Solving a Parabolic Identification Problem by Optimal Control Methods

An unknown coefficient of the interaction term of a parabolic system with a Neumann boundary condition in a multi-dimensional bounded domain is identified. The solution of the system represents the concentrations of prey and predator populations. Given partial (perhaps noisy) observations of a true solution in a subdomain, we seek to ``identify" the coefficient of the interaction term using an optimal control technique, involving Tikhonov's regularization. The existence and uniqueness of the optimal control approximating the desired coefficient are obtained, an optimality system is derived, the identification problem is discussed and an example illustrating how to find a solution numerically is presented.