Abstract:
The present research work deals with the existence of solutions and approximate
controllability of deterministic semilinear integer order systems with control delays
and fractional order systems without delay. To derive the existence and controllability
results, various techniques have been applied along with the semigroup, cosine
and sine families, fractional calculus, fractional cosine family, fractional resolvent,
xed point theory. Some examples are provided for the illustration of the obtained
results.
Some introductory matter along with literature survey on controllability of nonlinear
and linear control systems of fractional and integer orders are given in Chapter
1. Basic concepts and de nitions of control theory, semigroup theory, cosine family,
fractional calculus, fractional cosine family and nonlinear functional analysis which
are utilized in forthcoming chapters, are given in Chapter 2.
In Chapter 3, the existence of mild solutions of rst-order retarded semilinear
system with control delay is proved under the locally Lipschitz continuity of nonlinear
function and a xed point theorem. Then the approximate controllability of
semilinear system is proved provided that the associated linear system without delay
is approximately controllable. Controllability results are obtained by using the
method of steps and semigroup theory. The results of this chapter are illustrated
with controlled heat equation.
