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Control of Turbulent and Magnetohydrodynamic Channel Flows: by Rafael Vazquez, Miroslav Krstic

By Rafael Vazquez, Miroslav Krstic

This monograph offers new optimistic layout equipment for boundary stabilization and boundary estimation for numerous periods of benchmark difficulties in stream keep watch over, with strength purposes to turbulence keep watch over, climate forecasting, and plasma keep watch over. the root of the method utilized in the paintings is the lately constructed non-stop backstepping strategy for parabolic partial differential equations, increasing the applicability of boundary controllers for move platforms from low Reynolds numbers to excessive Reynolds quantity conditions.

Efforts in movement regulate during the last few years have ended in a variety of advancements in lots of varied instructions, yet so much implementable advancements to date were acquired utilizing discretized types of the plant types and finite-dimensional regulate strategies. by contrast, the layout equipment tested during this booklet are in keeping with the “continuum” model of the backstepping strategy, utilized to the PDE version of the movement. The postponement of spatial discretization till the implementation level bargains a variety of numerical and analytical advantages.

Specific issues and features:

* creation of keep an eye on and nation estimation designs for flows that come with thermal convection and electrical conductivity, specifically, flows the place instability might be pushed by means of thermal gradients and exterior magnetic fields.

* program of a distinct "backstepping" procedure the place the boundary keep watch over layout is mixed with a specific Volterra transformation of the circulation variables, which yields not just the stabilization of the move, but additionally the categorical solvability of the closed-loop system.

* Presentation of a end result unheard of in fluid dynamics and within the research of Navier–Stokes equations: closed-form expressions for the strategies of linearized Navier–Stokes equations less than feedback.

* Extension of the backstepping method of put off one of many well-recognized root motives of transition to turbulence: the decoupling of the Orr–Sommerfeld and Squire systems.

Control of Turbulent and Magnetohydrodynamic Channel Flows is a superb reference for a huge, interdisciplinary engineering and arithmetic viewers: regulate theorists, fluid mechanicists, mechanical engineers, aerospace engineers, chemical engineers, electric engineers, utilized mathematicians, in addition to examine and graduate scholars within the above components. The e-book can also be used as a supplementary textual content for graduate classes on keep an eye on of distributed-parameter structures and on movement control.

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Extra resources for Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation

Sample text

147) w 2 L2 . 148) is well defined for k and l. 1 for a reaction-diffusion equation, in the case that λ = 0, we get an estimate of exponential stability of w, namely that w(t) 2 L2 ≤ e−( /2)t 2 L2 . 2 Mathematical Preliminaries and Notation 29 thus showing exponential stability for u. 151) lx (x, ξ)w(t, ξ)dξ. 156) where m1 = max{2 1 + kx m2 = max{2 1 + lx L∞ (T ) 2 2 L∞ (T ) , 1+ k , 1+ l 2 L∞ (T ) L∞ (T ) 2 +2 k +2 l 2 L∞ (T ) }, 2 L∞(T ) }. 2 for w when λ = 0, which yields w(t) 2 H1 ≤ e−2( /5)t w(0) 2 H1 .

60) and call its matrix A: A= 1 8(R2 −R1 )2 2 − β1 +β 2 2 − β1 +β 2 1 R22 −γ . 62) Our interest is to find the maximum possible value of so A > 0. From Sylvester’s criterion, we get the condition for A to be positive definite: 0< 1 −γ R22 − 2(R2 − R1 )2 (β1 + β2 )2 . 63) Solving for 1/ , 1 2 > 2R22 (R2 − R1 )2 (β1 + β2 ) + R22 γ. 64) Substituting γ, we can define an upper bound for : 1 γ3 2 = 2R22 (R2 − R1 )2 (β1 + β2 ) + R22 γ1 + 2γ2 + 2 ∗ R2 . 65) Note that this bound is a function that depends exclusively on the geometry and physical parameters of the plant.

60) and call its matrix A: A= 1 8(R2 −R1 )2 2 − β1 +β 2 2 − β1 +β 2 1 R22 −γ . 62) Our interest is to find the maximum possible value of so A > 0. From Sylvester’s criterion, we get the condition for A to be positive definite: 0< 1 −γ R22 − 2(R2 − R1 )2 (β1 + β2 )2 . 63) Solving for 1/ , 1 2 > 2R22 (R2 − R1 )2 (β1 + β2 ) + R22 γ. 64) Substituting γ, we can define an upper bound for : 1 γ3 2 = 2R22 (R2 − R1 )2 (β1 + β2 ) + R22 γ1 + 2γ2 + 2 ∗ R2 . 65) Note that this bound is a function that depends exclusively on the geometry and physical parameters of the plant.

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