Adaptive Control of Parabolic PDEs by Andrey Smyshlyaev

By Andrey Smyshlyaev

This ebook introduces a entire technique for adaptive keep an eye on layout of parabolic partial differential equations with unknown practical parameters, together with reaction-convection-diffusion structures ubiquitous in chemical, thermal, biomedical, aerospace, and effort platforms. Andrey Smyshlyaev and Miroslav Krstic boost specific suggestions legislation that don't require real-time answer of Riccati or different algebraic operator-valued equations. The publication emphasizes stabilization by way of boundary keep watch over and utilizing boundary sensing for risky PDE structures with an enormous relative measure. The booklet additionally offers a wealthy selection of tools for process identity of PDEs, equipment that hire Lyapunov, passivity, observer-based, swapping-based, gradient, and least-squares instruments and parameterizations, between others. together with a wealth of stimulating rules and offering the mathematical and control-systems heritage had to keep on with the designs and proofs, the publication might be of significant use to scholars and researchers in arithmetic, engineering, and physics. It additionally makes a helpful supplemental textual content for graduate classes on disbursed parameter platforms and adaptive keep watch over.

Show description

Read or Download Adaptive Control of Parabolic PDEs PDF

Similar differential equations books

Numerical Methods for Ordinary Differential Equations

Researchers and scholars from numerical tools, engineering and different sciences will locate this e-book presents an obtainable and self-contained creation to numerical equipment for fixing usual differential equations. It sticks out among different books at the topic as a result of author's lucid writing variety, and the built-in presentation of idea, examples, and workouts.

Sobolev Spaces in Mathematics II: Applications in Analysis and Partial Differential Equations (International Mathematical Series)

Sobolev areas turn into the proven and common language of partial differential equations and mathematical research. between an important number of difficulties the place Sobolev areas are used, the subsequent vital themes are the focal point of this quantity: boundary price difficulties in domain names with singularities, greater order partial differential equations, neighborhood polynomial approximations, inequalities in Sobolev-Lorentz areas, functionality areas in mobile domain names, the spectrum of a Schrodinger operator with unfavourable capability and different spectral difficulties, standards for the entire integration of structures of differential equations with functions to differential geometry, a few elements of differential varieties on Riemannian manifolds regarding Sobolev inequalities, Brownian movement on a Cartan-Hadamard manifold, and so on.

Additional info for Adaptive Control of Parabolic PDEs

Example text

84) (with a1 = 0). 86) 1/4 v(y, t). 70) (written in v-variable) ¯ and going from v back to the original variable u, we get the following result. 5. 89) is exponentially stable at the origin in H 1 (0, 1) with the decay rate (c+ε0 π 2 /φ(1)2 ). 2. holds here as well. 71) is always zero, x0 should be chosen outside the region [0, 1] to keep ε(x) > 0 for x ∈ [0, 1]. 71) can approximate linear functions on [0, 1] very well. 4 the function ε(x) and the corresponding control gains are shown for different parameter values.

6), we can see that l(x, y) = −k(x, y) when λ is replaced by −λ. Therefore, using the properties of Bessel functions (Appendix C), we get l(x, y) = −λy J1 λ(x 2 − y 2 ) λ(x 2 − y 2 ) . 61) and get the following result. 1. 20) u(x, t) = √ λ + π 2 n2 0 n=1 where I1 1 ψn (ξ ) = sin(πnξ ) + ξ ξ λ(τ 2 − ξ 2 ) λ(τ 2 − ξ 2 ) sin(πnτ ) dτ. 7. 60) is solved explicitly with the help of [101]. 1), it is easy to repeat all the steps we have done for the Dirichlet case and get the following closed-form solution for the kernel: k(x, y) = −λx I1 λ(x 2 − y 2 ) λ(x 2 − y 2 ) .

92) where λ(t) is an analytic function of time. 96) w(1, t) = 0. 100) k(x, x, t) = − λ(t). 101) k(x, y, t) = − e− 0 λ(τ ) dτ f (z, t), z = x 2 − y 2 . 102) with boundary conditions fz (0, t) = 0, The 2,1 Cz,t f (0, t) = λ(t)e t 0 λ(τ ) dτ := F (t). (n + 1)! 2 2n F (n) (t). 104) This solution is rather explicit. 104). The controller is given by U (t) = − 1 2 1 y e− t 0 λ(τ )dτ 0 ∞ n=0 (1 − y 2 )n F (n) (t) u(y, t) dy. (n + 1)! 104) in closed form: when F (t) is a combination of exponentials (since it is easy to compute the nth derivative of F (t) in this case) or a polynomial (since the series is finite).

Download PDF sample

Rated 4.70 of 5 – based on 32 votes