Optimal control of time-delay systems with distributed parameters

An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.The author wishes to express his deep gratitude to Professors J. M. Sloss and S. Adali for the valuable guidance and constant encouragement during the preparation of this paper.     

Optimal control in systems with distributed parameters of hyperbolic type

On an example of transmission of a constant current, problems of optimal control in systems with distributed parameters of hyperbolic type are considered. A solution of corresponding problems is brought up to double extremal relations. Examples of application of the general theory to concrete control problems are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 71–78, 1988.     

Optimal control of parameter distributed systems with impulses

This paper concerns optimal control problems with impulses. The optimal magnitude of impulses and the spatial position of impulses are studied. We obtain maximum principles for these problems.     

Optimal control with a quadratic criterion in distributed gas-transport systems

On the basis of the method of generalized Lagrange multipliers we obtain optimality conditions in the problem of tracking a given variation in regime of a distributed pipeline gas transport system with an integral quadratic quality criterion. We propose a method and an algorithm for finding optimal controlling actions. We present the results of computation of optimal controls in a homogeneous pipeline. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Technical stability conditions for some dynamic systems with distributed and concentrated parameters

Sufficient conditions are derived for technical stability on a finite and an infinite time interval and for asymptotic technical stability (both local and global) of a class of processes described by a system of complex ordinary nonlinear differential equations, delayed dynamic processes, and some three-wave interacting processes. The Lyapunov direct method is used, combined with the comparison method and Gâteaux generalized differentiation.Institute of Mechanics of the Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 113–123, 1989.     

Optimal singular control for systems involving parameters

What we shall take into consideration is represented by the problems of singular optimal control when the index of performance, the differential constraints and the final conditions contain parameters. What is determined is the trajectory of the neighboring extremal for the initial point perturbed at the constraints of the perturbed final manifold. The present study evaluates the variation of the command on the extremal. This one allows obtaining the second variation in the singular case. The sufficient conditions of minimum are a result of imposing the non-negativity of the second variation. The application made analysis the optimality of the singular control for a system in the class of affine differential systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control for distributed linear systems subjected to null-controllability

In this article, we apply the notion of hierarchic control on a distributed system in which the state is governed by a parabolic equation. This notion assumes that we have two controls where one will be the Leader and the other, the Follower. The first control is of controllability type subjected to a constraint, while the second expresses that the state does not move too far from a given state. The results are achieved by means of an observability inequality of the Carleman type, which is ‘adapted’ to the constraint.     

Variational approach to modeling and control of dynamic systems with distributed parameters

Controlled motions of a thin elastic rod are studied. A variational statement of the initial-boundary value problem is given based on the integral formulation of constitutive relations. An approach relying on the Ritz method and finite element technique is proposed to design the boundary control that moves the rod with certain accuracy to a given final state and minimizes the mean energy. The results of numerical simulation and the estimates of solution quality are presented and discussed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control problems for distributed parameter systems in Banach spaces

We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf ₀ (u) defined in a metric spaceV subject to the constraintf(u) Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL ¹ andL spaces.This work was supported in part by the National Science Foundation under Grant DMS-9001793.     

Single-frequency oscillations of non-linear systems with distributed parameters

A weakly non-linear oscillatory system with distributed parameters is investigated An asymptotic method of constructing a solution, which describes the oscillatory motions of the single-mode (single-frequency) approximation, which is usually implemented in practical problems, is described and justified. Constructive sufficient conditions are formulated and the closeness of the approximate single-frequency solution to the exact solution in an asymptotically long time interval is proved. Possible extensions of the structure of the perturbing functions are considered and the case of the finite-mode approximation is investigated. Solutions of specific problems, which are of practical interest, are constructed to illustrate the effectiveness of the single-frequency approximation method.     

Optimal control of distributed parabolic systems with multiple time delays given in the integral form

Various optimal control problems for linear parabolic systemswith multiple time delays given in the integral form are considered.Necessary and sufficient conditions of optimality are derivedfor the Neumann problem. The optimal control is obtained inthe feedback form. Making use of the results of Schwartz, therepresentation of the optimal feedback control is given. A simpleexample of application is also provided.     

Optimal level of diagnostics in distributed communication systems

A unit (multiplexor, switch, terminal, processor, etc.) at a remote site is inspected occasionally from a central site to see if it is functioning properly. Two stochastic models are proposed for monitoring remote units whose operational status can be described by an alternating 0–1 process. Optimal inspection schedules are derived based on the objective of maximizing the overall steady state system availability while limiting inspections per-unit-time costs and customer inconvenience.     

Stabilization of linear distributed control systems with unbounded delay

In this paper we study the asymptotic stabilization of linear distributed parameter control systems with unbounded delay. Assuming that the semigroup of operators associated with the uncontrolled and nondelayed equation is compact and that the phase space is a uniform fading memory space, we characterize those systems that can be stabilized using a feedback control. As consequence we conclude that every system of this type is stabilizable with an appropriated finite dimensional control.     

Mobile control of distributed parameter systems

Book Review

Mobile control of distributed parameter systemsA. G. Butkovskiy and L. M. Pustyl'nikov: translated by L. W. Longdon, Ellis Horwood, Halsted Press, New York, Chichester, Brisbane, Toronto, 1987, 310 pp. US$45.00. ISBN 0-85312-507-4 (Ellis Horwood Limited), ISBN 0-470-20817-1 (Halsted Press).     

Infinite-horizon boundary control of distributed systems

For a boundary controlled dynamic system, algorithms for solving the problem of tracking reference motion and the problem of tracking reference control are described. The algorithms are robust to information noise and computational errors. The solution method is based on the extremal shift method from the theory of positional differential games.     

Destruction of solutions of wave equations in systems with distributed parameters

We consider two initial boundary value problems on an interval with homogeneous Dirichlet conditions. These problems were proposed by M. I. Rabinovich and D. I. Trubetskov and are given by nonlinear fourth-order Sobolev-type equations. We prove the local-in-time existence of a strong generalized solution of one problem, and for both problems, we obtain sufficient conditions for the destruction of their strong generalized solutions in a finite time.