Functional Lagrange expansion in state space and thes-domain

The functional Lagrange expansion was introduced by the authors [1], [2] as a general analytical technique for handling a wide variety of control problems in nonlinear systems. It has been shown [1], [2] how a large class of such problems may be cast into a form requiring the solution of a nonlinear functional equation of the type
y(t)=x(t)+\epsilonF[y](t)
. The functional Lagrange expansion provides a functional series expansion of the solution
y(t)
in powers of the expansion parameter ε. This paper presents generalizations of the functional Lagrange expansion to muitivariable nonlinear systems, i.e., state space, to functional equations in the
s
-domain, and to equations where the functional
F[y]
is expressible as a sum of other functionals. The results are applied to specific control problems.