Identification of parameters in delay differential equations
Mathematical modelling is the process of creating mathematical models describing the behavior of physical systems. Mathematical models of many physical systems (e.g., engineering, biological, economical, and environmental) are governed by ordinary differential equations [1]. These differential equations are usually nonlinear, and parameters might appear both linearly and nonlinearly in these equations. Delay differential equations are differential equations, where some delays exist in the forcing function; these delays are typical in engineering due to the inherent time lag between the sensing and control signals [2].
Recently we have proposed a homotopy optimization methodology [3] based on gradient algorithms to find the global minimum in certain problems of parameter identification for ordinary differential equations. We exploit symbolic computations by using MAPLE for efficient generation of sensitivity equations, which are required for optimization. In homotopy methods [4], the objective function to be minimized is modified by adding another function whose optimum is known, and a morphing parameter is used to transform the modified function into the original objective function. A series of optimizations is performed while slowly varying the morphing parameter until the modified function is transformed back into the original objective function and during this process we obtain the global minimum. This project involves developing morphing functions for delay differential equations (this work is closely related to observer design for delay differential equations [5]). The morphing function will be used in the homotopy optimization procedure for identifying the parameters for delay differential equations.
The student will be involved in programming and testing some of the algorithms that are being developed in our group.
