Semigroups
Before Defining what a semigroup is, one must recognize their global signifiance. We cannot really fully understand their Significance until we develop a
clear Definition and theory. However, in general, semigroups can be used to solve
a large class of problems commonly known as evolution equations. These types
of equations appear in many disciplines including physics, chemistry, biology,
engineering and economics. They are usually described by an initial value problem (IVP) for a di§erential equation which can be ordinary or partial. When we
view the evolution of a system in the context of semigroups we break it down
into transitional steps (i.e. the system evolves from state A to state B, and
then from state B to state C). When we recognize that we have a semigroup,
instead of studying the IVP directly, we can study it via the semigroup and
its applicable theory. The theory of linear semigroups is very well developed
[12]. For example, linear semigroup theory actually provides necessary and sufÖcient conditions to determine the well-posedness of a problem [5]. There is
also the theory for nonlinear semigroups which this section will not address. We
shall focus on a special class of linear semigroups called C
0
-semigroups which
are semigroups of strongly continuous bounded linear operators. The theory
of these semigroups will be presented along with some examples which tend to arise in many areas of applications.
Before Defining what a semigroup is, one must recognize their global signifiance. We cannot really fully understand their Significance until we develop a
clear Definition and theory. However, in general, semigroups can be used to solve
a large class of problems commonly known as evolution equations. These types
of equations appear in many disciplines including physics, chemistry, biology,
engineering and economics. They are usually described by an initial value problem (IVP) for a di§erential equation which can be ordinary or partial. When we
view the evolution of a system in the context of semigroups we break it down
into transitional steps (i.e. the system evolves from state A to state B, and
then from state B to state C). When we recognize that we have a semigroup,
instead of studying the IVP directly, we can study it via the semigroup and
its applicable theory. The theory of linear semigroups is very well developed
[12]. For example, linear semigroup theory actually provides necessary and sufÖcient conditions to determine the well-posedness of a problem [5]. There is
also the theory for nonlinear semigroups which this section will not address. We
shall focus on a special class of linear semigroups called C
0
-semigroups which
are semigroups of strongly continuous bounded linear operators. The theory
of these semigroups will be presented along with some examples which tend to arise in many areas of applications.
- معلم: Said Kouachi