Control of Wave and Beam PDEs is a concise, self-contained introduction to Riesz bases in Hilbert space and their applications to control systems described by partial differential equations (PDEs). The authors discuss classes of systems that satisfy the spectral determined growth condition, the problem of stability, and the relationship between fulfillment of the condition and stability. Using the (fundamental) Riesz-basis property, the book shows how controllability, observability, stability, etc., can be derived for a linear system. The text provides a crash course in the mathematical theory of Riesz bases so that a reader can quickly understand this powerful method of dealing with linear PDEs. It introduces several important methods for achieving the Riesz basis property through spectral analysis, as well as new approaches including treatment of systems coupled through boundary weak connections. The book moves from a discussion of mathematical preliminaries through bases in Hilbert Spaces to applications to Euler–Bernoulli and Rayleigh beam equations and hybrid systems. The final chapter expands the use of the book’s methods to applications in other systems. Many typical examples, representing physical systems, are discussed in the text. The book is suitable not only for applied mathematicians seeking a powerful tool to understand control systems, but also for control engineers interested in the mathematics of PDE systems.

The text's broad coverage includes parabolic PDEs; hyperbolic PDEs of first and second order; fluid, thermal, and structural systems; delay systems; PDEs with third and fourth derivatives in space (including variants of linearized Ginzburg-Landau, Schrodinger, Kuramoto-Sivashinsky, KdV, beam, and Navier-Stokes equations); real-valued as well as complex-valued PDEs; stabilization as well as motion planning and trajectory tracking for PDEs; and elements of adaptive control for PDEs and control of nonlinear PDEs.

A clear and concise introduction to backstepping, an elegant new approach to boundary control of partial differential equations (PDEs).

New adaptive and event-triggered control designs with concrete applications in undersea construction, offshore drilling, and cable elevators Control applications in undersea construction, cable elevators, and offshore drilling present major methodological challenges because they involve PDE systems (cables and drillstrings) of time-varying length, coupled with ODE systems (the attached loads or tools) that usually have unknown parameters and unmeasured states. In PDE Control of String-Actuated Motion, Ji Wang and Miroslav Krstic develop control algorithms for these complex PDE-ODE systems evolving on time-varying domains. Motivated by physical systems, the book’s algorithms are designed to operate, with rigorous mathematical guarantees, in the presence of real-world challenges, such as unknown parameters, unmeasured distributed states, environmental disturbances, delays, and event-triggered implementations. The book leverages the power of the PDE backstepping approach and expands its scope in many directions. Filled with theoretical innovations and comprehensive in its coverage, PDE Control of String-Actuated Motion provides new design tools and mathematical techniques with far-reaching potential in adaptive control, delay systems, and event-triggered control.

This book is devoted to analyze the vibrations of simpli?ed 1? d models of multi-body structures consisting of a ?nite number of ?exible strings d- tributed along planar graphs. We?rstdiscussissueson existence and uniquenessof solutions that can be solved by standard methods (energy arguments, semigroup theory, separation ofvariables,transposition,...).Thenweanalyzehowsolutionspropagatealong the graph as the time evolves, addressing the problem of the observation of waves. Roughly, the question of observability can be formulated as follows: Can we obtain complete information on the vibrations by making measu- ments in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems. UsingtheFourierdevelopmentofsolutionsandtechniquesofNonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total length of the network in a suitable Hilbert space that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree, we characterize these weights in terms of the eigenvalues of the corresponding elliptic problem. The resulting weighted observability inequality allows id- tifying the observable energy in Sobolev terms in some particular cases. That is the case, for instance, when the network is star-shaped and the ratios of the lengths of its strings are algebraic irrational numbers.

While rooted in controlled PDE systems, this 2005 AMS-IMS-SIAM Summer Research Conference sought to reach out to a rather distinct, yet scientifically related, research community in mathematics interested in PDE-based dynamical systems. Indeed, this community is also involved in the study of dynamical properties and asymptotic long-time behavior (in particular, stability) of PDE-mixed problems. It was the editors' conviction that the time had become ripe and the circumstances propitious for these two mathematical communities--that of PDE control and optimization theorists and that of dynamical specialists--to come together in order to share recent advances and breakthroughs in their respective disciplines. This conviction was further buttressed by recent discoveries that certain energy methods, initially devised for control-theoretic a-priori estimates, once combined with dynamical systems techniques, yield wholly new asymptotic results on well-established, nonlinear PDE systems, particularly hyperb These expectations are now particularly well reflected in the contributions to this volume, which involve nonlinear parabolic, as well as hyperbolic, equations and their attractors; aero-elasticity, elastic systems; Euler-Korteweg models; thin-film equations; Schrodinger equations; beam equations; etc. in addition, the static topics of Helmholtz and Morrey potentials are also prominently featured. A special component of the present volume focuses on hyperbolic conservation laws, to take advantage of recent theoretical advances with significant implications also on applied problems. in all these areas, the reader will find state-of-the-art accounts as stimulating starting points for further research.

This book is devoted to analyze the vibrations of simpli?ed 1? d models of multi-body structures consisting of a ?nite number of ?exible strings d- tributed along planar graphs. We?rstdiscussissueson existence and uniquenessof solutions that can be solved by standard methods (energy arguments, semigroup theory, separation ofvariables,transposition,...).Thenweanalyzehowsolutionspropagatealong the graph as the time evolves, addressing the problem of the observation of waves. Roughly, the question of observability can be formulated as follows: Can we obtain complete information on the vibrations by making measu- ments in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems. UsingtheFourierdevelopmentofsolutionsandtechniquesofNonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total length of the network in a suitable Hilbert space that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree, we characterize these weights in terms of the eigenvalues of the corresponding elliptic problem. The resulting weighted observability inequality allows id- tifying the observable energy in Sobolev terms in some particular cases. That is the case, for instance, when the network is star-shaped and the ratios of the lengths of its strings are algebraic irrational numbers.