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This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in practical applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- System theory. --- Mathematical physics. --- Vibration. --- Dynamical systems. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Systems Theory, Control. --- Mathematical Applications in the Physical Sciences. --- Vibration, Dynamical Systems, Control. --- Differential equations, Hyperbolic. --- Hyperbolic differential equations --- Differential equations, Partial --- Differential equations, partial. --- Differentiable dynamical systems. --- Systems theory. --- Cycles --- Mechanics --- Sound --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Physical mathematics --- Systems, Theory of --- Systems science --- Science --- Philosophy --- Differential equations, Partial.

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The term “control theory” refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010.  Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a friendly introduction to, and an updated account of, some of the most active trends in current research.

Control theory --- Differential equations, Partial --- Mathematics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Operations Research --- Calculus --- Control theory. --- Differential equations, Partial. --- Partial differential equations --- Mathematics. --- Partial differential equations. --- System theory. --- Numerical analysis. --- Fluids. --- Partial Differential Equations. --- Systems Theory, Control. --- Fluid- and Aerodynamics. --- Numerical Analysis. --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Math --- Philosophy --- Dynamics --- Machine theory --- Differential equations, partial. --- Systems theory. --- Differential equations. --- Continuum mechanics. --- Differential Equations. --- Systems Theory, Control . --- Continuum Mechanics. --- Mechanics of continua --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- 517.91 Differential equations --- Differential equations

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The goal of these proceedings and of the meeting of Gaeta was to celebrate and honor the mathematical achievements of Haim Brezis. The prodigious in?uence of histalentandhispersonalityinthedomainofnonlinearanalysisisunanimously- claimed!This impactis visible inthe huge number ofhis formerstudents (dozens), students of former students (hundreds) and collaborators (hundreds). Thus the Gaeta meeting was, to some extent, the family reunion of part of this large c- munity sharing a joint interest in the ?eld of elliptic and parabolic equations and pushing it to a very high standard. Italyhasa longtraditionandtasteforanalysisandwecouldnot?ndabetter placeneitheramorecompletesupportfortherealisationofourproject.Wehaveto thank here the university of Cassino, Napoli, Roma la Sapienza , the GNAMPA- Istituto di Alta Matematica, CNR-IAC, MEMOMAT, RTN Fronts-Singularities, the commune of Gaeta. Additional founding came from the universities of M- house and Zur ¨ ich. Finally, we are grateful to Birkh¨ auser and Dr. Hemp?ing who allowed us to record the talks of this conference in a prestigious volume. The organizers Progress in Nonlinear Di?erential Equations and Their Applications, Vol. 63, 1-12 c 2005 Birkh¨ auser Verlag Basel/Switzerland One-Layer Free Boundary Problems with Two Free Boundaries Andrew Acker Abstract. We studythe uniquenessand successive approximation of solutions of a class of two-dimensional steady-state ?uid problems involving in?nite periodic ?ows between two periodic free boundaries, each characterized by a ?ow-speed condition related to Bernoulli's law.

Partial differential equations --- differentiaalvergelijkingen

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Mathematics