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The dual system , and this is true since in .NET Development ECC200 in .NET The dual system , and this is true since




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6.2 The dual system , and this is true since use visual studio .net data matrix ecc200 maker tobuild barcode data matrix for .net QR Code Symbol Versions u(s), (D y )(s) ds = u(s), (D y )(s) ds (L (R;U ),L (R,U visual .net Data Matrix )) (L (R;Y ),L (R,Y )). p q p q = u, D y = Du, y = (Du)(s), y (s) ds. De nition 6.2.2 The system of . in Theorem 6.2.1 is called the anti-causal dual By reversing the direct Data Matrix 2d barcode for .NET ion of time in Theorem 6.2.

1 we get the following causal system. Theorem 6.2.

3 Let 1 p < , and let = A B be an L p -well-posed C D linear system on the re exive Banach spaces (Y, X, U ). De ne. Ad Bd Cd Dd C R RB RD R (6.2.3).

where A t = (At ) for all t 0. Then d is an L q -well-posed linear system on (U , X , Y ), where 1/ p + 1/q = 1 (and 1/ = 0). Proof This follows from Lemma 6.

1.2 and Theorem 6.2.

1. De nition 6.2.

4 The system .. in Theorem 6.2.3 is called the causal dual of Example 6.2.5 Let = A B be an L p -well-posed linear system on the C D re exive Banach spaces (Y, X, U ), with 1 p < .

(i) For each C, the dual Ad of the exponentially shifted system Example 2.3.5 is.

d . Ad Bd Cd Dd e A C Re e RB e RD Re (ii) For each > 0, the dual Ad of the time compressed system Example 2.3.6 is Ad Bd Cd Dd A . C R 1/ . RB RD R 1/ . Anti-causal, dual, and inverted systems (iii) For each (boundedly) invertible E B(X 1 ; X ), the dual Ad of the E similarity transformed system E in Example 2.3.7 is.

Ad Bd E E Cd Dd E E E A E E C R RB E RD R This follo visual .net datamatrix 2d barcode ws from Examples 2.3.

5 2.3.7 and Theorem 6.

2.3. Example 6.

2.6 We consider the systems in Examples 2.3.

10 2.3.13 in the L p setting with 1 p < and with re exive input spaces, state spaces and output spaces.

(i) The dual Ad of the cross-product of the systems 1 and 2 in Example d d 2.3.10 is the cross-product of the duals 1 and 2 .

(ii) The dual Ad of the sum junction of the systems 1 and 2 in Example d d 2.3.11 is the T-junction of the duals 1 and 2 .

d (iii) The dual A of the T-junction of the systems 1 and 2 in Example d d 2.3.12 is the sum junction of the duals 1 and 2 .

(iv) The dual Ad of the parallel connection of the systems 1 and 2 in d d Example 2.3.13 is the parallel connection of the duals 1 and 2 .

This follows from Examples 2.3.10 2.

3.13 and Theorem 6.2.

3. Example 6.2.

7 The dual of the delay line example 2.3.4 in L p with 1 < p < and re exive U is a similarity transformed version of the same delay line in L q , 1/ p + 1/q = 1, with the similarity transformation given in Lemma 3.

5.13(iv)(d). (The signal enters at the left end of the line, and leaves at the right end.

) This follows from Example 2.3.4, Theorem 6.

2.3, and Lemma 3.5.

13.. p Example 6.2.8 Let D TIC (U ; Y ) where 1 < p < , R and U and Y are re exive Banach spaces.

Then the exactly controllable shift realization of D in Example 2.6.5(i) and the exactly observable shift realization of Dd in Example 2.

6.5(ii) are duals of each other. Also the bilateral input shift realization of D in Example 2.

6.5(iii) and the bilateral output shift realization of Dd in Example 2.6.

5(iv) are duals of each other.. This is ob Data Matrix barcode for .NET vious (see Lemma 3.5.

13). The formal relationships between the nite time input/state/output maps Ats Bts and the corresponding maps for the anti-causal dual system are simCts Dts pler than the relationships with the causal dual system, due to the absence of re ection operators. Speci cally, we have the following result.

. 6.2 The dual system Lemma 6.2. .

net framework ECC200 9 Let 1 p < , and let = A B be an L p -well-posed linC D ear system on the re exive Banach spaces (Y, X, U ), and let be the corresponding anti-causal dual system. Then the (forward) maps introduced in De nition 2.2.

6 (applied to the original system ) and the (backward) maps introduced in De nition 6.1.7 (applied to ) are related in the following way for all < s < t < , (A )ts (B )ts (C )ts (D )ts (B )s (D )s.

(Ats ) (C ts ) (Bts ) (Dts ) (Cs ) (D ). , (6.2.4).

(C )t (D )t = (Bt ) (Dt ) . Proof This follows immediately from De nitions 2.2.

6, 6.1.7, and 6.

2.2. By combining this result with Lemma 6.

1.8 we get the corresponding relationships between the original system and the causal dual system d . Theorem 6.

2.10 Let 1 p < , 1/ p + 1/q = 1, s < t, xs X , xt X , u L p ((s, t); U ), and y L q ((s, t); Y ). Let x and y be the state trajectory and output function (restricted to (s, t)) of = A B with initial time s, initial C D state xs , and input function u, and let x and u be the backward state trajectory and output function (restricted to (s, t)) of the anti-causal dual system with initial time t, initial state xt , and input function y .

Then x(t), xt . (X,X ).
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