Boundary Stabilization and Observation of a Multi-dimensional Unstable Heat Equation ^††thanks: This work is supported by the National Natural Science Foundation of China (12071463,61873153,12130008). Corresponding author: Hongyinping Feng.

When there is at least one point spectrum in the right-half complex plane, the system is referred to as an unstable system. Owing to the pole assignment theorem, stabilization of the finite-dimensional unstable system is almost trivial. However, the problem may become pretty difficult for the infinite-dimensional unstable system. When there are only finite point spectrums in the right-half complex plane, the system can be stabilized by the finite spectral truncation technique [17], [18]. Since heat equation with the boundary or interior source term is usually an unstable system which has finite unstable point spectrums, the finite spectral truncation technique can be used to stabilize the unstable heat system. See, for instance, [4], [13] and [14], to name just a few.

The partial differential equation (PDE) backstepping method is another way to stabilize the infinite-dimensional unstable system. It has been used to stabilize the unstable heat equations in [15], [19] and [21]. The PDE backstepping method can also cope with other infinite-dimensional systems such as the first order hyperbolic equation system [11], the unstable wave equation system [10] and even for ODE-PDE cascade system [22]. Although the PDE backstepping is powerful, it is still hard to apply to the general multi-dimensional infinite-dimensional system. Very recently, [8] has considered the stabilization and observation of the unstable heat equation in a general multi-dimensional region. By using the dynamic compensation method [5], [6] and the finite dimensional spectral truncation method [4], both the full state feedback and the state observer are proposed for the unstable multi-dimensional heat equation under the assumption that the unstable point spectrums are algebraic simple. In this paper, we shall extend the results in [8] to the more general case.

Let $\Omega\subset{\mathbb{R}}^{n}(n\geq 2)$ be a bounded domain with $C^{2}$-boundary $\Gamma$. Suppose that $\Gamma$ consists of two parts: $\Gamma_{0}$ and $\Gamma_{1}$, $\Gamma_{0}\cup\Gamma_{1}=\Gamma$, with $\Gamma_{0}$ is a non-empty connected open set in $\Gamma$. Let $\nu$ be the unit outward normal vector of $\Gamma_{1}$ and let $\Delta$ be the usual Laplacian. We consider the following system

where $\mu>0$ is a positive constant, $u$ is the control and $y$ is the output. Owing to the source term $\mu w(x,t)$, system (1) is unstable provided $\mu$ is large enough. The main goal of this paper is to design an output feedback to stabilize system (1) exponentially. By the separation principle of the linear systems, the output feedback will be available once we address the following two problems: (i) stabilize system (1) by a full state feedback; (ii) design a state observer to estimate the state online. We will consider these two problems separately.

We consider system (1) in state space $L^{2}(\Omega)$. Define

Then $A$ generates an exponentially stable analytic semigroup on $L^{2}(\Omega)$. It is well known (e.g. [12, p.668]) that $D((-A)^{1/2})=H_{\Gamma_{0}}^{1}(\Omega)$ and $(-A)^{1/2}$ is a canonical isomorphism from $H^{1}_{\Gamma_{0}}(\Omega)$ onto $L^{2}(\Omega)$. Let $[D((-A)^{1/2})]^{\prime}=H^{-1}_{\Gamma_{0}}(\Omega)$ be the dual space of $H^{1}_{\Gamma_{0}}(\Omega)$ with the pivot space $L^{2}(\Omega)$. We obtain the following Gelfand triple compact inclusions:

An extension $\tilde{A}\in{\mathcal{L}}(H^{1}_{\Gamma_{0}}(\Omega),H^{-1}_{\Gamma_{0}}(\Omega))$ of $A$ is defined by

By a simple computation, the eigenpairs $\{(\phi_{j}(\cdot),\lambda_{j})\}_{j=1}^{\infty}$ of $A$ satisfy

Since the operator $A$ defined by (2) is self-adjoint and negative with compact resolvents, it follows from [20, p.76, Proposition 3.2.12] that the eigenvalues $\{\lambda_{j}\}_{j=1}^{\infty}$ are real and we can repeat each eigenvalue according to its finite multiplicity to get

Without loss of the generality, we assume that

Define the Neumann map $\Upsilon\in\mathcal{L}(L^{2}(\Gamma_{1}),$ $H^{3/2}(\Omega))$ [12, p.668] by $\Upsilon u=\psi$ if and only if

Using the Neumann map, one can write (1) abstractly in $H^{-1}_{\Gamma_{0}}(\Omega)$ as

where $\tilde{A}$ is the extension of $A$ given by (4), $B\in\mathcal{L}(L^{2}(\Gamma_{1}),H^{-1}_{\Gamma_{0}}(\Omega))$ is defined by

and $B^{*}$ is the adjoint of $B$, given by

The rest of the paper is organized as follows: In Section 2, we give some preliminaries for the full state feedback design. Section 3 is devoted to the full state feedback design. The exponential stability of the closed-loop system is also demonstrated in Section 3. Section 4 gives some preliminaries for the observer design which will be considered in Section 5. Section 6 concludes our paper and presents an outlook for future work. Some results that are less relevant to stabilizer and observer design are arranged in the Appendix.

Throughout of this paper, the space $\mathcal{L}(X_{1},X_{2})$ represents all the bounded linear operators from the space $X_{1}$ to $X_{2}$. The space of bounded linear operators from $X$ to itself is denoted by $\mathcal{L}(X)$. The spectrum, resolvent set and the domain of the operator $A$ are denoted by $\sigma(A)$, $\rho(A)$ and $D(A)$, respectively. The point spectrum of $A$ is represented by $\sigma_{p}(A)$. The set of positive integers is denoted by $\mathbb{Z}_{+}$. We define inner product in ${\mathbb{R}}^{N}$ by

In this section, we will give some preliminaries that are very important to the state feedback design for system (1). For any positive integer $i$, we can define, in terms of the function $\phi_{i}(x)$ of (5), the operator $P_{\phi_{i}}$: $\mathbb{R}\backslash\sigma(A)\rightarrow L^{2}(\Omega)$ by

where $\zeta_{\phi_{i}}$ is the solution of following elliptic equation:

Inspired by [8], the controller design is closely related to the following system:

where $P_{\phi_{i}}\theta$ is defined by (13), $\mu>0$, $N$ is a positive integer satisfies Assumption 1.1 and $u_{1},u_{2},\cdots,u_{N}$ are new controls. Since the sequence $\{\phi_{j}(\cdot)\}_{j=1}^{\infty}$ under the Assumption 1.1 forms an orthonormal basis for $L^{2}(\Omega)$, $P_{\phi_{i}}\theta$ and $z(\cdot,t)$ can be represented by

and

In view of (18), the function $z_{k}(t)$ in (20) satisfies

Since $\lambda_{k}+\mu<0$ provided $k>N$, $z_{k}(t)$ is stable for all $k>N$. Consequently, it is sufficient to consider $z_{k}(t)$ for $k\leq N$, which satisfy the following finite-dimensional system:

where $\Lambda_{N}$ and $F_{N}$ are defined by

This section is devoted to the full state feedback design for system (1). To this end, we first define, in terms of the eigenfunctions $\phi_{1},\phi_{2},\cdots,\phi_{N}$ that are given by (5), the operator $B_{v}\in\mathcal{L}\big{(}{\mathbb{R}}^{N},L^{2}(\Gamma_{1})\big{)}$ by following equation

Inspired by [5], we consider the following dynamics feedback

where $\alpha>0$ is a tuning parameter, $I$ is the identity operator in $L^{2}(\Gamma_{1})$ and

is a new control to be designed later. Under the controller (34), the control plant (1) becomes

where $\tilde{A}$ is the extension of $A$ given by (4) and $B\in\mathcal{L}\big{(}L^{2}(\Gamma_{1}),H^{-1}(\Omega)\big{)}$ is given by (10).

Since (36) is a cascade system, the “$v$-system” can be regarded as the actuator dynamics of the control plant “$w$-system”. Therefore, we can use the actuator dynamics compensation method proposed in [5] to stabilize the system (36). Actually, the controller (35) can be designed as

where $K_{i}$ are given by (24), $S\in\mathcal{L}\big{(}L^{2}(\Gamma_{1}),L^{2}(\Omega)\big{)}$ solves the Sylvester equation

Combining (24) and (43), the controller (37) turns to be

where $\varphi_{v}$ satisfies following equation

Combining (1), (33), (34), (35) and (39), we obtain the closed-loop system