Existence and comparison theorems for partial differential equations of Riccati type

In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.     

Necessary conditions for optimality of singular controls in systems governed by partial differential equations

In this paper, we present a method to obtain necessary conditions for optimality of singular controls in systems governed by partial differential equations (distributed-parameter systems). The method is based on the one developed earlier by the author for singular control problems described by ordinary differential equations. As applications, we consider conditions for optimality of singular controls in a Darboux-Goursat system and in control systems that describe chemical processes.This research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

A matrix Green's formula and optimal control of linear distributed-parameter systems

A matrix Green's formula is derived, and its application to the optimal control problem of distributed-parameter systems with both distributed and boundary inputs is presented.     

Time-domain optimization of input signals for distributed-parameter systems identification

The problem of the optimum signal choice for the purpose of distributed-parameter systems identification is considered. In contrast to earlier contributions to this topic, the time-domain approach leads to a nonconvex optimization problem. This difficulty is overcome by introducing an auxiliary convex problem, which is proved to have the same solution as the initial one. This approach allows one to obtain necessary and sufficient optimality conditions for the initial problem, which are then used to find analytical solutions for a wide class of systems. The so-calledD-Optimality criterion is used here, together with constraints on the signal energy.The author expresses his sincere thanks to the anonymous referees for their helpful comments and suggestions.     

Modelling of dynamic networks of thin thermoelastic beams

We derive a distributed-parameter model of a thin non-linear thermoelastic beam in three dimensions. The beam can also be initially curved and twisted. Our main task is to formulate the non-homogeneous initial, boundary and node value problem associated with the dynamics of a network of a finite number of such beams. The emphasis here is on a distributed-parameter modelling of the geometric and kinematic node conditions. The forces and couples appearing in the boundary and node conditions can then be viewed as control variables. The analysis of the resulting control systems and their controllability and stabilizability properties is the subject of [25] and of forthcoming papers.     

Successive approximation method in optimum distributed-parameter systems

The problem of controlling a linear distributed-parameter system with a nonquadratic error measure is discussed. The calculus of variations approach is used to derive an algorithm based on the first variation for theN-dimensional linear diffusion process. The procedure for determining whether the resulting solution is optimum is discussed. Extension of the algorithm for other linear distributed-parameter systems is indicated.This research was supported in part by the National Science Foundation, Grant No. GK-304.     

An LQR-control problem for a multicomponent flexible structure

This paper is concerned with modeling and control of a multicomponent distributed-parameter structure. General results from the theory of infinitedimensional systems are used to pose an LQR-control problem for the structure. Different variations of the structure are considered and, in particular, approximation schemes for two forms of the structure are presented in detail. Responses of the system to an elastic initial condition are presented. It is shown that convergent state feedback control laws can be obtained which result in significant suppression of vibrations and unwanted displacements throughout the structure.     

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Vibrations of mechanical systems in the presence of uncertainty

The minimax state estimation problem is solved for a linear mechanical system with a finite number of degrees of freedom. The proposed procedure is extended to distributed-parameter systems. As an example, we consider an elastic rod that executes small longitudinal oscillations under uncertainty in the initial-boundary conditions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 95–100, 1986.     

Application of the epsilon technique to the identification of distributed-parameter systems

The aim of this paper is to present and discuss the application of the epsilon technique to the identification of distributed-parameter systems. After the problem formulation, attention is focused on computational aspects. The numerical results obtained for two typical systems are presented.This work was supported by ENEL. The authors are indebted to Professor A. Ruberti for having suggested the topic and for useful advice.     

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.     

Modelling Fuzzy Control Problems with Toll Sets

We shall introduce in this paper toll sets and apply it to dynamical and control systems. We shall reformulate viability theorems for 'fuzzy controlled problem". 'Fuzzification" of dynamical and control systems led us to use indicators instead of characteristic functions for imbedding subsets into functions. Therefore, instead of obtaining the genuine fuzzy sets by a convexification procedure, we obtain the 'toll sets", the membership function being a cost function taking values between 0 and . Hence we can define 'fuzzy" or 'toll" – differential inclusions and control systems and solve the viability problem or the toll viability problem by building fuzzy controllers which regulate viable solutions (satisfying at each instant viability and toll viability constraints).     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

Feedback control for a class of pest management problems modeled by a noninteractively distributed system

A class of distributed-parameter optimal control problems whose system dynamics are more akin to those of systems of ordinary differential equations is presented together with a set of necessary conditions for optimality. It is shown that certain insect pest management problems fall within the ambit of this theory, which is then used to investigate the properties of the optimal pesticide application schedule and to synthesize a part-feedback control strategy. This latter strategy is presented in a way which makes it applicable to devising strategies for the application of pesticides in agricultural systems that are too complex to be easily modeled, but for which correlations are available between pest levels at certain points during the growing season and yield losses.     

Suboptimality and stability of linear distributed-parameter systems with finite-dimensional controllers

In order to implement feedback control for practical distributed-parameter systems (DPS), the resulting controllers must be finite-dimensional. The most natural approach to obtain such controllers is to make a finite-dimensional approximation, i.e., a reduced-order model, of the DPS and design the controller from this. In past work using perturbation theory, we have analyzed the stability of controllers synthesized this way, but used in the actual DPS; however, such techniques do not yield suboptimal performance results easily. In this paper, we present a modification of the above controller which allows us to more properly imbed the controller as part of the DPS. Using these modified controllers, we are able to show a bound on the suboptimality for an optimal quadratic DPS regulator implemented with a finite-dimensional control, as well as stability bounds. The suboptimality result may be regarded as the distributed-parameter version of the 1968 results of Bongiorno and Youla.This research was supported by the National Science Foundation under Grant No. ECS-80-16173 and by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0124. The author would like to thank the reviewer for many helpful suggestions.     

Control of non-self-adjoint distributed-parameter systems

Systems involving viscous damping forces, circulatory forces, and aerodynamic forces are non-self-adjoint. A method capable of controlling non-self-adjoint distributed systems is the independent modal-space control method, whereby the problem of controlling a distributed-parameter system is reduced to that of controlling an infinite set of independent, complex, second-order ordinary differential equations. In the case of optimal control, one must solve independent, complex, scalar Riccati equations. The transient solution of the Riccati equations can be found with relative ease and the steady-state solution can be found in closed form. A numerical example demonstrates the effectiveness of the method.This work was supported by AFOSR Research Grant No. 83-0017.     

Optimal feedback control policies for stochastic,distributed-parameter systems

This paper considers optimal feedback control policies for a class of discrete stochastic distributed-parameter systems. The class under consideration has the property that the random variable in the dynamic systems depends only on the time and possesses the Markovian property with stationary transition probabilities. A necessary condition for optimality of a feedback control policy, which has form similar to the Hamiltonian form in the deterministic case, is derived via a dynamic programming approach.     

On an optimal control problem for a hyperbolic partial differential equation

Existence of solutions is proved for a minimum problem for a distributed-parameter control system described by a linear, hyperbolic partial differential equation. The cost function is an integral depending on boundary controls.This research was supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Asymptotic Analysis of State Constrained Semilinear Optimal Control Problems

The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit. We gratefully acknowledge the support of the DAAD. The paper was prepared during the visit of the first author at the Institute of Applied Mathematics II, University Erlangen-Nuremberg in 2003.     

Polygonal quasiconformal maps and grunsky inequalities

There is a deep connection between the Grunsky coefficient inequalities for univalent functions and related extremal quasiconformal maps. In this paper, we develop the technique based on the Grunsky inequalities and apply it to solving a problem concerning polygonal quasiconformal maps. Dedicated to Edger Reich on the occasion of his 75th Birthday.