Fundamental Principles of Control Landscapes with Applications to Quantum Mechanics, Chemistry and Evolution

The concept of a landscape or response surface naturally arises in applications widely ranging over the sciences, engineering and other disciplines. A landscape is the desired output as a function of a set of input variables, often of very high dimension. The relationship between the features of a landscape and the input variables is usually unknown a priori and often thought to be highly complex due to the anticipated intricate interactions involved. This chapter reviews recent developments in the analysis of landscape topology with the input variables considered as controls. Taking a control perspective allows for the specification of particular assumptions whose satisfaction permits a general analysis of the landscape topology. Satisfaction of these conditions leads to the conclusion that control landscapes should be devoid of suboptimal critical point traps, thereby permitting ready excursions without hindrance to the highest values of the landscape. These principles are set out in a general framework and then specifically illustrated for applications involving control in quantum mechanics, chemical and material science, and in natural and directed evolution. Perspectives are given on the significance of these findings and potential future directions for additional analysis of landscape principles.

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