Systems and Control provide a paradigm that introduces many open problems of mathematical nature. The Determinantal Assignment Problem (DAP) belongs to the family of synthesis methods and has emerged as the abstract problem formulation to which the study of pole, zero assignment of linear systems may be reduced. This approach unifies the study of frequency assignment problems (pole, zero) of multivariable systems under constant, dynamic centralised, or decentralised control structure, has been developed. DAP is equivalent to finding solutions to an inherently non-linear problem and its determinantal character demonstrates the significance of exterior algebra and classical algebraic geometry for control problems. The overall goal of the current proposal is to develop those aspects of the DAP framework that can transform the methodology from a synthesis approach and solution of well defined problems to a design approach that can handle model uncertainty, capable to develop approximate solutions and further empower it with potential for studying stabilization problems. The research aims to provide solutions for non-generic frequency assignment problems and handle problems of model uncertainty. This is achieved by developing robust approximate solutions to the purely algebraic DAP framework and thus transforming existence results and general computational schemes to tools for control design. The research involves the computation of distances between Grassmann and families of Linear varieties, introducing a new robust methodology for Global Linearisation using homotopy theory and finally developing an integrated framework for approximate solutions of DAP and its extension to the case of stabilization problems. The research involves the study of challenging mathematical problems related to problems such as spectral analysis of tensors, homotopy methods, constrained optimization, theory of algebraic invariants and issues linked to the properties of the stability domain.