New Stochastic Processes, Partial Differential Equations, and Control Problems Arising in Models of Mechanical Structures Subjected to Vibrations

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This project concerns the study of randomly excited nonlinear dynamical systems with memory. Such systems represent viable models for predicting the response of mechanical structures when stressed beyond the elastic limit. A new mathematical approach is considered: stochastic variational inequalities (SVI). The stochastic process solutions of the underlying SVI are essentially continuous diffusion processes with degeneracies and state constraints, for which governing equations can be formulated to describe the evolution in time of their probability distributions. Dissipativity of the system can be used to show existence of the corresponding invariant measure. The uniqueness of this invariant measure and the ergodic property are conjectured to be true for classes of hysteretic systems appearing in applications. Based on the availability of the probability distribution as the solution of the SVI, many practical issues concerning reliability of structures modeled by hysteretic systems can be answered in a systematic fashion. Control problems for SVI are also considered, such as finding the lowest energy input excitation (the so-called critical excitation) that drives the system between prescribed initial and final states within a given time span. Obtaining necessary conditions is a non trivial problem in view of lack of differentiability of the state equations, i.e., optimality conditions are needed for non-smooth dynamical systems. The use of approximate penalty techniques (to remove the constraints) leads to the consideration of degenerate diffusion processes on infinite domains, with challenging questions concerning ergodicity of the corresponding process.The nonlinear behavior of mechanical structures subjected to vibrations is a major concern in designing buildings or plants that must resist earthquakes, ocean waves, the wind, and other random excitations. Many attempts have been made in the past decades, with some successes, in defining universally acceptable safety standards corresponding to potentially occurring environmental loadings. However, the analysis of the underlying mathematics of models involved in these problems is far from complete. The accumulated fatigue that explains the ruin of structures is a nonlinear phenomenon with memory. By providing a thorough understanding of stochastic processes related to the responses of nonlinear systems described by stochastic variational inequalities, this research will establish a solid theoretical foundation for the study of system reliability.