By Miroslav Krstic, Andrey Smyshlyaev
This concise and hugely usable textbook offers an creation to backstepping, a sublime new method of boundary keep an eye on of partial differential equations (PDEs). Backstepping presents mathematical instruments for changing complicated and risky PDE structures into basic, good, and bodily intuitive "target PDE platforms" which are general to engineers and physicists.
The textual content s vast insurance contains parabolic PDEs; hyperbolic PDEs of first and moment order; fluid, thermal, and structural platforms; hold up structures; PDEs with 3rd and fourth derivatives in area; real-valued in addition to complex-valued PDEs; stabilization in addition to movement making plans and trajectory monitoring for PDEs; and parts of adaptive keep watch over for PDEs and keep an eye on of nonlinear PDEs.
it's acceptable for classes on top of things thought and comprises homework workouts and a options handbook that's to be had from the authors upon request.
Audience: This publication is meant for either starting and complex graduate scholars in a one-quarter or one-semester path on backstepping innovations for boundary keep watch over of PDEs. it's also obtainable to engineers with out previous education in PDEs.
Contents: checklist of Figures; record of Tables; Preface; creation; Lyapunov balance; targeted ideas to PDEs; Parabolic PDEs: Reaction-Advection-Diffusion and different Equations; Observer layout; Complex-Valued PDEs: Schrodinger and Ginzburg Landau Equations; Hyperbolic PDEs: Wave Equations; Beam Equations; First-Order Hyperbolic PDEs and hold up Equations; Kuramoto Sivashinsky, Korteweg de Vries, and different unique Equations; Navier Stokes Equations; movement making plans for PDEs; Adaptive keep watch over for PDEs; in the direction of Nonlinear PDEs; Appendix: Bessel services; Bibliography; Index
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Additional info for Boundary Control of Pdes: A Course on Backstepping Designs (Advances in Design and Control)
In  we recently extended the construction in  by showing that the set of smooth positive weight functions for which local injectivity does not hold is also dense. In  we proved local injectivity for a class of smooth weight functions that was introduced by Gindikin . In this note we shall extend the result from  by replacing the family of straight lines by a two-dimensional family of curves satisfying a certain condition, thus obtaining a statement that is invariant under local diffeomorphisms in the xy space and in the ξ η space of curves.
It says that if the boundary values on the translations ∂B + te1 of a function f coincide, to 42 M. Agranovsky and L. Zalcman order m, with a solution Ft in B + te1 of a given elliptic equation LFt = 0, then f itself is a solution, Lf = 0 in the union of the spheres ∂B + te1 . In , the problem of generalizing such results from Laplace’s equation to more general PDE was posed. The following theorem presents a reasonably general result for elliptic equations. Before formulating this result, we need to introduce some notation.
5) We may assume that x0 = (x 0 , y 0 ) = (0, 0) and γ 0 = (ξ 0 , η0 ) = (0, 0). We can also rotate the coordinate systems in X and Γ so that Fx = 0, Fy = 0, Fξ = 0, Fη = 0 at the origin. (6) Then we can solve y or η from (3) and obtain respectively y = u(ξ, η, x), η = ρ(x, y, ξ ) . (7) Differentiating the identity F (x, u(ξ, η, x), ξ, η) = 0, we obtain Fx + Fy ux = 0, Fξ + Fy uξ = 0, F η + F y uη = 0 (8) 48 J. Boman in a neighborhood of the origin. From here and from (6) we obtain ux = uξ = 0 and uη = 0 at the origin, (9) and similarly, ρx = ρξ = 0 and ρy = 0 at the origin.