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Abstract

The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations.

A unified infinite-dimensional controllability/observability theory]A unified controllability/observability theory for some stochastic and deterministic partial differential equations X. Zhang]Xu Zhang thanks: This work is supported by the NSFC under grants 10831007, 60821091 and 60974035, and the project MTM2008-03541 of the Spanish Ministry of Science and Innovation. \contact[xuzhang@amss.ac.cn]School of Mathematics, Sichuan University, Chengdu 610064, China; and Key Laboratory of Systems Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. \numberwithinequationsection

Mathematics Subject Classification (2000). Primary 93B05; Secondary 35Q93, 93B07.

Keywords. Controllability, observability, parabolic equations, hyperbolic equations, weighted identity.

1 Introduction

We begin with the following controlled system governed by a linear Ordinary Differential Equation (ODE for short):

In (1), , (), is the state variable, is the control variable, and are the state space and control space, respectively. System (1) is said to be exactly controllable at a time if for any initial state and any final state , there is a control such that the solution of (1) satisfies .

The above definition of controllability can be easily extended to abstract evolution equations. In the general setting, it may happen that the requirement has to be relaxed in one way or another. This leads to the approximate controllability, null controllability, and partial controllability, etc. Roughly speaking, the controllability problem for an evolution process is driving the state of the system to a prescribed final target state (exactly or in some approximate way) at a finite time. Also, the above can be unbounded for general controlled systems.

The controllability/observability theory for finite dimensional linear systems was introduced by R.E. Kalman ([19]). It is by now the basis of the whole control theory. Note that a finite dimensional system is usually an approximation of some infinite dimensional system. Therefore, stimulated by Kalman’s work, many mathematicians devoted to extend it to more general systems including infinite dimensional systems, and its nonlinear and stochastic counterparts. However, compared with Kalman’s classical theory, the extended theories are not very mature.

Let us review rapidly the main results of Kalman’s theory. First of all, it is shown that: System (1) is exactly controllable at a time if and only if . However, this criterion is not applicable for general infinite dimensional systems. Instead, in the general setting, one uses another method which reduces the controllability problem for a controlled system to an observability problem for its dual system. The dual system of (1) reads:

It is shown that: System (1) is exactly controllable at some time if and only if the following observability inequality (or estimate) holds

(0)

Here and henceforth, denotes a generic positive constant, which may be different from one place to another. We remark that similar results remain true in the infinite dimensional setting, where the theme of the controllability/observability theory is to establish suitable observability estimates through various approaches.

Systems governed by Partial Differential Equations (PDEs for short) are typically infinite dimensional. There exists many works on controllability/observability of PDEs. Contributions by D.L. Russell ([40]) and by J.L. Lions ([29]) are classical in this field. In particular, since it stimulated many in-depth researches on related problems in PDEs, J.L. Lions’s paper [29] triggered extensive works addressing the controllability/observability of infinite dimensional controlled system. After [29], important works in this field can be found in [1, 4, 8, 11, 13, 17, 21, 25, 26, 43, 46, 55, 56]. For other related works, we refer to [18, 28] and so on.

The controllability/observability of PDEs depends strongly on the nature of the underlying system, such as time reversibility or not, and propagation speed of solutions, etc. The wave equation and the heat equation are typical examples. Now it is clear that essential differences exist between the controllability/observability theories for these two equations. Naturally, one expects to know whether some relationship exist between the controllability/observability theories for these two equations of different nature. Especially, it would be quite interesting to establish, in some sense and to some extend, a unified controllability/observability theory for parabolic equations and hyperbolic equations. This problem was initially studied by D.L. Russell ([39]).

The main purpose of this paper is to present the author’s and his collaborators’ works with an effort towards a unified controllability/observability theory for stochastic/deterministic PDEs. The crucial analytic tool we employ is a class of elementary pointwise weighted identities for partial differential operators. Starting from these identities, we develop a unified approach, based on global Carleman estimate. This universal approach not only deduces the known controllability/observability results (that have been derived before via Carleman estimates) for the linear parabolic, hyperbolic, Schrödinger and plate equations, but also provides new/sharp results on controllability/observability, global unique continuation, stabilization and inverse problems for some stochastic/deterministic linear/nonlinear PDEs.

The rest of this paper is organized as follows. Section 2 analyzes the main differences between the existing controllability/observability theories for parabolic equations and hyperbolic equations. Sections 3 and 4 address, among others, the unified treatment of the controllability/observability problem for deterministic PDEs and stochastic PDEs, respectively.

2 Main differences between the known theories

In the sequel, unless otherwise indicated, stands for a bounded domain (in ) with a boundary , denotes an open non-empty subset of , and is a given positive number. Put , and .

We begin with a controlled heat equation:

(0)

and a controlled wave equation:

(0)

In (2), and are the state variable and control variable, the state space and control space are chosen to be and , respectively; while in (2), and are the state variable and control variable, and are respectively the state space and control space. System (2) is said to be null controllable (resp. approximately controllable) in if for any given (resp. for any given , ), one can find a control such that the weak solution of (2) satisfies (resp. ). In the case of null controllability, the corresponding control is called a null-control (with initial state ). Note that, due to the smoothing effect of solutions to the heat equation, the exact controllability for (2) is impossible, i.e., the above cannot be zero. On the other hand, since one can rewrite system (2) as an evolution equation in a form like (1), it is easy to define the exact controllability of this system. The dual systems of (2) and (2) read respectively

(0)

and

(0)

The controllability/observability theories for parabolic equations and hyperbolic equations turns out to be quite different. First of all, we recall the related result for the heat equation.

Theorem 2.1.

([25]) Let be a bounded domain of class . Then: i) System (2) is null controllable and approximately controllable in at time ; ii) Solutions of equation (2) satisfy

(0)

Since solutions to the heat equation have an infinite propagation speed, the “waiting” time can be chosen as small as one likes, and the control domain dose not need to satisfy any geometric condition but being open and non-empty. On the other hand, due to the time irreversibility and the strong dissipativity of (2), one cannot replace in inequality (2.1) by .

Denote by the eigenvalues of the homogenous Dirichlet Laplacian on , and the corresponding eigenvectors satisfying . The proof of Theorem 2.1 is based on the following observability estimate on sums of eigenfunctions for the Laplacian ([25]):

Theorem 2.2.

Under the assumption of Theorem 2.1, for any , it holds

(0)

Note that Theorem 2.2 has some other applications in control problems of PDEs ([32, 34, 44, 49, 55, 56]). Besides, to prove Theorem 2.1, one needs to utilize a time iteration method ([25]), which uses essentially the Fourier decomposition of solutions to (2) and especially, the strong dissipativity of this equation. Hence, this method cannot be applied to conservative systems (say, system (2)) or the system that the underlined equation is time-dependent.

As for the controllability/observability for the wave equation, we need to introduce the following notations. Fix any , put

(0)

where is the unit outward normal vector of at . For any set and , put .

The exact controllability of system (2) is equivalent to the following observability estimate for system (2):

(0)

Note that the left hand side of (2) can be replaced by (because (2) is conservative). The following classical result can be found in [29].

Theorem 2.3.

Assume and . Then, inequality (2) holds for any time .

The proof of Theorem 2.3 is based on a classical Rellich-type multiplier method. Indeed, it is a consequence of the following identity (e.g. [47]):

Proposition 2.4.

Let be a vector field of class . Then for any , it holds that

The observability time in Theorem 2.3 should be large enough. This is due to the finite propagation speed of solutions to the wave equation (except when the control is acting in the whole domain ). On the other hand, it is shown in [4] that exact controllability of (2) is impossible without geometric conditions on . Note also that, the multiplier method rarely provides the optimal control/observation domain and minimal controll/observation time except for some very special geometries. These restrictions are weakened by the microlocal analysis ([4]). In [4, 5, 6], the authors proved that, roughly speaking, inequality (2) holds if and only if every ray of Geometric Optics that propagates in and is reflected on its boundary enters at time less than .

The above discussion indicates that the results and methods for the controllability/observability of the heat equation differ from those of the wave equation. As we mentioned before, this leads to the problem of establishing a unified theory for the controllability/observability of parabolic equations and hyperbolic equations. The first result in this direction was given in [39], which showed that the exact controllability of the wave equation implies the null controllability of the heat equation with the same controller but in a short time. Further results were obtained in [32, 49], in which organic connections were established for the controllability theories between parabolic equations and hyperbolic equations. More precisely, it has been shown that: i) By taking the singular limit of some exactly controllable hyperbolic equations, one gives the null controllability of some parabolic equations ([32]); and ii) Controllability results of the heat equation can be derived from the exact controllability of some hyperbolic equations ([49]). Other interesting related works can be found in [34, 36, 43]. In the sequel, we shall focus mainly on a unified treatment of the controllability/observability for both deterministic PDEs and stochastic PDEs, from the methodology point of view.

3 The deterministic case

The key to solve controllability/observability problems for PDEs is the obtention of suitable observability inequalities for the underlying homogeneous systems. Nevertheless, as we see in Section 2, the techniques that have been developed to obtain such estimates depend heavily on the nature of the equations, especially when one expects to obtain sharp results for time-invariant equations. As for the time-variant case, in principle one needs to employ Carleman estimates, see [17] for the parabolic equation and [47] for the hyperbolic equation. The Carleman estimate is simply a weighted energy method. However, at least formally, the Carleman estimate used to derive the observability inequality for parabolic equations is quite different from that for hyperbolic ones. The main purpose of this section is to present a universal approach for the controllability/observability of some deterministic PDEs. Our approach is based on global Carleman estimates via a fundamental pointwise weighted identity for partial differential operators of second order (It was established in [13, 15]. See [27] for an earlier result). This approach is stimulated by [24, 20], both of which are addressed for ill-posed problems.

3.1 A stimulating example

The basic idea of Carleman estimates is available in proving the stability of ODEs ([27]). Indeed, consider an ODE in :

(0)

where . A well-known simple result reads: Solutions of (3.1) satisfy

(0)

A Carleman-type Proof of (3.1). For any , by (3.1), one obtains

(0)

Choosing large enough so that for a.e. , we find that

which proves (3.1).             

Remark 3.1.

By (3.1), we see the following pointwise identity:

(0)

Note that is the principal operator of the first equation in (3.1). The main idea of (3.1) is to establish a pointwise identity (and/or estimate) on the principal operator in terms of the sum of a “divergence” term and an “energy” term . As we see in the above proof, one chooses to be big enough to absorb the undesired terms. This is the key of all Carleman-type estimates. In the sequel, we use exactly the same method, i.e., the method of Carleman estimate via pointwise estimate, to derive observability inequalities for both parabolic equations and hyperbolic equations.

3.2 Pointwise weighted identity

We now show a fundamental pointwise weighted identity for general partial differential operator of second order. Fix real functions and satisfying (). Define a formal differential operator of second order: , . The following identity was established in [13, 15]:

Theorem 3.2.

Let and . Put and . Let be parameters. Then

where

As we shall see later, Theorem 3.2 can be applied to study the controllability/observability as well as the stabilization of parabolic equations and hyperbolic equations. Also, as pointed by [13], starting from Theorem 3.2, one can deduce the controllability/observability for the Schrödinger equation and plate equation appeared in [23] and [48], respectively. Note also that, Theorem 3.2 can be applied to study the controllability of the linear/nonlinear complex Ginzburg-Landau equation (see [13, 15, 38]).

3.3 Controllability/Observability of Linear PDEs

In this subsection, we show that, starting from Theorem 3.2, one can establish sharp observability/controllability results for both parabolic systems and hyperbolic systems.

We need to introduce the following assumptions.

Condition 3.3.

Matrix-valued function is uniformly positive definite.

Condition 3.4.

Matrix-valued function is uniformly positive definite.

Also, for any , we introduce the following

Condition 3.5.

Matrix-valued functions for some , and .

Let us consider first the following parabolic system:

where takes values in . By choosing and in Theorem 3.2, one obtains a weighted identity for the parabolic operator. Along with [27], this identity leads to the existing controllability/observability result for parabolic equations ([9, 17]). One can go a little further to show the following result ([10]):

Theorem 3.6.

Let Conditions 3.3 and 3.5 hold. Then, solutions of (3.3) satisfy

Note that (3.6) provides the observability inequality for the parabolic system (3.3) with an explicit estimate on the observability constant, depending on the observation time , the potential and . Earlier result in this respect can be found in [9] and the references cited therein. Inequality (3.6) will play a key role in the study of the null controllability problem for semilinear parabolic equations, as we shall see later.

Remark 3.7.

It is shown in [10] that when , and , the exponent in (for the case that in the inequality (3.6)) is sharp. In [10], it is also proved that the quadratic dependence on is sharp under the same assumptions. However, it is not clear whether the exponent in is optimal when .

Next, we consider the following hyperbolic system:

where takes values in .

Compared with the parabolic case, one needs more assumptions on the coefficient matrix as follows ([10, 16]):

Condition 3.8.

There is a positive function satisfying

i) For some constant , it holds

ii) There is no critical point of in , i.e., ;

iii) .

We put

(0)

By choosing and in Theorem 3.2 (and noting that only the symmetry condition is assumed for in this theorem), one obtains the fundamental identity derived in [16] to establish the controllability/observability of the general hyperbolic equations. One can go a little further to show the following result ([10]).

Theorem 3.9.

Let Conditions 3.4, 3.5 and 3.8 hold, and for some . Then one has the following conclusions:

1) For any , the corresponding weak solution of system (3.3) satisfies

2) If () and , then for any , the weak solution of system (3.3) satisfies

As we shall see in the next subsection, inequality (3.9) plays a crucial role in the study of the exact controllability problem for semilinear hyperbolic equations.

Remark 3.10.

As in the parabolic case, it is shown in [10] that the exponent in the estimate in (3.9) (for the special case ) is sharp for and . Also, the exponent in the term in (3.9) is sharp. However, it is unknown whether the estimate is optimal for the case that .

By the standard duality argument, Theorems 3.6 and 3.9 can be applied to deduce the controllability results for parabolic systems and hyperbolic systems, respectively. We omit the details.

3.4 Controllability of Semi-linear PDEs

The study of exact/null controllability problems for semi-linear PDEs began in the 1960s. Early works in this respect were mainly devoted to the local controllability problem. By the local controllability of a system, we mean that the controllability property holds under some smallness assumptions on the initial data and/or the final target, or the Lipschitz constant of the nonlinearity.

In this subsection we shall present some global controllability results for both semilinear parabolic equations and hyperbolic equations. These results can be deduced from Theorems 3.6 and 3.9, respectively.

Consider first the following controlled semi-linear parabolic equation:

For system (3.4), the state variable and control variable, state space and control space, controllability, are chosen/defined in a similar way as for system (2). Concerning the nonlinearity , we introduce the following assumption ([9]).

Condition 3.11.

Function is locally Lipschitz-continuous. It satisfies and