Optimal boundary control for hyperdiffusion equation

summary:In this paper, we consider the solution of optimal control problem for hyperdiffusion equation involving boundary function of continuous time variable in its cost function. A specific direct approach based on infinite series of Fourier expansion in space and temporal integration by parts for analytical solution is proposed to solve optimal boundary control for hyperdiffusion equation. The time domain is divided into number of finite subdomains and optimal function is estimated at each subdomain to obtain desired state with minimum energy. Proposed method has high flexibility so that decision makers are able to trace optimal control in a prescribed subinterval. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by some numerical examples

Optimal sensor placement

In this thesis we explore the problem of finding optimal sensor/actuator locations to achieve the minimum square error/least effort. The solution for the optimal sensor/actuator is often combinational, this means in order to solve for the solution we have to look at many parameters in the system and those parameters change frequently. In this thesis, we propose two methods to achieve this goal: the first one is based on gradient flow differential in which it provides the global optimal solution for the placement, and the second one is based on the evaluation of the Hessian matrix at the critical points. The optimal sensor/actuator location found using the gradient flow or Hessian matrix is usually not sparse. However in practical settings, the optimal sensor/actuator locations are often determined by discrete numbers such as ones and zeroes. We then propose some methods of relating the optimal sensor/actuator locations found using the gradient flow or Hessian matrix to the optimal sensor locations in the practical settings. Next we test the performance of the method we proposed by comparing the square error/effort of the system using the optimal sensor/actuator locations we found and the optimal sensor locations in the practical settings in multiple dimensions and with selected number of sensors/actuators

Optimal placement of controls for a one-dimensional active noise control problem

summary:In this paper, we investigate the optimal location of secondary sources (controls) to enhance the reduction of the noise field in a one-dimensional acoustic cavity. We first formulate the active control strategy as a linear quadratic tracking (LQT) problem in a Hilbert space, and then formulate the optimization problem as minimizing an appropriate performance criterion based on the LQT cost function with respect to the location of the controls. A numerical scheme based on the Legendre–tau method is used to approximate the control and the optimization problems. Numerical examples are presented to illustrate the effect of location of controls on the reduction of the noise field