Optimal control of physical system governed by partial differential equations

Abstract

This thesis consists of two parts: An optimal vibration control of two linear beam models governed by hyperbolic partial differential equations and a bilinear optimal control of heat transfer process governed by a parabolic partial differential equation. In the first part of the thesis, an optimal boundary control of two different beam models, Mindlin-type beam and second strain gradient theory-based beam, are examined. The beam models to be controlled are described by linear higher order distributed parameter systems. When the Mindlin-type beam is considered, the partial differential equation that governs dynamic behavior of the model must be of the same order with respect to both spatial and time parameters to match up with Einstein’s causality. As for the second strain gradient theory-based beam, it captures the size effects of the structures in micro and nanoscale. In this theory, the potential energy is dependent of strains, gradient of strains and also second gradient of strains. The constitutive equation is a hyperbolic partial differential equation of sixth order. For both beam models, controllability and well-posedness of the system are analyzed. Pontryagin’s maximum principle is used to obtain an optimal control function and its trajectories. By using Pontryagin’s maximum principle, the control problem is turned to solving a new partial differential equation system involving adjoint and state variables providing initial, boundary and terminal conditions. Analytical solutions are formed with the help of an eigenfunction expansion method. Simulations are presented to confirm the theoretical results. In the second part of the thesis, an optimal control of a parabolic system governed by a bilinear control is discussed. The parabolic system describes heat transfer process. For the heat transfer process, coolant flow rate is a bilinear control variable and temperature is a state variable. As a result of implementation of a bilinear control to a uniform rod with non-uniform temperature, the temperature on the metal rod is homogeneously distributed when the temperature reaches equilibrium. The performance index to be minimized indicates that the goal is to keep the temperature close to the steady-state value without consuming control effort in large quantities. The proposed approach uses a reduction of order in the model, Pontryagin’s maximum principle and numerical techniques. Using a modal space expansion method the distributed parameter system is transformed into a lumped parameter system. The obtained system corresponds to a bilinear system in the temporal term. Pontryagin’s maximum principle is used to obtain the optimal control function that leads to a nonlinear two-point boundary value problem. Two iterative numerical techniques for determining optimal trajectories and optimal control of the system are discussed. Steepest descent and quasilinearization are procedures for solving this nonlinear two-point boundary value problem. The steepest descent method is used as a beginning procedure and quasilinearization to confirm the solution. Comparisons of some features such as what the initial guess is and importance of initial guess, number of iterations and convergence of these two iterative methods in the control problem are presented. Numerical simulation studies show the effectiveness and applicability of the proposed approach.