Brauer third main theorem

integro-power series

A series containing the powers of the unknown function under the integral sign. Let $ K ( s , t _ {1} \dots t _ {k} ) $ be a continuous function in all variables in a cube $ [ a , b ] ^ {k+} 1 $ and let $ U ( s) $ be an arbitrary continuous function on $ [ a , b ] $. The expression

$$ U ^ {\alpha _ {0} } ( s) \int\limits _ { a } ^ { b } \dots \int\limits _ { a } ^ { b } K ( s , t _ {1} \dots t _ {k} ) U ^ {\alpha _ {1} } ( t _ {1} ) \dots U ^ {\alpha _ {k} } ( t _ {k} ) d t _ {1} \dots d t _ {k} , $$

where $ \alpha _ {0} \dots \alpha _ {k} $ are non-negative integers and $ \alpha _ {0} + \dots + \alpha _ {k} = m $, is called a Volterra term of degree $ m $ in $ U $. Two Volterra terms of degree $ m $ belong to the same type if they differ only in their kernels $ K $. The finite sum of Volterra terms (of all types) of degree $ m $ is called a Volterra form of degree $ m $ in the function $ U $. It is denoted by

$$ W _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) . $$

Let

$$ | W | _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) $$

denote the Volterra form in which the kernel $ K $ is replaced by $ | K | $, and let

$$ \widetilde{U} = \ \max _ {[ a , b ] } | U ( s) | ,\ \ \widetilde{W} _ {m} = \ \max _ {[ a , b ] } | W | _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) ; $$

then

$$ \left | W _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) \right | \leq \ \widetilde{W} _ {m} \widetilde{U} {} ^ {m} . $$

The expression

$$ W _ {0} \left ( \begin{array}{c} s \\ U \end{array} \right ) + W _ {1} \left ( \begin{array}{c} s \\ U \end{array} \right ) + W _ {2} \left ( \begin{array}{c} s \\ U \end{array} \right ) + \dots $$

is called a Volterra series. If the series of numbers $ \widetilde{W} _ {0} + \widetilde{W} _ {1} \widetilde{U} + \widetilde{W} _ {2} \widetilde{U} {} ^ {2} + \dots $ converges, then the Volterra series is called regularly convergent. In this case the Volterra series converges absolutely and uniformly, and its sum is continuous on $ [ a , b ] $.

Analogously one introduces Volterra series in several functional arguments, and Volterra series in which $ [ a , b ] $ is replaced by some closed bounded set in a finite-dimensional Euclidean space. Volterra series are a particular case of the more general concept of an abstract power series.

References

[1]	A.M. Lyapunov, "On equilibrium figures deviating slightly from ellipsoids of rotation of homogeneous fluid masses" , Collected Works , 4 , Moscow (1959) (In Russian)
[2]	E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen III" Math. Ann. , 65 (1908) pp. 370–399
[3]	M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian)

Comments

A non-linear input-output dynamical system with input $ u $ and output $ y $ gives rise to a Volterra series of the form

$$ y( t) = \int\limits _ {- \infty } ^ { {+ } \infty } h _ {1} ( \tau _ {1} ) u ( t - \tau _ {1} ) d \tau _ {1} + $$

$$ + \int\limits _ {- \infty } ^ { {+ } \infty } \int\limits _ {- \infty } ^ { {+ } \infty } h _ {2} ( \tau _ {1} , \tau _ {2} ) u ( t - \tau _ {1} ) u( t - \tau _ {2} ) d \tau _ {1} d \tau _ {2} + \dots + $$

$$ + \int\limits _ {- \infty } ^ { {+ } \infty } \dots \int\limits _ {- \infty } ^ { {+ } \infty } h _ {n} ( \tau _ {1} \dots \tau _ {n} ) u( t- \tau _ {1} ) \dots u ( t - \tau _ {n} ) $$

$$ d \tau _ {1} \dots d \tau _ {n} + \dots , $$

in which $ h _ {n} ( \tau _ {1} \dots \tau _ {n} ) = 0 $ if $ \tau _ {j} < 0 $ for some $ j $. Such series were first introduced by V. Volterra, [a1], and first applied to questions of system theory by N. Wiener, leading to Wiener integrals, [a2]. Cf. [a3] for an extensive discussion of Volterra series in system theory.

References

[a1]	V. Volterra, "Theory of functionals and of integral and integro-differential equations" , Dover, reprint (1959) (Translated from French)
[a2]	N. Wiener, "Nonlinear problems in random theory" , M.I.T. (1958)
[a3]	M. Schetzen, "The Volterra and Wiener theories of nonlinear systems" , Wiley (1980)