Optimal Control of Dissipative Solids: Viscosity Limits and Non-Smooth Algorithms

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The proposed project concerns the optimal control of dissipative solids. Our point of departure is a thermodynamically consistent material model which takes into account damage effects as well as thermo-elastoplasticity. A modern solution theory for such systems with rate-independent components is based on so-called balanced viscosity solutions whose existence is proved by means of viscous regularization and a subsequent passage to the limit in the viscosity parameters.Within the proposed project, we intend to analyze the optimization of damage and thermo-plastic deformation processes under this solution concept. Besides the existence of optimal controls, we are mainly interested in the approximability of locally optimal controls by viscous regularization.The rate-dependent, viscous problems have a physical meaning in their own right, and they are still non-smooth in the sense that the associated control-to-state operator is, in general, not Gateaux differentiable. Moreover, the viscous problems serve as a basis for the development of an efficient optimization algorithm, a bundle method in function space. To apply it, elements of the subdifferential in the sense of Clarke are to be determined on the basis of directional derivatives for the viscous model problems.By using a path-following approach for vanishing viscosity, we expect to be able to compute optimal solutions even of the associated rate-independent problems.