Regional stabilisation for infinite bilinear systems E. Zerrik, M. Ouzahra and K. Ztot Abstract: An output stabilisation technique for infinite dimensional bilinear systems is presented. It consists in studying the asymptotic behaviour of such a system only on a subregion of its geometrical domain, so we give sufficient conditions to obtain a stabilising control. Also, we concentrate on the determination of the control which ensures regional stabilisation by minimising a given performance cost. The obtained results are illustrated by numerical examples. 1 Introduction Recently, the question of regional stabilisation for linear systems has been tackled and developed by Zerrik and Ouzahra [1, 2], and consists in studying the asymptotic behaviour of a distributed system only within a subregion v interior or in the boundary of its evolution domain V: The principal reason for introducing this notion is that, it makes sense for the usual concept of stabilisation taking into account the spatial variable and then it becomes closer to real world problems, where one wishes to stabilise a system on a critical subregion of its geometrical domain. Defi- nitions, properties, and characterisation of a feedback control ensuring regional stabilisation on a subregion of the closure " V are given in [1] and [2] We consider the question of regional stabilisation for an infinite bilinear system defined on a domain V  R n by: @z @ t ¼ Az þ vðtÞBz zð:; 0Þ¼ z 0 ð1Þ where A is the infinitesimal generator of a linear strongly continuous semi-group S(t), t  0 on the state space H ¼: L 2 ðVÞ endowed with its natural complex inner product h; i and the corresponding norm k.k. B is a linear bounded operator from H to H. The complex valued function v(t) is a control and the problem of regional stabilisation of (1) on a subregion v of V consists in choosing a control in such a way that for all solutions z(t) of (1), we have that x v zðtÞ converges to zero in some sense, where x v indicates the operator restriction to v: However, it is clear that: 1. The regional stabilisation problem may be seen as a special case of an output stabilisation one for infinite dimensional systems with partial observation y ¼ x v z: 2. If the system (1) is regionally stabilisable on v  V; then it is regionally stabilisable on v 0  v using the same control. 3. Stabilising a system regionally is cheaper than stabilising it on the whole domain. Indeed if we consider the following functional cost qðvÞ¼ Z þ1 0 jvðtÞj 2 dt where v 2U ad ðvÞ¼fv; kx v zðtÞk! 0; as t !þ1 and qðvÞ < 1g: Then we have min U ad ðvÞ qðvÞ min U ad ðVÞ qðvÞ and the inequality may be strict, indeed let us consider the system defined on V ¼0; þ1½ by: @ z @t ¼ Az þ vðtÞzzð:; 0Þ¼ z 0 ð2Þ where A ¼@=@x; and DðAÞ¼fz 2 H 1 ðVÞ; zð0Þ¼ 0g: For vð:Þ¼ 0; the state of the system (2) verifies kzðtÞk¼ kz 0 k so 0 62U ad ðVÞ; for any z 0 6¼ 0: But for v ¼0; 1½ we have kx v zðtÞk! 0; as t !þ1: Then min U ad ðvÞ qðvÞ¼ 0 < min U ad ðVÞ qðvÞ The problem of stabilising a distributed bilinear system on its geometrical domain V was resolved in the case of a semi-group of contraction by considering the quadratic feedback control: vðtÞ¼hBzðtÞ; zðtÞi ð3Þ If B is compact (see [3]), the control (3) stabilises the system (1) weakly on V provided that hBSðtÞz 0 ; SðtÞz 0 i¼ 0; 8t  0 ) z 0 ¼ 0 ð4Þ On the other hand, if B is a monotone and self-adjoint operator, then the control (3) strongly stabilises the system (1) provided that (4) holds and the resolvent of A is compact [4]. If in addition the following holds Z T 0 kB 1 2 SðtÞz 0 k 2 dt  akz 0 k 2 ; z 0 2 L 2 ðVÞ; for some a > 0 and T > 0 then we have the decay estimate kzðtÞk 2 ¼ Oð1=tÞ; as t !þ1 (see [5]). In the finite dimensional case, the system (1) is strongly stabilisable by the feedback law (3) if (4) holds (see [6]). In [7], the following functional cost is considered: q IEE, 2004 IEE Proceedings online no. 20040017 doi: 10.1049/ip-cta:20040017 The authors are with the MACS Group - AFACS UFR, University of Moulay Ismail, Faculty of Sciences, Meknes, Morocco Paper received 17th February 2003 IEE Proc.-Control Theory Appl., Vol. 151, No. 1, January 2004 109