A.1 Proof of Expression (8.29)

From (8.28) we get

By taking into account (8.18), we obtain

A.2 Proof of Expression (8.42)

By taking ? ? 0 in (8.41), we obtain

In accordance with (8.43), we get

A.3 Proof of Expression (8.65)

By taking into account (8.62), (8.63), and (8.64), we obtain

From (8.57) we get

The above expression may be rewritten in the following form:

Hence,

Consequently,

Therefore,

A.4 Proof of Expression (11.37)

1. From (11.34) and (11.38), we get

   (A.1)

2. Denote

   (A.2)

   From (11.27), (11.28) and (11.29), we get

   (A.3)

3. By substituting (11.22) and (11.36) into (11.32), we obtain

   
   

The above expression can be rewritten in the form:

By (A.1) and (A.2) we have that the above expression is identical to (11.37).

A.5 Proof of Expressions (11.40) (11.41)

From (11.37) and (11.39), we obtain

By taking into account (11.42) and (A.2), the above expression can be rewritten in the form:

Hence,

By taking into account (11.8) and (11.42), we get

A.6 Proof of Expression (11.47)

1. From (11.43) we obtain

   (A.4)

2. Let us consider a stationary behavior of control variable

   (A.5)

   given that the conditions (11.43), (11.44), and (11.45) are satisfied. Hence, we have

   (A.6)

   (A.7)

   From closed-loop system equations (11.32) (11.33), by taking into account (11.48)-(11.49) and (A.6) (A.7), we obtain

   (A.8)

   (A.9)

   Since (A.8) and (A.9) are satisfied for all k, the equations (A.8) and (A.9) can be decomposed into

   (A.10)

   (A.11)

   and

   (A.12)

   (A.13)

   From (A. 10) we get

   (A.14)

   From (11.43) and (A.14) it follows that (A.11) holds...