Distributed Parameter Systems Controllability And Unique Continuation Problems

This thesis is mainly devoted to two problems of distributed parameter systems. One is the controllability for heat equations with measurable controllers, the other is the unique continuation for Lame systems and elastic wave equations.This thesis is divided into five chapters.In Chapter 1, we will study the controllability for heat equations with measur-able controllers. When the controller is an open set, the theory of controllability for linear evolution equations is nearly complete. However cases of the controller being measurable sets are hard to deal with since neither the method of useing the Carleman inequality to prove observability or the spectrum method is valid. To study this problem, we have to find a new method. Using the theory of control transmutation, we obtain the null controllability for heat equations with measurable controllers in one dimension. In cases of high dimensions, we will by Harnack inequality derive an observable type inequality which holds for all positive solutions.In Chapter 2, we will consider the problem of data assimilation for the heat equa-tion (?)ty-Δy= 0, in (0, T)×Ω, which we would like to "predict" on a time interval (T0, T) but for which the initial value of the state variable is unknown. However, unlike the case of Puel's [60], "measures" of the solutions are known only on (0, To) x{x0} where 0< T0< T and x0 is a point ofΩ. The classical approach in data assimilation is to look for the initial value at time 0 and this is known to be an ill-posed problem for heat equations. In this paper, by the property of null controllability of heat equations we give a result of approximate reconstruction of the value at To. The approximation needs a sharp estimation of the cost of the null controllability of the heat equation. Since the assumption of measure conditions is different from Puel's paper, the difficulties are totally different.Starting from Chapter 3, we will study the unique continuation.In Chapter 3 and Chapter 4, we will consider the Lame systems. The three spheres inequalities and strong unique continuation for the Lame systems with Lipschitz coef-ficients in three dimensions are two open questions. Recently, they have been solved by C.-L. Lin etc. [39] and us separately. To our best knowledge, the classical methods which derive three spheres inequalities and strong unique continuation are invalid in this case. In order to prove three spheres inequalities, following Higashimori [23] we use the Carleman inequality given by Eller [11] to obtain the conditional stability which can be used to prove the three spheres inequality. To prove strong unique continuation, we firstly divide the Lame operator into two first order operators. Then we can obtain a Carleman estimate with a polynomial type Carleman weight by proving the the same type estimate for the two first order operators. Then we can follow Regbaoui's way [61] to prove the strong unique continuation.Chapter 5 is an extension of the method to prove the Carleman estimate with a polynomial type Carleman weight for Lame operator. We will study the Carleman estimate for an elastic wave operator in Chapter 5. Accordingly, we will also prove the unique continuation and approximately controllability for elastic wave equations.