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55
Well Posedness For Damped Second Order Systems With Unbounded Input Operators
 DIFFERENTIAL AND INTEGRAL EQUATIONS
, 1995
"... We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a ..."
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Cited by 27 (17 self)
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We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a weak or variational setting are given. A semigroup formulation is presented and conditions under which the variational solutions and semigroup solutions are the same are discussed.
Adaptive Learning with Nonlinear Dynamics Driven by Dependent Processes
 Econometrica
, 1994
"... ..."
Diffusion in PoroElastic Media
 Jour. Math. Anal. Appl
, 1998
"... . Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of se ..."
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Cited by 15 (7 self)
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. Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of secondary consolidation and pore fluid exposure on the boundary are included. This quasistatic system is resolved as an application of the theory of linear degenerate evolution equations in Hilbert space, and this leads to a precise description of the dynamics of the system. 1. Introduction We shall consider a system modeling diffusion in an elastic medium in the case for which the inertia effects are negligible. This quasistatic assumption arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. The fluid pressure is denoted by p(x; t) and the displacement of the structure by u(x; t). ...
Approximation in LQR Problems for Infinite Dimensional Systems With Unbounded Input Operators
, 1990
"... We present a variational framework based on sesquilinear forms for Galerkin approximation techniques for state feedback control in problems governed by infinite dimensional dynamical systems. Both parabolic and second order in time, hyperbolic partial differential equations with unbounded input and ..."
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Cited by 13 (6 self)
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We present a variational framework based on sesquilinear forms for Galerkin approximation techniques for state feedback control in problems governed by infinite dimensional dynamical systems. Both parabolic and second order in time, hyperbolic partial differential equations with unbounded input and unbounded observation operators are included as special cases of our treatment. 1 Introduction In this paper we discuss the linear quadratic regulator (LQR) problem for a class of (essentially parabolic) unbounded input or boundary control problems. A variational framework using sesquilinear forms is developed to treat Dirichlet and Neuman boundary control problems for parabolic equations and strongly damped elastic systems. Using such a framework, convergence of Galerkin approximations to solutions of Riccati equations is also established. The boundary control problem for parabolic systems has been studied extensively over the last two decades, inspired by the monograph of J.L. Lions [21] ...
Single Phase Flow In Partially Fissured Media
"... . Totally fissured media in which the individual cells are isolated by the fissure system are effectively described by double porosity models with microstructure. Such models contain the geometry of the individual cells in the medium and the flux across their interface with the fissure system which ..."
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Cited by 13 (5 self)
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. Totally fissured media in which the individual cells are isolated by the fissure system are effectively described by double porosity models with microstructure. Such models contain the geometry of the individual cells in the medium and the flux across their interface with the fissure system which surrounds them. We extend these results to a dualpermeability model which accounts for the secondary flux arising from direct celltocell diffusion within the solid matrix. Homogenization techniques are used to construct a new macroscopic model for the flow of a single phase compressible fluid through a partially fissured medium from an exact but highly singular microscopic model, and it is shown that this macroscopic model is mathematically well posed. Preliminary numerical experiments illustrate differences in the behaviour of solutions to the partially fissured from that of the totally fissured case. 1. Introduction. The bulk characteristics of laminar flow through porous media are det...
Analytic semigroups: applications to inverse problems for flexible structures
 IN DIFFERENTIAL EQUATIONS WITH APPLICATIONS
, 1991
"... We present new convergence and stability results for least squares inverse problems involving systems described by analytic semigroups. The practical importance of these results is demonstrated by application to several examples from problems of estimation of material parameters in flexible structur ..."
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Cited by 10 (9 self)
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We present new convergence and stability results for least squares inverse problems involving systems described by analytic semigroups. The practical importance of these results is demonstrated by application to several examples from problems of estimation of material parameters in flexible structures using accelerometer data.
Model Reference Adaptive Control of Distributed Parameter Systems
 SIAM J. CONTROL OPTIM
, 1995
"... A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop sys ..."
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Cited by 8 (1 self)
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A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop system is shown to be well posed. The tracking error is shown to converge to zero, and regularity results for the control input and the output are established. With an additional richness, or persistence of excitation assumption, the parameter error is shown to converge to zero as well. A finite dimensional approximation theory is developed. Examples involving both first (parabolic) and second (hyperbolic) order systems and linear and nonlinear systems are discussed, and numerical simulation results are presented.
An Approximation Theory of Solutions to Operator Riccati Equations for ... Control
, 1994
"... As in the finitedimensional case, the appropriate statefeedback for the infinitedimensional H disturbance attenuation problem may be calculated by solving a Riccati equation. This operator Riccati equation can rarely be solved exactly. We approximate the original infinitedimensional system by a s ..."
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Cited by 7 (1 self)
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As in the finitedimensional case, the appropriate statefeedback for the infinitedimensional H disturbance attenuation problem may be calculated by solving a Riccati equation. This operator Riccati equation can rarely be solved exactly. We approximate the original infinitedimensional system by a sequence of finitedimensional systems and consider the corresponding finitedimensional disturbanceattenuation problems. We make the same assumptions required in approximations for the classical linear quadratic regulator problem, and show that the sequence of solutions to the corresponding finite dimensional Riccati equations converge strongly to the solution to the infinitedimensional Riccati equation. Furthermore, the corresponding finitedimensional feedback operators yield performance arbitrarily close to that obtained with the infinitedimensional solution. 1 Introduction In this paper we discuss H 1 control problems for the linear system in a Hilbert space X (1:1) d dt x(t) = Ax...
Design of Finitedimensional Controllers for Infinitedimensional Systems by Approximation
, 1994
"... Several difficulties in controller design for infinitedimensional systems arise from using an approximation for the state of the system. In this paper it is demonstrated that the graph topology is an appropriate framework in which to discuss convergence of approximations used for controller design. ..."
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Cited by 7 (2 self)
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Several difficulties in controller design for infinitedimensional systems arise from using an approximation for the state of the system. In this paper it is demonstrated that the graph topology is an appropriate framework in which to discuss convergence of approximations used for controller design. It is also shown that Galerkin type approximations to a large class of problems possess the required convergence properties and can be used to design controllers which will perform as designed when implemented on the original infinitedimensional system. An H1controller design problem is used to illustrate this approach. Key words: infinitedimensional systems, Galerkin approximations, coprime factorizations, control theory, graph topology 1 Introduction There are computational difficulties, apart from the theoretical problems, to designing controllers for systems whose dynamics are described by partial differential equations or integraldifferential equations. Consider the following on ...
Error estimates for the discontinuous Galerkin methods for implicit parabolic equations
 NO6, 2006, PP 24782499. MR2206444 (2006J:65280
"... We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spac ..."
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Cited by 6 (5 self)
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We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at each time step and do not require the spatial operator to be self adjoint or independent of time. Our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate. These projections are local and enjoy optimal approximation properties when the solution is sufficiently regular.