ABSTRACT
In this chapter, structural perturbations are the cause of ill-posed nature (in the sense of Hadamard), and the infrastructures are ill-posed because the optimal control problems are ill-posed. To formulate the regularized problems with the real-world measurements on the structural perturbations, we begin this chapter with an introduction to the regularization techniques for ill-posed optimal control problems. Then, we construct a family of partial differential equations in linear elasticity to model the act of adding or removing the structural perturbations imposed at the nodes of 25-bar truss systems. Through the modeling, we define the structural perturbations as continuous functions in Hilbert space and as the inverse solution to finding structural perturbations. This approach requires well-chosen Tikhonov and sparse regularization parameters on the structural perturbations in order to formulate the regularized ill-posed optimal control problems with state and structural perturbation constraints. The numerical applications show that the structural perturbations can change in size, shape, and topology with any, fewer, or many zeros. Finally, they are interpreted as the control instruments which govern the optimal design of 25-bar truss systems. Any deviation can be penalized and treated to have the measured state as close as possible to the optimal design.
