Optimal Control Of Differential Equations

Optimal control theory of distributed parameter systems mainly includes: Pontryagin's maximum principle; controllability; Hamilton-Jacobi equation (i.e., dynamic programming equation); time optimal control, etc.In this dissertation, we establish Pontryagin's maximum principle of optimal control problems governed by some nonlinear differential equations (parabolic differential equations, elliptic differential equations and 3-dimensional Navier-Stokes equations), which in particular could have local solution only or could admit more than one solution (we shall call such systems as non-well-posed systems and the corresponding optimal control problems as non-well-posed optimal control problems), and some well-posed nonlinear evolution systems; we study time optimal control of phase-field system and we obtain local internal controllability of Boussinesq system.This dissertation consists of four chapters. Chapter 2 is preliminary in which we give all definitions and main results used in Chapter 3 and Chapter 4. Chapter 3 presents Pontryagin's maximum principle of optimal control problems governed by nonlinear differential equations. Chapter 4 is concerned with Carleman inequality and its applications to optimal control problems.