Homological structure of optimal systems
Pure mathematics is often classified as continuous or discrete, that is into topology
and combinatorics. Classical topology is the study of spaces in the small, modern
topology or homology theory is the study of their large scale structure. The latter
and its applications to General Systems Theory and implications on computer
programming are the subject of our investigations.
A general homology theory includes boundary and adjoint operators defined over a
graded category. Singular homology theory describes the structure of high dimensional
Simplicial complexes, and is the basis of Kron's tearing of electrical networks. De
~ham Cohomology Theory describes the structure of exterior differential forms used to
~nalyse distributed fields in high dimensional spaces. Likewise optimal control
~roblems can be described by abstract homology theories. Ideas from tensor theory are
~sed to identify the homological structure of Leontief's economic model as a real
~xample of an optimal control system. The common property of each of the above
~ystems is that of optimisation or equivalently the mapping of an error to zero. The
~~iterion may be a metric in space, or energy in an electrical or mechanical network
~~ system, or an abstract cost function in state space or money in an economic system
~~d is always the product of a covariant and a contravariant variable.
~e axiomatic nature of General Homology Theory depends on the definition of an
~~missable category, be it group, ring or module structure. Similarly real systems
~~e analysed in terms of mutually recursive algebras, vector, matrix or polynomial.
~~rther the group morphisms or mode operators are defined recursively. An orthogonal
~~mputer language, Algo182, is proposed which is capable of manipulating the objects
~~scribed by homological systems theory, thus alleviating the tedium and insecurity
t~curred in iDtplementing computer programs to analyse engineering systems.