Optimization of the Design and Control of an Optical System with Structural Inhomogeneities

This study investigates the optimal parameter selection for mechanical devices that excite and shape wave motion, which are crucial for generating, transforming, and transmitting information and wave energy. The focus is on modulating a mirror affixed to active supports. The primary objective is to determine the control forces—their application points, amplitudes, and phases—that best approximate a desired shape and phase of mirror oscillations, while considering structural inhomogeneities (defects) with uncertain geometric and mechanical properties. These uncertainties include the physical characteristics, locations, sizes, and shapes of the defects. The research is divided into deterministic and stochastic analyses. In the deterministic analysis, the characteristics of the defects are assumed to be known. The problem simplifies to minimizing a functional based on the root-mean-square deviation of the plate’s deflections from a specified wave profile. The optimization parameters are the application points, amplitudes, and phases of the control forces from the active supports. To expedite solving the boundary value problem for plate equilibrium, defects are modeled using high-order point singularities. This approach ensures accuracy of a specified order relative to a small parameter—the area of the inhomogeneous region. The deterministic problem employs methods like the generalized Green’s function and harmonic analysis, with controlled harmonic distribution along the plate’s circular coordinate. In the stochastic analysis, the number of defects and their characteristics are uncertain or partially known. Therefore, a stochastic optimization method based on the Monte Carlo approach is utilized. Traditionally, risk in reliability theory is quantified by the probability of failure. However, this measure has undesirable mathematical properties, such as incoherence and discontinuity in sampling distributions. To overcome these issues, an alternative risk measure called the buffered probability of failure (BPF) is used. The buffered probability of exceedance (BPOE) generalizes BPF for cases where the system failure threshold can be any value, not just zero. These risk measures are grounded in the properties of the Conditional Value-at-Risk (CVaR) measure. Under general conditions, BPOE exhibits exceptional mathematical properties, such as quasi-convexity with respect to a random variable.