ([ - h,0],R^n )$, of general linear retarded systems are obtained. It is shown that approximate controllability is equivalent to two conditions: a) spectral controllability, and b) the existence of linear feedback which transforms the original system into a system with a complete set of generalized eigenfunctions. Both conditions are expressed in algebraic form. The proof of this result is based on recently obtained criteria of completeness of generalized eigenfunctions associated with retarded systems and on an algebraic approach to functional differential equations. Practical verifiability of the new conditions is demonstrated on several examples.

Analysis of the Asymptotic Behavior of Optimal Control Trajectories: The Implicit Programming Problem

Abstract: The asymptotic behavior of the optimal trajectories of the infinite horizon control problem with discounting, is characterized by a static optimization problem. In the undiscounted case, the limit point of the optimal dynamic trajectory is the steady-state that minimizes the kernel of the objective functional. The corresponding static characterization of the limit point in the discounted case, called the implicit programming problem, is derived. The implicit programming problem is a mathematical programming problem with the special feature that part of the solution is contained in the definition of the problem. All results are achieved in the context of a sufficient maximum principle, which is shown to be equivalent to the other approaches taken in the literature to perform the dynamic analysis. The equivalence is based on convexity conditions assumed in the current dynamic theory. The class of problems that satisfy such convexity conditions is characterized in terms of a property of vector-valued mapping...

Causal Factorization and Linear Feedback

Abstract: An algebraic framework for the investigation of linear dynamic output feedback is introduced. Pivotal in the present theory is the problem of causal factorization, i.e. the problem of factoring two systems over each other through a causal factor. The basic issues are resolved with the aid of the new concept of latency kernels.

Estimation of delays and other parameters in nonlinear functional differential equations

Abstract: A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

Stochastic Control on Hilbert Space for Linear Evolution Equations with Random Operator-Valued Coefficients

Abstract: We consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and B...

Lower Semicontinuity of Integral Functionals with Nonconvex Integrands by Relaxation-Compactification

Abstract: A new approach to the lower semicontinuity of integral functionals is presented. By a topological embedding of the “control” and “state” spaces in the Hilbert cube and a simultaneous relaxation of the “control functions,” a powerful approach emerges whose main features include: (i) A generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case. (ii) In the embedded setting the integrand can be supposed Lipschitz-continuous in “control” and “state” arguments without loss of generality. (iii) Convergence in measure of the “trajectories,” metamorphoses into $L_1 $-norm convergence in the embedded setting.

Recursive Estimation in Diffusion Model

Abstract: This paper is concerned with the nonlinear identification of dynamical systems disturbed by white noise, an important problem in control engineering. A nonparametric identification procedure for such systems which are usually described by a diffusion model is given in Banon (1977), (1978), where weak consistency of estimators has been obtained and simulation study has been carried out successfully. In this paper, we prove a stronger result concerning asymptotic properties of the estimators of the drift term, namely, strong consistency, and also related results.

Extreme Points and Basic Feasible Solutions in Continuous Time Linear Programming

Abstract: This paper studies the extreme points arising in continuous time linear programming. The main result is for the case of constant coefficients where all so-called right analytic extreme points are characterized, analogously to the result for linear programming, in terms of certain full rank conditions. Examples of continuous time linear programs with time-varying constraints are given to show that this kind of characterization cannot hold in general.

On the Adjoint Process for Optimal Control of Diffusion Processes

Abstract: For the Markovian control problem \[\begin{gathered} \mathop {\min }\limits_u E\left\{ {\int_0^T {l(t,x,u)dt + c(x(T))} } \right\}, \hfill \\ dx = f(t,x,u)dt + \sigma (t,x)dw, \hfill \\ \end{gathered} \] it is shown that the adjoint process appearing in the maximum principle has the form \[p(t,x) = - E_{tx} \left\{ {\frac{{\partial c}}{{\partial x}}(x(T))\Phi (T, t) + \int_t^T {\frac{{\partial l}}{{\partial x}}(s,x(s),\hat u(s,x(s))\Phi (s,t)ds} } \right\},\] where $\hat u$ is the optimal feedback control, $E_{tx} $ denotes conditional expectation, and $\Phi $ is the fundamental matrix solution of \[dy = \frac{{\partial f}}{{\partial x}}(t,x(t)\hat u(t,x(t))),ydt + \sum_k {\frac{{\partial \sigma ^k }}{{\partial x}}(t,x))y} dw_k \] Here, $\sigma ^k $ is the kth column of $\sigma $ and $w_k $ is thekth component of the Brownian motion w, It is also shown that $p(t,x(t))$ satisfies ($ * $ denotes transpose) \[\begin{gathered} dp^ * = \left\{ - {\frac{{\partial f^ * }}{{\partial x}}(t,x(t),\hat u(t,x(t)))p^ *...

A Characterization of Well-Posed Optimal Control Systems

Abstract: A constrained optimal regulator problem is considered. Continuous dependence of the optimal control on the desired trajectory (Hadamard well-posedness) or convergence toward the optimal control of any minimizing sequence (Tykhonov well-posedness) are proved when the dynamics are affine (linear plus constant). Dense well-posedness is obtained in the non-affine case. A necessary and sufficient condition for well-posedness for all desired trajectories is shown to be the affine structure of the plant.

Perturbations of Nonlinear Controllable Systems

Abstract: In the class of nonlinear, nonautonomous control systems we consider the property of controllability to a compact set on a fixed time interval, and we give a sufficient condition for this property to be preserved under small perturbations of the control system. Our results are formulated in terms of control vector fields on a differentiable manifold.

Selecting Subsets from the Set of Nondominated Vectors in Multiple Objective Linear Programming

Abstract: In this paper, we develop methods for selecting certain subsets from the set N of nondominated points for multiple-objective linear programming problems. One such subset is the set of a points x in N for which the maximum deviation of the objective function values $C_x $ from some ideal vector M is as small as possible. This subset can be obtained as the set of nondominated points for a multiple-objective problem that is considerably smaller than the original problem and the proposed method does not require that the set N be calculated explicitly. The method is extended to obtain another subset of N called the trade-off compromise set that has some interesting properties and that gives valuable information about possible trade-offs amongst the objectives.

The Equation $XR + QY = \Phi $: A Characterization of Solutions

Abstract: In this paper we consider the solutions of the equation $XR + QY = \Phi $. Here Q , R , $\Phi $ are given $p \times q$, $m \times t$ and $p \times t$ polynomial matrices over a field k. X and Y are $p \times m$ and $q \times t$ polynomial matrices which are unknown. Using certain recent results on the realization of matrix fraction descriptions of transfer matrices, we give a characterization (parametrization) of all possible $(X,Y)$ which solve this equation. This also provides a system theoretic interpretation for this equation.