Stabilization and robustness of constrained linear systems

In this paper, we consider the feedback stabilization of linear systems in a Hilbert state space. The paper proposes a class of nonlinear controls that guarantee exponential sta- bility for linear systems. Applications to stabilization with saturating controls are provided. Also the robustness of constrained stabilizing controls is analyzed.


Introduction
In this paper, we consider the following linear system : where the state space is a Hilbert H with inner product •, • and corresponding norm ., the Hilbert space U with norm • U is the space of control and u(t) ∈ U is a control subject to the constraint u(t) U ≤ u max , u max > 0. The operator B : U → H is linear and bounded, and the unbounded operator A : D(A) ⊂ H → H is an infinitesimal of a semigroup of contractions S(t) on H.The radial projection onto the unit ball enables us to define the following bounded control : .
This control guarantees weak and strong stabilization for a class of linear systems under the approximate controllability assumption : B * S(t)y = 0, ∀t ≥ 0 ⇒ y = 0 (see [13,14]).Furthermore, under the following exact controllability assumption : U dt ≥ α y 2 , ∀y ∈ H, (T, α > 0), strong and exponential stabilization results have been established by [3], using the feedback u 1 (t) and the following smooth control : The purpose of this paper is to give necessary and sufficient conditions for exponential stabilization of an autonomous nonlinear systems.Then we give applications to problems of local and global exponential stabilization and robustness for constrained control systems.The plan of the paper is as follows : in the second section, we give necessary and sufficient conditions for exponential stability of an autonomous nonlinear system.The third section is devoted to problems of stabilization of the linear system (1) using bounded controls.The robustness problem is considered in the fourth section.Finally, an illustrating example is given in the fifth section.

Exponential stability
In this section, we discuss the stabilization question of the following autonomous system : where the state space is a Hilbert H with inner product •, • and corresponding norm ., the dynamic A is an unbounded operator with domain D(A) ⊂ H and generates a semigroup of contractions S(t) on H, and N is a nonlinear operator from H into H such that N(0) = 0, so that 0 is an equilibrium for (2).EJQTDE, 2013 No. 35, p. 2

Definitions and notations
Let us give the following definition regarding the stability of system (2).

Definition 1
We say the origin is exponentially stable on a set Y ⊂ H if, for all initial states z 0 in Y, there exist M, σ > 0 (depending on z 0 ) such that the mild solution z(t) starting at z 0 satisfies The origin is said to be uniformly exponentially stable on Y if (3) holds for some σ and M, which are independent of z 0 .It is said to be globally exponentially (resp.globally uniformly exponentially) stable if it is exponentially (resp.uniformly exponentially) stable on Y = H.
To state stabilization results for (2) we consider, for ρ > 0, the assumption : where T, δ ρ > 0 and B ρ = {y ∈ H/ y ≤ ρ}.In this case we set We also consider the following strong controllability assumption : where T, δ > 0, and let us set : On the other hand if N is Lipschitz on B ρ , then there exists L ρ > 0 such that In this case, we can set : and when N is Lipschitz we set L(N) = sup

Sufficient conditions for exponential stability
Our first result concerns the local exponential stability and is stated as follows : Theorem 1 Let (i) A generate a semigroup S(t) of contractions on H, (ii) N be dissipative (i.e., Ny, y ≤ 0, ∀y ∈ H) and Lipschitz on any bounded set, and let (iii) (4) hold.Then 2 ) 1 2 we have z(t) → 0, exponentially, as t → +∞.Proof. 1) Since N is locally Lipschitz, the system (2) has a unique local mild solution z(t), and since N is dissipative, then z(t) is bounded in time and hence it is defined for all t ≥ 0. Furthermore, z(t) is given by the variation of constants formula : Since S(t) is a semigroup of contractions (so that A is dissipative), then by using approximation techniques and proceeding as in [1], we obtain the following inequality : It follows that For all z 0 ∈ B ρ and t ≥ 0, we have the relation where y(t) = t 0 S(t − s)Nz(s)ds.
Then, using ( 6) and ( 9) and the fact that the semigroup S(t) is of contractions, we deduce that By virtue of (9), the inequality (4) also holds for y = z(t).Then, integrating (10), yields It follows from the inequality (8) that for all k ∈ IN, we have Then using (11), we get z(kT where which is, by virtue of (12), positive and from the assumption on L 2 z 0 (N) we have which gives (since z(t) decreases) the following exponential decay z(t) ≤ M z 0 e −σt , 2) Under the assumption 1 2 , we obtain from the above development the estimate : , so the parameters M and σ are independent of z 0 , which gives the uniform stability.
The following result concerns the global stabilization.

Corollary 1 Let (i) A generate a semigroup S(t) of contractions on H, (ii) N be dissipative and Lipschitz and let (iii
, then ( 2) is uniformly globally exponentially stable.
Proof.From the proof of the above theorem, we have the estimate : 2T are independent of z 0 , which means that the stability is global and uniform.
Remark 1 Note that (5) implies that ( 4) holds for all ρ > 0, but the converse is not true as we can see taking Az = 0 and Nz = −z z 2 + 1 , ∀z ∈ H := R.

5.
The assumption ( 16) does not guarantee the exponential stability of ( 2), as we can see for A = 0 and Nz = −z 3 , ∀z ∈ H := IR.Indeed, for all 0 ≤ s ≤ t, we have S(t − s)NS(s)y = Ny = y 3 and hence (16) holds.However, for all initial state z 0 = 0, the solution is given by z In this section, we will study the problem of exponential stabilization and robustness of the system (1).For this end, we consider (for some T, α > 0) the following exact controllability assumption : and let us set α(B) = inf y =1 B * S(•)y 2 L 2 (0,T ;U ) , (so that α ≤ α(B)).

Nonlinear controls
In order to study various kinds of control saturation, it would be more appropriate to consider the general feedback : where r : H → R * + is an appropriate function and c is positive constant.
Remark 2 If r(y) ≥ ν B * y U , for all y ∈ H (for some ν > 0), then we have : for all t ≥ 0.
The following result gives sufficient conditions for the control (18) to guarantee local and global stabilization of (1).

Theorem 3 Let (i)
A generate a semigroup S(t) of contractions on H, (ii) B ∈ L(U, H) such that (17) holds and let (iii) r be Lipschitz on any bounded set.
2) If r is Lipschitz and 0 < m ≤ r(y) ≤ M, for all y ∈ H, (for some m, M > 0), then there exists c > 0 for which the control (18) exponentially globally stabilizes the system (1).
Proof. 1) To study the stabilizability of (1) using the control (18), we introduce the , which is clearly dissipative.Moreover, since S(t) is of contractions, then for all z ∈ B ρ , we have S(t)z ≤ z ≤ ρ and so Then for all z ∈ B ρ , we have . In other words, N verifies (4) with δ ρ (N) ≥ cα(B) M(ρ) .Furthermore, the operator N is locally Lipschitz.
Indeed, let x ∈ H and let R, L R,x (r) > 0 such that for all z, y ∈ H; x − y , x − z ≤ R, we have r(y) − r(z) ≤ L R,x (r) y − z .Then, letting R x = R + x , we obtain This shows that N is locally Lipschitz.Now, taking x = 0, R = ρ, and letting L ρ,0 (N) = L ρ (N) in the last inequality, we get We have This, together with (20), implies that The result of Theorem 1 implies the uniform exponential stabilizability of the system (1) on B ρ with the control (18).
2) Let ρ > z 0 and let c be such that : It follows from the first point that the control (18) exponentially stabilizes the system (1) on B ρ .The choice of ρ implies that z 0 ∈ B ρ , and hence the solution of system (1) with z 0 as initial state exponentially converges to 0, as t → +∞.This achieves the proof.EJQTDE, 2013 No. 35, p. 8

Constrained controls
Let us consider the two bounded controls and where c > 0 is the gain control.
As applications to constrained stabilization of the system (1), we have the following result Theorem 4 Let A generate a semigroup S(t) of contractions on H and let B ∈ L(U, H) such that (17) holds.Then 1) for all ρ > 0, there exists c > such that both the controls ( 22) and ( 23) uniformly exponentially stabilizes (1) on B ρ .
2) It follows from the same techniques as in 1) by taking ρ > z 0 .
In other words, the controls ( 22) and ( 23) are uniformly bounded with respect to the initial states.EJQTDE, 2013 No. 35, p. 9

Necessary conditions for exponential stabilization
In the case of the system (1), the results of Theorem 2 can be reformulated as follows : Theorem 5 1) The condition ∀y ∈ H, S(t − s)BB * S(s)y = 0, ∀t ≥ 0, ∀s ∈ [0, t] implies y ∈ Λ, is necessary for the exponential stability of ( 1) with the control (18).
Proof.It follows from Theorem 2 by taking N = −cBB * r .

Remark 4
1.The results of Theorem 5 can be applied to avoid the "bad" actuators, i.e, the ones that do not guarantee the exponential stability.
We recall that an actuator can be defined as a couple (ω, a(•)) of a function f , which indicates the spatial distribution of the action on the support ω which is a part of the closure Ω of the domain Ω (see [5,6,7,10]).18) is that all the modes of A corresponding to eigenvalues λ such that Re(λ) ≥ 0 are actives.In other words, for all λ ∈ Sp(A); Re(λ) ≥ 0 and for all corresponding eigenfunction ϕ ∈ ker (A − λI) − (0), we have BB * ϕ = 0.As an example; for H = L 2 (0, 1) and

As a consequence of the above theorem, a necessary condition for exponential stabilization of the system (1) with the control (
, a necessary condition for exponential stability is BB * (1) = 0.In term of actuators, if we take B : u ∈ U = IR → (a(•)χ ω )u ∈ L 2 (0, 1), i.e, the action applies in the subregion ω of Ω with the spatial repartition a(x), then we have B * y = ω a(x)y(x)dx.Thus, an actuator (ω, a(•)) such that ω a(x)dx = 0 is a "bad" one.

Robustness of constrained controls
Let us now proceed to robustness question of the controls ( 22) and (23) to small perturbations of the parameters system.Consider the following perturbed system : where A and B are as in (1) and the perturbation a is a nonlinear operator from H to itself.EJQTDE, 2013 No. 35, p. 10 Consider the nominal system : where Let us define the set of admissible perturbations : Note that the assumption a(0) = 0 implies that 0 remains an equilibrium for (25).
We have the following result Theorem 6 Let assumptions of Theorem 4 hold.Then for any perturbation a ∈ Ω A , the controls ( 22) and ( 23) uniformly exponentially stabilize the system (25) on B ρ .If a is Lipschitz, then the controls ( 22) and ( 23) globally exponentially stabilize (25).
Proof.First let us note that from Theorem 4, one deduce that Ω A = ∅.Let a ∈ Ω A and let Ñi = a − N i .We have Clearly the operator Ñi is dissipative, locally Lipschitz and verifies : , for all a ∈ Ω A .
which holds for c small enough.The global stability follows then from Theorem 1.
The system (25) may be seen as a perturbation of (1) in its dynamic A. Next, we consider the problem of robustness of controls ( 22) and (23) with respect to perturbations of B. Let us consider the linear system where b ∈ L(U, H).We have the following result.From Theorem 4, we deduce that the controls : globally exponentially stabilize the perturbed system (27) for some c > 0; uniformly on B ρ .Now let us see the problem of robustness associated to linear perturbations acting, jointly, on the dynamic and the operator of control.Consider the perturbed system : where a ∈ L(H) and b ∈ L(U, H).We have the following result.

Proof.
Under the assumptions on a, the operator A + a is the infinitesimal generator of a semigroup of contractions S a (t) (see [11]), and for all t ≥ 0 and y ∈ H, we have S a (t)y = S(t)y + t 0 S(t − s)aS a (s)yds. (29) The system (28) may be seen as a perturbation of the system (25) in its control operator B by b.Then from Theorem 7, it is sufficient to show that for some α a > 0.
Based on (29), we obtain the following relation where φ(t) is a scalar function such that where Integrating this last inequality and using (17), we deduce T .Then we conclude by Theorem 7.
Let us now consider the perturbed system : The system (32) may be seen as the system (1), perturbed in its dynamic by a = 0 0 0 λI and in its operator of control by b = 0 µ .Applying results of Theorem

Conclusion
In this work, sets of necessary and sufficient conditions for exponential stability of nonlinear systems are obtained.Then we have studied the exponential stabilization of distributed linear systems using bounded feedbacks.The established results can be applied to systems which are subject to constraint on the control input.Also sets of allowed perturbations of the parameters system that maintain the exponential stabilization of the considered systems are given.
EJQTDE, 2013 No. 35, p. 11 Theorem 7 Let A generate a semigroup S(t) of contractions on H and let B ∈ L(U, H) such that (17) holds.Then the controls (22) and (23) are globally exponentially robust to any perturbation b ∈ L(U, H) of B such that b * < α(B) 2T B * .Furthermore, the robustness is uniform on B ρ .S(t)y 2 U − 2 B * S(t)y b * S(t)y + b * S(t)y 2 ≥ B * S(t)y 2 U − 2 B * b * y 2 Integrating this inequality and using (17), we get U ≥ B *