Time Optimal Control Problems For Some Evolution Equations Of Parabolic Type

This thesis is concerned with time optimal control problems for some evolution e-quations of parabolic type. The main purpose of this thesis is to prove the bang-bang property and uniqueness of time optimal controls. We also establish the associated ob-servability inequalities from measurable sets of positive measure.The thesis consists of six chapters. In the first chapter, we simply take the heat equation for example, to recall the concepts of the null controllability and the observability inequality, as well as the time optimal control problem. We also review the previous results in the literature and give some notations which will be frequently used throughout this paper.The second chapter from [Z2] studies the necessary and sufficient conditions for the optimal time, as well as time optimal controls for some time-varying ordinary differential equations, where the control constraint set is of the rectangular type. We also provide an algorithm for the optimal time and the optimal control for this problem. The basis consists of an equivalence theorem between the time optimal control problem and an associated norm optimal control problem, and the variational characterization of the latter problem.The third chapter from [WZ] deals with the observability inequality from measurable sets in time for some abstract evolution equations. We utilize the quantitative estimate for analytic functions, as well as a telescoping series method, to obtain an observability inequality from measurable sets of positive measure. We also provide some applications to the time optimal control problem in a Hilbert space, and show two specific examples of the latter.The fourth chapter from [PWZ] shows the bang-bang property of the time optimal control problem for semilinear heat equations. By making use of the weighted frequency function method and a telescoping series method, we can prove an observability estimate from a measurable set of positive measure in time for the linear heat equation with bounded potentials. Also, we utilize the Kakutani fixed point theorem to present the null controllability for semilinear heat equations.The fifth chapter from [Zl] and [AEWZ] presents two observability inequalities from measurable sets for the heat equation in a bounded domain Ω. In the first one, the observation is from a subset of positive measure in Ω x (0,T), while in the second, the observation is from a subset of positive surface measure on (?)Ω x (0, T). They are mainly based on the the Lebeau-Robbiano spectral inequality and a telescoping series method, as well as the quantitative estimate of analytic functions on measurable sets with positive measure. We also prove the Lebeau-Robbiano spectral inequality holds when Ω is a bounded Lipschitz and locally star-shaped domain. As applications, we show the bang-bang property for the time (or norm) optimal control problem for heat equations.The last chapter from [EMZ] focuses on observability inequalities over measurable sets for parabolic equations associated to the m-th (m(?)N) power of the Laplace operator, as well as for the coupled parabolic equations associated to the second order self-adjoint elliptic systems with time-independent analytic coefficients. In particular, we present an observability inequality for two coupled parabolic equations but with only one component. The proofs are mainly based on the analyticity of solutions and the quantitative estimates of analytic functions on measurable sets of positive measure, as well as a telescoping series method. Some applications for the above-mentioned observability inequalities are provided in control theory.