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This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems, our new method does not require solving non-standard Riccati-type or Klein-Gorden-type partial differential equations (PDEs). Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradient of the system cost functional (which penalizes both kernel and output magnitude) is computed by solving a so-called costate PDE instandard form. Special constraints are imposed on the kernel coefficients to ensure that, under mild conditions, the optimized kernel yields closed-loop stability. Numerical simulations demonstrate the effectiveness of the proposed approach.

This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems, our new method does not require solving non-standard Riccati-type or Klein-Gorden-type partial differential equations (PDEs). Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradient of the system cost functional (which penalizes both kernel and output magnitude) is computed by solving a so-called costate PDE instandard form. Special constraints are imposed on the kernel coefficients to ensure that, under mild conditions, the optimized kernel yields closed-loop stability. Numerical simulations demonstrate the effectiveness of the proposed approach.

This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems, our new method does not require solving non-standard Riccati-type or Klein-Gorden-type partial differential equations (PDEs). Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradient of the system cost functional (which penalizes both kernel and output magnitude) is computed by solving a so-called costate PDE instandard form. Special constraints are imposed on the kernel coefficients to ensure that, under mild conditions, the optimized kernel yields closed-loop stability. Numerical simulations demonstrate the effectiveness of the proposed approach.

This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems, our new method does not require solving non-standard Riccati-type or Klein-Gorden-type partial differential equations (PDEs). Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradient of the system cost functional (which penalizes both kernel and output magnitude) is computed by solving a so-called costate PDE instandard form. Special constraints are imposed on the kernel coefficients to ensure that, under mild conditions, the optimized kernel yields closed-loop stability. Numerical simulations demonstrate the effectiveness of the proposed approach.

This research seeks to unify certain problems of distributed parameter control: model reduction, control, sensor/actuator selections/failure, and decentralized control. These topics are all related and are to be unified through the quadric performance metric with use of cost decomposition methods. The final research topic on model error estimation is required to male vernier adjustments after best models and controllers are developed, to absorb remaining modeling errors. During the period covered by the grant, 16 conference and journal research papers were written. Titles include: Computer Aided Design of Suboptimal LQG Controller, Cost Equivalent Realizations of Stochastic Processes, Component Cost Analysis of Large Scale Systems, Comments on Realizations and Reduction of Markovian Models for Nonstationary Data, and On the Selection of Controller Order for the Control of Linear Dynamic Systems.

This final report describes the purpose and details of the equipment purchased under a University Research Instrumentation Program (URIP) grant. Equipment includes instruments for remote sensing of vibrations, for modal analysis of vibrating structures, and for active vibration suppression of viscoelastic beams.

Controllability of distributed-parameter control systems is mathematically studied. A general theory for control systems includes those that cannot be described by ordinary differential equations.

Methods for estimating system parameters are discussed for a class of partial differential equations. We develop schemes based on modal subspace approximations in some detail and include numerical examples. (Author)

The main purpose of this research is to develop a rigorous mathematical framework for the design of control laws for feedback stabilization and for controllability of the transient behavior of flexible structures based on distributed parameter models of such structures. This work has entailed deriving accurate distributed parameter models for elastic structures and understanding the implications of the various models for the controllability and stabilizability of structures. Substantial progress has been made for models of multiple-link constructions that are composed of elastic beams, plates., shells or combinations of such elastic elements. Such structures are representative of trusses, frames, robot arms, solar panels, antennae, deformable mirrors, etc., currently in use.

In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset $\Omega$ of IR n. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in $\Omega$, over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L 2-energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional parabolic operators, but we also provide several examples in higher dimensions. We consider two types of controllers: either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results.

The present work is an attempt to put together the most relevant aspects of the engineering problems involving distributed parameter systems (D.P.S.'s). Simulation and optimal control are explained in detail in Chapters II and III. The original contribution of this thesis is given in Chapters V and VI, where modal control theory and a gradient subroutine that searches for the optimal reference coefficients are used. As a result, it was possible to obtain an output distribution better than the one achievable by the known methods. This technique works in situations of strongly nonlinear control and compensates the effect of having the analyzer and synthesizer approximated by low order matrices. It also makes it possible to give higher weight to some zones of the output distribution in order to have a better local fit. The necessary background for understanding Chapters V and VI is given in Chapter IV.

Parabolic partial differential equations for optimal pointwise feedback control of distributed parameter systems

A Chandrasekhar-type factorization method is applied to the linear-quadratic optimal control problem for distributed parameter systems. An aeroelastic control problem is used as a model example to demonstrate that if computationally efficient algorithms, such as those of Chandrasekhar-type, are combined with the special structure often available to a particular problem, then an abstract approximation theory developed for distributed parameter control theory becomes a viable method of solution. A numerical scheme based on averaging approximations is applied to hereditary control problems. Numerical examples are given.

Here an original idea is suggested to prove the existence of optimal control for some types of non- linear problems. The obtained results can be considered as individual existence theorems (in some sense).

Cyber-physical systems (CPSs) are man-made complex systems coupled with natural processes that, as a whole, should be described by distributed parameter systems (DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.

A series of numerical examples is reported and several algorithms compared for estimation of coefficients in differential equation models. Unconstrained, constrained and Tikhonov regularization methods are tested for their behavior with regard to both convergence (of approximation methods for the states and parameters) and stability (continuity of the estimates with respect to perturbations in the data or observed states).

A series of numerical examples is reported and several algorithms compared for estimation of coefficients in differential equation models. Unconstrained, constrained and Tikhonov regularization methods are tested for their behavior with regard to both convergence (of approximation methods for the states and parameters) and stability (continuity of the estimates with respect to perturbations in the data or observed states).

We developed methodologies for the bifurcation detection and stability analysis of microscopic timesteppers (LB-based. lattice gas or Monte Carlo) based on the Recursive Projection Method (RPM). We demonstrated RPM-based 'coarse' (macroscopic) bifurcation and stability computations for systems for which only microscopic evolution rules are available (LB. KMC, Brownian Dynamics -BD-). We were able to perform the coarse bifurcation analysis of KMC models of surface reactions, of LB models of multiphase-bubbly and of reaction diffusion problems, as well as of BD non-Newtonian rheological problems. New micro-Galerkin simulation methods based on Projective Integrators and Telescopic Projective Integrators were implemented. The 'gaptooth' scheme and the 'patch dynamics' schemes were developed and demonstrated. Methods for doing effective bifurcation analysis using timesteppers (thus sidestepping the necessity of deriving homogenized equations) were developed and demonstrated. Furthermore we implemented RPM around a state-of-the-art massively parallel finite-element based code (codename: MPSalsa) at Sandia, as well as a state of the art chemical plant simulation code (gPROMS) which was used in problems of interest to UTRC (fuel cell reactor train modeling). We also developed and demonstrated methods for the computation of self-similar and 'coarsely' self similar solutions, as well as methods for the coarse control of microscopic simulations.

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonliner systems and a summary of numerical experiments with the techniques proposed is given. (Author)

Efforts to develop computational methods for the identification and optimal control of linear and nonlinear systems governed by distributed parameter systems are reported on. Specifically, approximation methods for determining Optimal LOG compensators (feedback control and estimator gains) and functional parameters in linear and nonlinear partial differential equations and hereditary systems were developed, analyzed and tested. The study included theoretical, experimental and numerical components. Covergence theories for spline-based and modal finite element schemes were established and extensive numerical studies on both conventional (serial) and vector supercomputers were carried out. A parameter estimation scheme was tested using experimental data taken from the RPL structure, a laboratory experiment designed to test control algorithms for the large angle slewing of spacecraft with flexible appendages, and other projects involving the identification of flexible structures based upon experimental data were initiated.

Efforts to develop computational methods for the identification and optimal control of linear and nonlinear systems governed by distributed parameter systems are reported on. Specifically, approximation methods for determining Optimal LOG compensators (feedback control and estimator gains) and functional parameters in linear and nonlinear partial differential equations and hereditary systems were developed, analyzed and tested. The study included theoretical, experimental and numerical components. Covergence theories for spline-based and modal finite element schemes were established and extensive numerical studies on both conventional (serial) and vector supercomputers were carried out. A parameter estimation scheme was tested using experimental data taken from the RPL structure, a laboratory experiment designed to test control algorithms for the large angle slewing of spacecraft with flexible appendages, and other projects involving the identification of flexible structures based upon experimental data were initiated.

The solution of the sparse algebraic system for a point controlled elliptic distributed parameter system by a multiple shooting and sweep algorithm is discussed. The multiple shooting and sweep algorithm enhances the convergence rate of the ordinary shooting method while achieving a significant reduction in the dimension of the linear equations to be solved. The optimization algorithm for a special problem involving the minimum cost selection of source intensities with the state satisfying a specified constraint set is presented. An application of the techniques involving the analysis and management design for water quality control in Corpus Christi Bay, an estuary on the Texas Gulf Coast in the United States, is discussed. The resulting algorithm is more than twice as fast as a corresponding successive overrelaxation method. (Author)

This theses considers the problem of robustly stabilizing infinite-dimensional systems using finite-dimensional controllers. The controllers are assumed to be linear quadratic Gaussian (LQG) based controllers. This research uses a direct approach to demonstrate the existence of finite-dimensional LQG-based controllers that stabilize the nominal system. Once existence is proven, the research focuses on ways to analyze the robustness of the controller. Several types of perturbations are considered, including bounded, relatively bounded, additive, and multiplicative. Several approaches to analyzing robustness are developed. Direct analysis using results from functional analysis is accomplished, followed by an approach called the optimal projection equation approach, and then H-infinity techniques are used to develop a sufficient condition for robustness in the presence of multiplicative perturbations of the plant transfer function. A new interpretation of the linear quadratic Gaussian/loop transfer recovery technique (LQG/LTR) is made for the case of reduced order controllers.

An approximation and convergence theory for the identification of nonlinear damping in abstract wave equations is developed. It is assumed that the unknown dissipation mechanism to be identified can be described by a maximal monotone operator acting on the generalized velocity. The stiffness is assumed to be linear and symmetric. Functional analytic techniques are used to establish that solutions to a sequence of finite dimensional (Galerkin) approximating identification problems in some sense approximate a solution to the original infinite dimensional inverse problem. Accretive operator, Galerkin approximation, Inverse problems, Nonlinear damping, Nonlinear evolution equation.

Models of complex systems can be differentiated by their ability to reproduce or generate system behavior, by their prediction power, by their robustness, or, conversely, by their sensitivity to inputs and parameters; by their uncertainty (if captured); and by their intelligence. Even the term prediction is not unique. First, a first-principle (physically based) distributed parameter model could be an excellent predictor if (a) it captures the main system behavior, and (b) its parameters and inputs are known accurately; otherwise, it would fail, possibly drastically. Second, predictive power depends on the data, on the goal, and on the time scale. For example, scheduling of pumping and injection in an oil field for maximum profit over the next 5 years; or pumping from a contaminated aquifer in order to maintain certain (low) concentration at a compliance point for the next 20 years, vs. prediction of plume migration in groundwater towards a nearby river, over time: in each case, the model has a slightly different expected function, as well as different intelligence type. The paper reviews the recent developments in subsurface fluid flow management such as optimization of oil production and groundwater remediation (both sharing similar practices, though for different purposes) as a continuous struggle to increase intelligence by (a) adapting new tools such as artificial intelligence and dynamic stochastic control; (b) attempting to integrate these tools; and (c) reducing uncertainty. Although the systems discussed seem specific to the (mathematical) geosciences (specifically to oil reservoirs and contaminated aquifers), and although these systems are very different from man-made machines, similar rigid structure and reliance on differential-integral calculus, as well as the serial processing, knowledge evolution, and uncertainty propagation from one discipline to the next exist in most science and engineering fields, and so does the need for a paradigm shift.

An abstract approximation framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original finite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.

The problem of estimating spatially varying coefficients of partial differential equations is considered from observation of the solution and of the right hand side of the equation. It is assumed that the observations are distributed in the domain and that enough observations are given. A method of discretization and an efficient multigrid method for solving the resulting discrete systems are described. Numerical results are presented for estimation of coefficients in an elliptic and a parabolic partial differential equation.

An effective modeling method for nonlinear distributed parameter systems (DPSs) is critical for both physical system analysis and industrial engineering. In this Rapid Communication, we propose a novel DPS modeling approach, in which a high-order nonlinear Volterra series is used to separate the time/space variables. With almost no additional computational complexity, the modeling accuracy is improved more than 20 times in average comparing with the traditional method.

An abstract approximation framework and convergence theory for the identification of first and second order nonlinear distributed parameter systems developed previously by the authors and reported on in detail elsewhere are summarized and discussed. The theory is based upon results for systems whose dynamics can be described by monotone operators in Hilbert space and an abstract approximation theorem for the resulting nonlinear evolution system. The application of the theory together with numerical evidence demonstrating the feasibility of the general approach are discussed in the context of the identification of a first order quasi-linear parabolic model for one dimensional heat conduction/mass transport and the identification of a nonlinear dissipation mechanism (i.e., damping) in a second order one dimensional wave equation. Computational and implementational considerations, in particular, with regard to supercomputing, are addressed.

An abstract approximation framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original infinite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.

This report concerns several aspects of damping and control in distributed parameter systems, with emphasis on applications to elastic and viscoelastic structures. For viscoelastic bodies, the ineffectiveness of boundary feedback for damping nonoscillatory 'creep' decay was demonstrated, and a precise energy space formulation was developed for the viscoelastic wave equation. In addition, a reachability result was proved for a second-order linear integral equation. For elastic beams and plates, exact controllability and the exponential decay of energy were established in new settings. These include uniform exponential decay of energy by means of locally distributed damping in a one-dimensional nonhomogeneous medium, and, under certain boundary conditions, for longitudinal vibrations in a thermoelastic rod.

The titles include: On the Dirichlet Neumann Boundary Control Problem Associated with Maxwell's Equations in a Cylindrical Region, A Floquet decomposition for Volterra Equations with Periodic Kernel and a transform Approach to Linear Recursion Equations, and Mathematical Models for the Elastic Beam and their Control Theoretic Implications, and Mathematical Models for the Elastic Beam with Frequency Proportional Damping.

Citing data collected and analyzed with the University of Wisconsin MIPAC (Modelling, Information Processing and Control) Facility electronic measurement and analysis equipment, the investigators make some general comments concerning mathematical models which appear to be appropriate for modelling certain vibratory systems of distributed parameter type. Aspects discussed include: location of vibrational spectra, damping rates, and spectral displacement due to mass density and/or elasticity variations. Particular emphasis is placed on some properties of segmented beams. The article ends with a preliminary mathematical discussion of the feasibility of parameter identification, from vibrational spectrum data alone, in the wave and Euler-Bernoulli beam equations.

Finite dimensional approximation schemes that work well for distributed parameter systems are often not suitable for the analysis and implementation of feedback control systems. The relationship between approximation schemes for distributed parameter systems and their application to optimal control problems is discussed. A numerical example is given.

This instrumentation award funded the purchase of a laser vibrometer system, mass computer data storage and data acquisition equipment. This equipment used in conjunction with existing vibration testing and control facilities provides a sophisticated low frequency velocity measurement system for use in identifying the coefficients in partial differential equation models of distributed mass structures. In addition, the vibrometer system provides straightforward and direct velocity feedback for such systems. These flexible structures characteristically have very low natural frequencies which cannot be detected by accelerometers. This system has and is being used to perform tests on models and sub-assemblies of large space structures for the purpose of evaluating existing identification and control strategies as well as to stimulate new research in the area of control, observers (estimators) and identification. Several intense experiments using the laser vibrometer were performed to measure the response of a quasi isotropic cantilevered beam with a removable tip mass excited by an impulse at various locations. This data was collected, stored, and sent to AFOSR researchers at Brown University transmitted using BITNET. The data was analyzed using a spline based estimation procedure, starting with a partial differential equation model of the structure. A clear advantage over modal methods based on a finite dimensional model of the same system was observed. (JHD)

In this project questions related to stabilization of vibrating system by feedback were studied. A vibrating system is uniformly stabilized by a feedback if there is a uniform rate of decay for all vibrational modes of the closed-loop system, which is the system with feedback. Necessary conditions were found for certain kinds of feedback to uniformly stabilize a system which is originally undamped and has the feedback applied at the boundary. Sufficient conditions were found for the feedback to lead to a closed-loop system which is mathematically well-posed. It was also shown that in a large class of systems which are stabilized by feedback, if a small delay is introduced in the computation of the feedback, then the stability of the closed-loop system is destroyed. Keywords: Robust stabilization.

This research was concerned with analytical results related to control of distributed parameter systems. Existence of optimal controls was established for a variety of systems governed by partial differential equations, most of which involved boundary control. Much of the work also dealt nonlinearity. The work on switching systems promises to provide a theoretical baais for dealing under this grant, including A class of stabilizing feedback laws for marginally stable systems answers the question of how costly it is (in terms of control energy) to control quickly. Keywords: Reports; Abstracts.

This research was concerned with analytical results related to control of distributed parameter systems. Existence of optimal controls was established for a variety of systems governed by partial differential equations, most of which involved boundary control. Much of the work also dealt nonlinearity. The work on switching systems promises to provide a theoretical baais for dealing under this grant, including A class of stabilizing feedback laws for marginally stable systems answers the question of how costly it is (in terms of control energy) to control quickly. Keywords: Reports; Abstracts.

We report here on several research projects funded by contract F49620-93-1-0153 have continued investigations in Bayesian analysis and empirical distribution approaches for parameter estimation problems. We are also developing computational methods for simulating subsurface transport and remediation strategies of interest at AFESC, Tyndall AFB and AL/OES Brooks AFB. (AN)

We report here on several research projects funded by contract F49620-93-1-0153 have continued investigations in Bayesian analysis and empirical distribution approaches for parameter estimation problems. We are also developing computational methods for simulating subsurface transport and remediation strategies of interest at AFESC, Tyndall AFB and AL/OES Brooks AFB. (AN)

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

A theoretical framework is presented that can be used to treat approximation techniques for very general classes of parameter estimation problems involving distributed systems that are either first or second order in time. Using the approach developed, one can obtain both convergence and stability (continuous dependence of parameter estimates with respect to the observations) under very weak regularity and compactness assumptions on the set of admissible parameters. This unified theory can be used for many problems found in the recent literature and in many cases offers significant improvements to existing results.