An Introduction to the Formation Theory of Active Materials

Abstract

We will use the term formation theory to describe a certain class of optimal design and control problems for elastic systems. The subject matter concerns the deliberate alteration of the equilibrium configuration of an elastic structure by means of attached or embedded actuators responding to an external control signal. To the extent that the controlled configuration is maintained constant over an extended time interval, or varies slowly enough so that inertial effects can be ignored, we have what we designate as static formation theory. It will be seen as we proceed that, from the mathematical point of view, the subject has much in common with parameter identification problems in a similar context, the optimal design part of the work primarily involving the specification of parameter functions in a particular class of partial differential equations.