Regularization Methods For Trajectory Controllability Of Heat Equation And Numerical Experiments

In order to drive the state to our desired one in system represented by partial differentialequation,we usually need to compel a control on it, for example, the controlled system ofheat equation. Then the control function not only can act on internal of state domain butalso on boundary. The latter is called the boundary control of heat equation. In this paper,the main work is studying the trajectory control acts on internal of the state domain.The problem of trajectory control in heat equation raised by Zuazua in [1],which giventhe detailed introduction and analysis. Because it is an open issue,there is even no perfecttheory system. To the problem raised in the paper, for that we need to set out from theobserved ideal state to find an optimal control function, so it is an inverse problem.Andbecause the observed value usually comes by observation or calculation, which may existsome error, therefore we often consider the stability of solutions.However, the solutionssolved directly is not stable, hence the problem is required to be regularized.In this paper,the regularization methods of trajectory controllable to heat equation aremainly considered inL2space. Before regularization methods are given,I will illustrate theill-posedness of trajectory control in heat equation from two aspects.Then the operationalprocedure and analysis of convergence and error in regularization methods of T ikhonovand one based on optimal control theory are given.At last,according to the specificproblem, this paper presents numerical experiments corresponding to the two methods inone dimension,and states quality of the two methods by comparing results in numericalexperiments.Furthermore the regularized solutions in this paper is helpful to explore andfind new regularization methods.