By Professor Wilson Rugh II
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Extra info for Nonlinear System Theory: The Volterra / Wiener Approach
11 - 13. 51 F [φ1 ], F [φ2 ], . . converges to zero in R. 3 It should be noted that it is a purely technical exercise to show that a continuous linear functional on L 2 (0,∞) also is a continuous linear functional on K. Thus a functional of the form ∞ F [φ] = ∫ f (σ)φ(σ) dσ 0 where f ε L 2 (0,∞), is a continuous linear functional on K. But F [φ] = φ(0) also is a continuous linear functional on K, although there is no corresponding way to write an integral representation. The established, though very confusing, terminology is to call any continuous linear functional on K a generalized function or distribution.
Dtn = 0 ∫ | hn (t 1 , . . , tn ) + gn (t 1 , . . , tn ) | dt 1 . . dtn 0 ≤ hn + gn so that the bound function for H + G satisfies bH +G (x) ≤ bH (x) + bG (x), x ≥ 0 and the radius of convergence satisfies rH +G ≥ min [rH ,rG ]. Also, the kernels for the Volterra system representation for F = HG in (56) satisfy ∞ ∫ 0 ∞ n 0 j =0 | fn (t 1 , . . , tn ) | dt 1 . . dtn = ∫ Σ h j (t 1 , . . , t j )gn −j (t j +1 , . . , tn ) | dt 1 . . dtn ≤ n Σ h j gn −j j =0 Thus, bHG (x) ≤ bH (x)bG (x), x ≥ 0 and rHG ≥ min [rH ,rG ].
M )v (t −σ1 ) . . v (t −σm ) dσ1 . . dσm −∞ where, for j = 1, . . ,m, 22 (42) ∞ v (t−σ j ) = ∫ hn (σm +(j −1)n +1 , . . , σm +jn )u (t −σ j −σm +(j −1)n +1 ) −∞ . . u (t −σ j −σm +jn ) dσm +(j −1)n +1 . . 11. Cascade connection of two systems. Of course, I have chosen the labeling of variables in (43) to make the end result look nice. Substituting (43) into (42) gives y (t) = ∞ ∞ −∞ −∞ ∫ gm (σ1 , . . , σm )[ ∫ hn (σm +1 , . . , σm +n )u (t −σ1 −σm +1 ) . . u (t −σ1 −σm +n ) dσm +1 .
Nonlinear System Theory: The Volterra / Wiener Approach by Professor Wilson Rugh II