By Roland Glowinski, Jacques-Louis Lions, Jiwen He

ISBN-10: 0511721595

ISBN-13: 9780511721595

ISBN-10: 0521885728

ISBN-13: 9780521885720

This e-book investigates how a consumer or observer can effect the habit of structures mathematically and computationally. an intensive mathematical research of controllability difficulties is mixed with an in depth research of tools used to unravel them numerically; those tools being demonstrated through the result of numerical experiments. within the first a part of the e-book, the authors speak about the math and numerics when it comes to the controllability of structures modeled via linear and non-linear diffusion equations; half is devoted to the controllability of vibrating platforms, normal ones being these modeled through linear wave equations; and eventually, half 3 covers move keep watch over for platforms ruled via the Navier-Stokes equations modeling incompressible viscous move. The publication is available to graduate scholars in utilized and computational arithmetic, engineering and physics; it's going to even be of use to extra complex practitioners.

**Read Online or Download Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications (No. 117)) PDF**

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**Additional info for Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications (No. 117))**

**Sample text**

64) ∂ψ ∗ − + A ψ = 0 in Q, ψ(T ) = k(yT − y(T )), ψ = 0 on . 64) (two points with respect to the time variable, the two points being t = 0 and t = T ). 5). 63). Given ε > 0, there exists a control w such that y(T ; w) − yT ≤ ε. 2) and it is not constructive. 67) w2 dx dt + ε2 . 63). w2 dx dt ≤ 1 2 β . ). 64) if we have a way to choose k. 69) (where the solution has been denoted by f ∗ , instead of f ). 62) reads as follows: (k −1 I + )f = yT . 70) by f we obtain ( f , f ) + k −1 f 2 = (yT , f ).

152b) where ψ n ≈ ψ(n t) (ψ(n t) : x → ψ(x, n t)). 153a) φ 0 = 0; then, assuming that φ n−1 is known, we solve the following Dirichlet problem for n = 1, . . , N , φ n − φ n−1 + Aφ n = ψ n χO in t Finally, we approximate t by , φ n = 0 on . 153b) defined by t g = φN . 153b) have a unique solution; we have, furthermore, the following. 37 Operator L2 ( ). is symmetric and positive semidefinite from L2 ( ) into Proof. Consider a pair {g, g} ˜ ∈ L2 ( )×L2 ( ). We have then (with obvious notation) t ( g, g) ˜ L2 ( ) = φ N ψ˜ N +1 dx.

49). 55) with ψ obtained from f via the solution of − We have then ∂ψ + A∗ ψ = 0 in Q, ∂t ψ(T ) = f , ψ = 0 on y(0) = 0, y = 0 on . 56) f = y(T ), where, ∂y + Ay = ψχO×(0,T ) in Q, ∂t . 53). 56) to compute the corresponding value of ψ. 54). 8. Before that, several remarks are in order. 56). Therefore, ψ is smooth (indeed, the smoother the coefficients of operator A, the smoother will be ψ). In other words, the control u is a smooth function of x and t. This observation excludes the possibility of finding an optimal control of the “bangbang” type.

### Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications (No. 117)) by Roland Glowinski, Jacques-Louis Lions, Jiwen He

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