By Miroslav Krstic, Andrey Smyshlyaev

This concise and hugely usable textbook provides an advent to backstepping, a chic 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 ordinary, strong, and bodily intuitive "target PDE platforms" which are well-known to engineers and physicists.

The textual content s large insurance comprises parabolic PDEs; hyperbolic PDEs of first and moment order; fluid, thermal, and structural structures; hold up platforms; 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 an eye on for PDEs and keep an eye on of nonlinear PDEs.

it really is acceptable for classes on top of things idea and comprises homework routines and a suggestions guide that's to be had from the authors upon request.

**Audience: This e-book is meant for either starting and complex graduate scholars in a one-quarter or one-semester direction on backstepping recommendations for boundary regulate of PDEs. it's also obtainable to engineers without past education in PDEs. **

**Contents: record of Figures; record of Tables; Preface; creation; Lyapunov balance; particular recommendations 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 an eye on for PDEs; in the direction of Nonlinear PDEs; Appendix: Bessel services; Bibliography; Index
**

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**Extra info for Boundary control of PDEs: a course on backstepping designs**

**Sample text**

13) that k(0, 0) = 0, which gives us the following: kxx (x, y) − kyy (x, y) = λk(x, y) , k(x, 0) = 0 , λ k(x, x) = − x . 15) It turns out that these three conditions are compatible and in fact form a well-posed PDE. This PDE is of hyperbolic type: one can think of it as a wave equation with an extra term λk (x plays the role of time). In quantum physics such PDEs are called Klein–Gordon PDEs. 1. The boundary conditions are prescribed on two sides of the triangle and the third side (after solving for k(x, y)) gives us the control gain k(1, y).

7 Respectively, we mean the transport equation, the heat and wave equations, the Korteweg–de Vries equation, and the Euler–Bernoulli beam and Kuramoto–Sivashinsky equations. 8 Respectively, we mean the heat equation, the Schrödinger equation. 29 ✐ ✐ ✐ ✐ ✐ ✐ ✐ n48 main 2008/4/7 page 30 ✐ 30 Chapter 4. Parabolic PDEs: Reaction-Advection-Diffusion and Other Equations The main feature of backstepping is that it is capable of eliminating destabilizing effects (“forces” or “terms”) that appear throughout the domain while the control is acting only from the boundary.

45). Top: open-loop response. 46) implemented. 9) (which does not depend on the particular plant). 59) x kyy (x, y)u(y) dy − λk(x, y)u(y) dy (integration by parts). 0 ✐ ✐ ✐ ✐ ✐ ✐ ✐ n48 main 2008/4/7 page 40 ✐ 40 Chapter 4. 4. 45). 59), we get wt − wxx = λ + 2 x + d k(x, x) u(x) − ky (x, 0)u(0) dx kxx (x, y) − kyy (x, y) − λk(x, y) u(y) dy . 60) 0 For the right-hand side of this equation to be zero for all u(x), the following three conditions must be satisfied: kxx (x, y) − kyy (x, y) − λk(x, y) = 0 , ky (x, 0) = 0 , d λ + 2 k(x, x) = 0 .