Sustainable Optimal Controls for Nonlinear Partial Differential Equations with Applications

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Optimization and Control under constraints typically leads to full exploitation of the resources. Therefore, as a rule, the resulting optimal controls exert significant inputs to the system under consideration. In particular, in the context of continuum mechanics and, say, time-minimum final profile control, large control forces may, in the long run, introduce damage to the system until complete failure occurs. We are, thus, faced with a dilemma: On the one hand we want to achieve our objective, say, in minimum time, while on other hand we would like to save the infrastructure. In other words, we would like to apply controls that maintain or enhance sustainability. Obviously, the interaction of controls and the evolution of damage in the corresponding material or structure heavily depends on the particular type of process and the material characteristics. The project, therefore focuses on a detailed analysis of different aspects (such as relaxation, approximation, regularization) of optimal controls problems for parabolic and hyperbolic initial-boundary value problems for elastic bodies arising, e.g. in contact mechanics, coupled systems, composite materials, where 'life-cycle-optimization' appears as a challenge. Instead of the standard statements of optimal control problems for PDEs, the effect of sustainability in the setting of the optimal control problems for elastic materials typically leads to the consideration of degeneration in hyperbolic (or parabolic) equations, where the evolution of the damage field is described by a parabolic inclusion or equation with a damage source function depending on the mechanical compression or tension. Material damage typically is reflected by degenerating coefficients in the leading coefficients of the underlying differential operator. In case of strong degeneration, the corresponding mixed initial-boundary value problem may lack uniqueness (or even the existence) of weak solutions due to e.g. the Lavrentieff-gap phenomenon. However, damage may also take place in coupling conditions, such that the degenerating coefficients are not factors of the highest order differential expressions, but rather appear in connection with lower coupling terms at the coupling interface. This is true e.g. for plate and beam structures where the coupling between two structural elements, realized via welding or glueing, may suffer deterioration to due excessive stresses at the coupling interface.The project is concentrated on the existence of sustainable optimal controls, deriving of the corresponding optimality conditions, development of the scheme for their approximation, study of the possible ways for the relaxation of the original statements of optimal control problems. In cases, where the strict fulfilment of the constraints appears to be infeasible, we proposed approximate solutions to the sustainable optimal control problems under rather general assumptions on the evolution of damage field.