Constrained Feedback Stabilization for Bilinear Parabolic Systems

In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.


Introduction
Bilinear systems represent a small, but important subset of nonlinear systems within which linear systems coexist as a special subclass. Adopting a bilinear model retains a well structured framework, which contains the wellknown notional concepts such as time constants and steady-state behaviour. When adopting a bilinear approach, these concepts become operation-dependent quantities which can be appropriately modelled. Bilinear system models represent an important class of nonlinear models that are defined to be linear in both state and control when considered independently, with the nonlinearity (or bilinearity) arising from coupled terms involving products of system state and control (see [1] [2]). By formulating the model appropriately, the bilinear term could also be represented by products of system output and control input, i.e. the output is defined as a system state. There are numerous combinations of product terms that could be considered, thus potentially increasing the model complexity. However, it has been found in practice that a minimal number of product terms can provide an adequate model for the purpose of control. Bilinear model structures are able to represent nonlinear phenomena more accurately than linear models, and thereby extend the range of satisfactory performance. In this paper, we are concerned with the question of the stabilization by a constrained feedback control for bilinear parabolic systems that can be described in the following form: (see [3]). By choosing an adequate feedback control ( ) p t in such a way, the corresponding solution ( ) y t of the system (1) converges to zero when t → +∞ , for all 0 y in H . For finite-dimensional bilinear systems associated to a skew-adjoint matrix A , the question of stabilization has been treated in [4], under the condition: Using the following assumptions: the problem of stabilization has been studied in [5]. In [3], when the linear operator B is compact and is a contraction semigroup, then using the quadratic feedback control a weak stabilization result is obtained under the weak observability condition: In the case where B is sequentially continuous from w H ( H endowed with the weak topology) to H , the quadratic feedback control (2) weakly stabilizes the system (1), provided that the following weak observability assumption (4) holds (see [3]). Under the exact observability assumption The strong stabilization result with the following decay estimate M > for t large enough, has been obtained using the quadratic feedback control (5) (see [6]). However, in this way the convergence of the resulting closed loop state is not better than (8). In [7] the rational decay rates are established i.e. using the following feedback control: in H , then the system (1) is strongly stable with the explicit decay estimate (8), using the control (9), provided that the estimate (7) holds. Here, we will establish an explicit decay estimate of the stabilized state and the robustness of the control (9) for a large class of bilinear systems as considered in [3] [8] [9]. The method used in this paper is based on decomposing the system (1) into two suitable subsystems: the stable part and the unstable one. Then, we will show that one can concentrate on the determination of a stabilizing control for the so-called unstable part which maintains the exponential stability of the stable part. The rest of this article is as follows: in Section 2, we will give the main hypotheses that allow the decomposition of the system (1) into two subsystems. Then, under the compactness hypothesis of the operator B , we will give a weaker variant of the condition (6) which achieves strong stabilization of the system (1). In Section 3, we will show that under a weaker version of (6), we obtain the stabilization with the decay estimate (8). Section 4 concerns the robustness of the stabilizing controls. The last section is devoted to an illustrating example and simulations.

Stabilization Results
Let us now recall the following definition concerning the asymptotic behavior of the system (1).

Definition
The system (1) is weakly (resp. strongly) stabilizable if there exists a feedback control ( ) ( ) ( ) : y t → , weakly (resp. strongly), as t → +∞ , for all 0 y H ∈ . In the sequel of this section, we will present an appropriate decomposition of the state space H and the system (1) via the spectral properties of the operator A , and we apply this approach to study the stabilization problem of the system (1). In [10]- [12], it has been shown that if the spectrum ( ) where C is a curve surrounding Which is equivalent to: denotes the semigroup generated by s A in s H , then stabilizing the whole system turns out to stabilizing its projection on u H (see [13]). In the sequel, we suppose that the operator B satisfies It is easily verified that the condition (14) is equivalent to the fact that the linear operator B commutes with u P . We note that the condition (14) also holds in the special case: Let us consider that the system (1) can be decomposed in the following two subsystems: ⊕ . It has been proved that stabilizing a linear system turns out to stabilizing its unstable part (see [13]).

Remark
For finite-dimensional systems, the conditions (6) and (7) are equivalent (see [5] [10]). However, in infinitedimensional case, and if B is compact, then the condition (7) is an orthonormal basis of H , then applying (7) for j y ϕ = and using the fact that 0 j ϕ → , weakly as j → +∞ , we obtain the contradiction: 0. δ = The following result concerns the strong stabilization of the system (1).

2.
A allows the decomposition (10) of H with dim u H < +∞ such that (13) holds, 3. B be compact such that Then, the constrained feedback control law: (18) strongly stabilizes the system (1).

Proof
The system (1) controlled by (18) corresponds to (18) (see [9]). Since is a contraction semigroup, we get: It follows from (20) [14]). Now, let us show that ( ) 0 y t → , weakly as t → +∞ . Let n t → +∞ such that ( ) Then for all 0 0 t t ≤ ≤ , we have: By y t S t t y t S t B y y By It follows from (13)

A Decay Rate Estimate of the Stabilized State
In what follows, we will study the strong stabilizability of the system (1) with the decay estimate (8).
Before we state our main result, the following lemmas will be needed (see [15]).

Lemma
Let ( ) 0 k k s ≥ be a sequence of positive real numbers satisfying Let us now recall the following existing result (see [9]).

Lemma
Let A generate a contraction semigroup ( ) S t on H and let B be linear operator from H into itself.
Hence, the system (29) admits a unique mild solution defined for almost all 0 t ≥ (see [9]). Integrating now the inequality (30) over the interval ( )   = , then using the same techniques as in [9], we can obtain the result of Theorem 3.3, if the estimate (7) is changed to (33).

Robustness
In this section, we study the robustness of the controls (18) and (28), under a class of perturbations of the system (1).

Strong Robustness
In this part, we consider the strong robustness of the feedback (18). Then, we will show that the stability property of the system (1) remains invariant under a certain class of bounded perturbations.
Let us consider the following perturbed system The following main result concerns the strong stability of the system (37).  Then the system (37) is strongly stabilizable.

Proof
First, let us note that 0 remains an equilibrium state of the perturbed system (37), which can be written in the form: Since f and ξ are locally Lipschitz, so is g . Also g is dissipative: The assumption ( ), It follows from (13)  strongly converges to 0, as t → +∞ .

A Polynomial Decay Estimate for the Perturbed System
Our second main result in this section is stated as follows:

An Application
In this part, we will give an illustrating example of the established results.

Example
Let us consider the following 1-d bilinear heat equation: where ( ) y t is the temperature profile at time t . We suppose that the system is controlled via the flow of a liquid ( ) p t in an adequate metallic pipeline. Here we take the state space

Conclusion
In this work, we have considered the problem of strong stabilization with polynomial decay rate of the stabilized state for bilinear parabolic systems that can be decomposed in the stable and unstable parts (15) and (16) under a weaker condition (27). We have also considered the problem of using a stabilizing feedback control for the unstable part (15) only that can make the whole system (1) stable. Various questions remain open. This is the case of stabilization for nonlinear systems. Finally, we have studied the robustness problem of the stabilizing controls with respect to a class of perturbations, but a confrontation to more realistic situations remain done. This leads us to consider the stabilization problem for stochastic bilinear system.