Exponential Stability of Linear Systems Under a Class of Desch

Home , Exponential stability

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II. Exponential stability

In this section, we provide our two main stabilization results. We start by introducing the nec- essary tools regarding the notion of admissibility in connection with the generation results and then provide some a priori estimations of the solution. i. Preliminary As pointed out in the introduction, the unbounded aspect of the operator B do not guarantee the existence of an X valued solution x(t) of (1). However, one may extend the system at hand in a − larger (extrapolating) space X 1 of the state space X in which the existence of the solution x(t) is − ensured and then give the required admissibility conditions of B so that the solution x(t) lies in X. Classically, the spaces X1 and X 1 are deﬁned as follows: X1 :=(D(A), 1), where x 1 := − k · k k k (λI A)x X, x D(A) for some λ in the resolvent set ρ(A) of A, and X 1 is the completion of X k − k ∈ 1 − with respect to the norm x 1 := (λI A)− x X, x X. These spaces are independent of the k k− k − k ∈ .choice of λ and are related by the following continuous and dense embedding: X1 ֒ X ֒ X 1 → → − That way the unbounded operator B becomes bounded from X to the extrapolating space X 1, i.e, − B (X, X 1). Thus, in order to give a meaning to solutions of (1), we have to use the fact that ∈ L − the semigroup (S(t))t 0 can be extended to a C0 semigroup (S 1(t))t 0 on X 1 whose generator ≥ − − ≥ − A 1 has D(A 1) = X as domain and is such that A 1x = Ax, for any x D(A). Moreover, if the − − − ∈ semigroup S(t) is a contraction, then so is S 1(t). Indeed, having in mind that for all λ ρ(A 1), 1 − 1 1 ∈ − we have (λI A 1)− x X, x X 1 and (λI A 1)− x = (λI A)− x, x X, we can see − − ∈ ∀ ∈ − − − − ∀ ∈ that all for x X we have ∈ 1 1 S 1(t)x 1 = (λI A)− S(t)x X (λI A)− x X = x 1, k − k− k − k ≤ k − k k k− thus, by density of X in X 1, we conclude that the semigroup S 1(t) is a contraction on X 1. Recall that for any given initial− state x X, a mild solution of− (1) is an X valued continuous− 0 ∈ − function x on [0, T] satisfying the following variation of parameters formula:

t x(t) = S(t)x0 ρ S 1(t s)Bx(s)ds, t 0, − Z0 − − ∀ ≥ which always makes sense in X 1. The system (1) can be rewritten in the large space X 1 in the following abstract form: − − x˙(t) = A 1x(t) ρBx(t), x(0) = x0. (2) − − which is well-posed in X whenever A ρB is a generator of a C semigroup on X (cf. [6], − 0− Section II.6). The well-posedeness of systems like (1) has been studied in many works using different approaches (see e.g. [1, 3, 5, 9, 13]). The next result provides sufﬁcient conditions on a Desch-Schappacher perturbation B to guarantee the existence and uniqueness of the mild solution of (1) (see [1] & ([6], p. 183)).

2 Theorem II.1 Let A be the generator of a C0 semigroup S(t) onX andlet B (X, X 1) be p admissible − ∈L − − for some 1 p < ∞, i.e., there is a T > 0 such that ≤ T p S 1(T t)Bu(s)ds X, u L (0, T; X). (3) Z0 − − ∈ ∀ ∈ Then for any ρ, the operator (A 1 ρB) X defined on the domain D((A 1 ρB) X) := x X : (A 1 ρB)x X − − | − − | { ∈ − − ∈ } by (A 1 ρB) Xx := A 1x ρBx, x D((A 1 ρB) X) (4) − − | − − ∀ ∈ − − | is the generator of a C0 semigroup (T(t))t 0 on X, which verifies the variation of parameters formula − ≥ t T(t)x = S(t)x ρ S 1(t s)BT(s)xds, t 0, x D((A 1 ρB) X). − Z0 − − ∀ ≥ ∀ ∈ − − | An operator B (X, X 1) satisfying the condition (3) is called a Desch-Schappacher opera- ∈ L − tor or perturbation. Moreover, the operator defined by (4) is the part (A 1 ρB) X of (A 1 ρB) − − | − − on X see ([16], p. 39) and ([6], p. 147) . Remark 1 Notice that since W1,p(0, T; X) is dense in Lp(0, T; X), the rang condition (3) is equivalent to the existence of some M > 0 such that

T 1,p S 1(T s)Bu(s)ds M u Lp(0,T;U), u W (0, T; X), (5) − − ≤ k k ∀ ∈ Z0 X

1 T p p with u Lp(0,T;X) = u(t) Xdt . k k 0 k k Z

Remark 2 Note that if the operator B (X, X 1) is p admissible in [0, T], then it is so in [0, t] for ∈ L − − any t [0, T]. In other words, if (5) holds then for all t [0, T] we have the following inequality ∈ ∈ t S 1(t r)Bu(r)dr M u Lp(0,t;X) (6) 0 − − ≤ k k · Z X

Indeed, for all t [0, T] and u Lp(0, T; X) we deﬁne u Lp(0, T; X) by ∈ ∈ t ∈ 0, for r [0, T t] ∈ − u (r) := t  u(r + t T), for r (T t, T]  − ∈ − t  T Then, observing that 0 S 1(t r)Bu(r)dr = 0 S 1(T r)But(r)dr X, it comes from the p admissibility t − − − − ∈ t − of B that S 1(t r)Bu(r)dr M ut Lp(0,T;X) Thus for all t [0, T], we have S 1(t r)Bu(r)dr 0 − −R X ≤ k kR · ∈ 0 − − X ≤ M u p L ( 0,Rt;X) R k k · Let us now show the two following lemmas that will be needed in the sequel. < < 1 Lemma II.2 Let assumptions of Theorem II.1 hold. Then for any 0 ρ 1 , the mild solution x(t) T p M of the system (1) satisﬁes the following estimate

1 T p p x(.) L (0,T;X) 1 x0 X, x0 X. (7) k k ≤ 1 ρT p M k k ∀ ∈ −

3 Proof 1 Let x0 D((A 1 ρB) X). From Theorem II.1, we know that the system (1) admits a unique ∈ − − | mild solution x(t) which is given by

t x(t) = S(t)x0 ρ S 1(t s)Bx(s)ds, t 0. (8) − Z0 − − ∀ ≥ Let us estimate x( ) p . From (8), we get via Minkowski’s inequality k · kL (0,T;X) 1 1 T p T t p p p x(.) Lp(0,T;X) S(t)x0 dt + ρ S 1(t s)Bx(s)ds dt k k ≤ k kX − − Z0 Z0 Z0 X

Then from Remark 2 we derive 1 1 x( ) p T p x + ρT p M x(.) , k · kL (0,T;X) ≤ k 0kX k kp 1 T p p where x( ) p := x(τ) dτ , which gives the estimate k · kL (0,T;X) k kX Z0 1 T p p x(.) L (0,T;X) 1 x0 X, (9) k k ≤ 1 ρT p M k k − < < 1 for any 0 ρ 1 . T p M Now, since the mapping x x(t) deﬁnes a C semigroup T(t) on X, the mapping x x( ) = 0 7→ 0− 0 7→ · T( )x is continuous from X to Lp(0, T; X). Then the estimate (9) holds by density for any x X. · 0 0 ∈ Remark 3 It follows directly from Lemma II.2 that for all t 0, we have ≥ 1 ρMT p x(t) X 1 + 1 x0 X, x0 X (10) k k ≤  1 T p ρM  k k ∀ ∈ −   < < 1 for any 0 ρ 1 . T p M < < 1 Lemma II.3 Let assumptions of Theorem II.1 hold and let 0 ρ 1 . Then the mild solution x(t) of T p M the system (1) satisﬁes the following estimate

t S 1(t s)Bx(s)ds Mρ x0 X, t [T,2T], x0 X, 0 − − ≤ k k ∀ ∈ ∀ ∈ Z X

1 p 1 MT 2 p with Mρ := 1 2 + ρM T . 1 ρT p M − Proof 2 Let x X, and let us write for any t [T,2T], 0 ∈ ∈ t T t S 1(t s)Bx(s)ds = S 1(t s)Bx(s)ds + S 1(t s)Bx(s)ds := L1 + L2 Z0 − − Z0 − − ZT − − Then we consider the two terms of the sum separately. For the ﬁrst one, the admissibility of B together with the contraction property of S 1(t) yields − T L1 X = S 1(t T) S 1(T s)Bx(s)ds M x( ) Lp(0,T;X) (11) k k − − − − ≤ k · k Z0 X

4 t T For the second term, observing that L2 = − S 1(t T τ)Bx(τ + T)dτ, we obtain again from the 0 − − − admissibility of B R L M x(. + T) p k 2kX ≤ k kL (0,T;X)· Using (10), we derive the following inequalities:

1 1 1 2 p p p ρM T x(. + T) LP(0,T;X) T x(T) X T 1 + 1 x0 X. (12) k k ≤ k k ≤  1 ρMT p  k k −   Summing up (11) and (12) we obtain the desired estimate:

t S 1(t s)Bx(s)ds Mρ x0 X 0 − − ≤ k k · Z X

ii. A direct approach In the following theorem, we provide sufﬁcient conditions for exponential stability in terms of admissibility and observation conditions.

Theorem II.4 Suppose that A is the inﬁnitesimal generator of a linear C semigroup of contractions 0− (S(t))t 0 on X and that for some T > 0, ≥ (i) there exists 1 < p < ∞ such that for all u Lp(0, T; X), we have T ∈ S 1(T s)Bu(s)ds D(B X), (B X being the part of B on X), Z0 − − ∈ | | (ii) for any t > 0, Range(BS(t)) X and for some L > 0 we have ⊂ T BS(s)x Xds L x X, x X, (13) + Z0 k k ≤ k k ∀ ∈ (iii) there exists δ > 0 such that T Re BS(t)x, S(t)x dt δ S(T)x 2 , x D(A). (14) + X X Z0 h i ≥ k k ∀ ∈ Then there is a ρ > 0 such that the system (1) is exponentially stable on X for all ρ (0, ρ ). 1 ∈ 1 Proof 3 Let us set AρB :=(A 1 ρB) X. According to assumption (i), we have − − | T p S 1(T s)Bu(s)ds X, u L (0, T; X). Z0 − − ∈ ∀ ∈ Then for any ρ > 0, the system (1) admits a unique mild solution which is given, for x D((A ) ), 0 ∈ ρB |X by the variation of parameters formula (see [5]):

t x(t) = S(t)x0 ρ S 1(t s)Bx(s)ds, t 0. (15) − Z0 − − ∀ ≥ Let x D(A) be ﬁxed. Then it comes from assumption (i) and the fact that Range(BS(t)) X, t > 0 0 ∈ ⊂ that for any t > 0, x(t) D(B ) and so x(t) D(B ) D(A), t > 0. Thus for all t > 0 we have ∈ |X ∈ |X ∩ ∀ x(t) D(AρB), i.e. (A 1 ρB)x(t) X. Thus, we have ∈ − − ∈ d x(t) 2 = 2Re A x(t), x(t) , t > 0 (16) dt k kX h ρB iX ∀ ·

5 For all t > 0, we have the following equality

BS(t)x , S(t)x = BS(t)x , S(t)x x(t) h 0 0iX h 0 0 − iX + BS(t)x Bx(t), x(t) + Bx(t), x(t) h 0 − iX h iX Let us estimate each term of this last expression. By virtue of the closed graph theorem, we deduce from (i) that for some constant M > 0 and for all u Lp(0, T; X), we have ∈ T S 1(T s)Bu(s)ds M u Lp(0,T;X) (17) 0 − − ≤ k k Z X and T B S 1(T s)Bu(s)ds M u Lp(0,T;X). (18) − − ≤ k k Z0 X

The formula (15) combined with the estimate (6), gives

t S(t)x0 x(t) X = ρ S 1(t s)Bx(s)ds ρM x(.) Lp(0,T;X), t [0, T]. k − k 0 − − ≤ k k ∀ ∈ Z X

Then, according to Lemma II.2, we conclude that

1 ρMT p S(t)x0 x(t) X 1 x0 X. (19) k − k ≤ 1 ρT p M k k − By similar arguments as in Remark 2, we can see that (18) implies that for all t [0, T] we have ∈ t p B S 1(t s)Bu(s)ds M u Lp(0,T;X), u L (0, T; X), (20) 0 − − ≤ k k ∀ ∈ Z X

from which it comes via (15)

B(S(t)x x(t)) ρM x(.) p , t [0, T], k 0 − kX ≤ k kL (0,T;X) ∀ ∈ and hence

1 ρMT p B(S(t)x0 x(t)) X 1 x0 X, t [0, T]. (21) k − k ≤ 1 ρT p M k k ∀ ∈ − For every t > 0, we have

Re BS(t)x , S(t)x BS(t)x S(t)x x(t) h 0 0iX ≤ k 0kXk 0 − kX + x(t) B S(t)x x(t) + Re Bx(t), x(t) k kXk 0 − kX h iX · Using (10), (19) and (21), we deduce that

1 ρMT p Re BS(t)x0, S(t)x0 X 1 x0 X BS(t)x0 X h i ≤ 1 ρT p M k k k k − 1 1 p p ρMT ρMT 2 + 1 1 + 1 x0 X + Re Bx(t), x(t) X , t (0, T] 1 ρT p M  1 T p ρM  k k h i ∀ ∈ · − −   6 Then, integrating the last inequality and using (13), yields

1 T p ρLMT 2 Re BS(t)x0, S(t)x0 dt x0 + X 1 X Z0 h i ≤ 1 ρT p M k k − 1 1 1+ p p T ρMT ρMT 2 + 1 + x0 + Re Bx(t), x(t) dt. 1  1  X + X 1 ρT p M 1 T p ρM k k Z0 h i − −   Thus

T T 2 Re BS(t)x0, S(t)x0 dt ρC1 x0 + Re Bx(t), x(t) dt + X X + X Z0 h i ≤ k k Z0 h i

1 1 +1 MT p ρMT p with C1 = 1 T + L + 1 . 1 ρT p M 1 ρT p M ! − − Applying the inequality (iii), we deduce that

t+T 2 2 δ S(T)x(t) X ρC1 x(t) X Re Bx(s), x(s) X ds (22) k k − k k ≤ Zt h i Using Lemma II.3 we deduce via the variation of constants formula (15) that for all t [T,2T], we have ∈ t x(t) X S(T)x0 X + ρ S 1(t s)Bx(s)ds X k k ≤ k k k 0 − − k R S(T)x + ρM x . ≤ k 0kX ρk 0kX By reiterating the processes for t [kT, (k + 1)T], k 1, we deduce that ∈ ≥ x(t) S(T)x(kT) + ρM x(kT) . k kX ≤ k kX ρk kX Then for all k 1, we have ≥ x((k + 1)T) 2 2 S(T)x(kT) 2 + 2ρM x(kT) 2 . (23) k kX ≤ k kX ρk kX Integrating (16) and using the dissipativeness of A gives

(k+1)T 2 2 2ρ Re Bx(τ), x(τ) Xdτ x(kT) X x((k + 1)T) X, k 0. ZkT h i ≤ k k − k k ≥ This together with (22) and (23) implies

ρδ x((k + 1)T) 2 2ρM x(kT) 2 2C ρ2 x(kT) 2 k kX − ρk kX − 1 k kX ≤ x(kT) 2 x((k + 1)T) 2 . k kX − k kX Hence (1 + ρδ) x((k + 1)T) 2 2δρ2M + 2C ρ2 + 1 x(kT) 2 , k 0 k kX ≤ ρ 1 k kX ≥ · This implies x ((k + 1)T) 2 C x(kT) 2 k kX ≤ 2k kX

7 2ρ2(δM +C )+1 where C = ρ 1 , which is in (0,1) for ρ 0+. 2 1+ρδ → Since x(t) decreases, we get for k = E t (where E(.) is the integer part function). k kX T x(t) 2 (C )k x 2 , k kX ≤ 2 k 0kX which gives the following exponential decay

σt x(t) Ke− x , t 0 k kX ≤ k 0k ∀ ≥ · 1 (C ) 2 ln 2 where K =(C2)− and σ = − 2T . This estimate extends by density to all x0 X. < < ∈ < < 1 Hence the uniform exponential stability hold for any 0 ρ ρ1, where ρ1 is such that 0 ρ1 1 T p M 2ρ2(δM +C )+1 and ρ 1 (0,1). 1+ρδ ∈ Remark. Note that the assumption BS(t)x X, x D(A), t > 0 is fulfilled if in particular ∈ ∀ ∈ D(A) D(B ). This last property is satisfied if for example B is the extension to X of a ⊂ |X Miyadera-Voigt operator. iii. A range decomposition method X X X Let 1 be a direct sum in X 1, where = i(X)(i being the canonical injection of X in ⊕ − − X 1), so we can write X = X. Then for any C (X, X 1) such that rg(C) X X 1, we set − X ∈ L −X X ⊂ ⊕ − XC =: PXC, where PX is the projection of according to 1. Now, given a pair of operator X X ⊕ − (K, L) (X, X 1) (X, X 1), the decomposition 1 is said to be admissible for (K, L) if ∈L − ×L − ⊕ − the three following properties hold: X X X X (a) rg(K) 1 and rg(L) 1, ⊂ ⊕ − ⊂ ⊕ − (b) K is dissipative on D((K + L) ) := x X : Kx + Lx X , X |X { ∈ ∈ } (c) L (X). X ∈L For our stabilization problem, we will be interested with admissible decompositions for the pairs (A 1, ρB) for ρ > 0 small enough. Note that if the domain of the operator AρB) X is indepen- − − | dent of ρ > 0 (small enough), which is equivalent to D((A ) ) = D(A) D(B ), then for the ρB |X ∩ |X sum X X 1 to be admissible for the pairs (A 1, ρB), ρ > 0 it suffices to be admissible for the ⊕ − − − pair (A 1, B). − We are ready to state our second main result.

Theorem II.5 Let A be the inﬁnitesimal generator of a linear C0 semigroup of contractions (S(t))t 0 on − ≥ X and let B (X, X 1). Let X X 1 be an admissible decomposition for the pair (A 1, ρB) for any ∈L − ⊕ − − − ρ > 0 small enough, and assume that for some T > 0, theoperator B is p admissible for some 1 < p < ∞ − and satisﬁes the estimate:

T 2 Re X BS(t)x, S(t)x X dt δ S(T)x X, x X, (24) Z0 h i ≥ k k ∀ ∈ for some T, δ > 0. Then there is a ρ > 0 such that the system (1) is exponentially stable on X for all ρ (0, ρ ). 1 ∈ 1 < < 1 Proof 4 Let 0 ρ 1 , and let x(t) be the unique mild solution of the system (1) given for x0 T p M ∈ D((A ) ) by the formula (15). ρB |X

8 The admissibility assumption on B together with Lemma II.2 implies the following estimate for t [0, T] : ∈ 1 ρMT p x(t) S(t)x0 X 1 x0 X (25) k − k ≤ 1 ρT p M k k −

Moreover, observing that AρBx(t) =X(AρB)x(t), we can write

d 2 > x(t) X = 2Re X(A 1)x(t) ρ X Bx(t), x(t) X, t 0 dt k k h − − i ∀ ·

Integrating this last equality and using the dissipativeness of X(A 1) gives − t 2 2 2ρ Re X Bx(τ), x(τ) Xdτ x(s) X x(t) X, t s 0. (26) Zs h i ≤ k k − k k ≥ ≥ We have the following equality

BS(t)x , S(t)x = BS(t)x Bx(t), S(t)x hX 0 0iX hX 0 − X 0iX + Bx(t), S(t)x x(t) + Bx(t), x(t) hX 0 − iX hX iX ·

Using the the fact that the operator X B is bounded, it comes

Re X BS(t)x0, S(t)x0 X X B (X) x0 X S(t)x0 x(t) X h i ≤ k kL k k k − k + X B (X) x(t) X S(t)x0 x(t) X + Re X Bx(t), x(t) X k kL k k k − k h i The estimate (25) combined with 10, implies

1 p ρMT 2 Re XBS(t)x0, S(t)x0 X X B (X) 1 x0 X h i ≤ k kL 1 T p ρM k k − 1 1 p p ρMT ρMT 2 + X B (X) 1 1 + 1 x0 X k kL 1 T p ρM  1 T p ρM  k k − − + Re Bx(t), x(t) , t [0, T]  hX iX ∀ ∈ · Integrating this inequality and using the inequality (24), we deduce that

(k+1)T 2 2 δ S(T)x(kT) X ρC1 x(kT) X Re X Bx(s), x(s) X ds k k − k k ≤ ZkT h i

1 1 MT p ρMT p with C1 = T 1 X B (X) 2 + 1 . 1 ρT p M k kL 1 ρT p M ! − − By using Lemma II.3 we derive

ρδ x((k + 1)T) 2 2ρM x(kT) 2 2C ρ2 x(kT) 2 k kX − ρk kX − 1 k kX ≤ x(kT) 2 x((k + 1)T) 2 , k kX − k kX or equivalently x ((k + 1)T) 2 C x(kT) 2 , k kX ≤ 2k kX

9 2ρ2(δM +C )+1 where C = ρ 1 , which is in (0,1) for ρ 0+. 2 1+ρδ → Hence using again the decreasing of x(t) we deduce the following exponential decay k kX σt x(t) Ke− x , t 0, k kX ≤ k 0k ∀ ≥ 1 (C ) 2 ln 2 where K = (C2)− and σ = − 2T . This estimate extends by density to all x0 X. The result of the > ∈ < < 1 theorem follows, by taking ρ1 0 small enough so that the following constraints hold: 0 ρ1 1 T p M and C (0,1). 2 ∈

III. Examples

Example 1 Let Ω be an open and bounded subset of Rd, d 1, and let us consider the following bilinear ≥ equation of diffusion type

∂ 1 x = ∆x + gx + ν(t)( ∆) 2 x on (0, ∞) ∂t − x(t) = 0 on ∂Ω (0, ∞) (27)  × x(0) = x0 on Ω  where g L∞(Ω), ν is a real valued bilinear control and x(t) = x(ζ, t) L2(Ω) is the state. Let us ∈ ∈ observe that system (27) can be written in the form of (1) if we close it by the switching feedback control ν(t) = ρ1 . The system (27) is an example of fractional equation of diffusion equations type, − t 0 / x(t)=0 and may describe{ ≥ transport6 } processes in complex systems which are slower than the Brownian diffusion. As practical situations displaying such anomalous behaviour, let us mention the charge carrier transport in amorphous semiconductors, the nuclear magnetic resonance diffusometry in percolative and porous media etc (see [3, 10, 13, 12]). Here, we aim to show the exponential stabilization of (27). For this end we will verify the assumptions of Theorem II.4. Let us take the state space X = L2(Ω) (endowed with its 1 natural scalar product , ), and consider the control operator B = ( ∆) 2 and the system’s operator h· ·iX − A = ∆ + gI with D(A) = H2(Ω) H1(Ω). The operator A generates an analytic semigroup S(t) on X ∩ 0 (see [6], p. 107 and p. 176) which is given by the following variation of constants formula:

t S(t)x = S0(t)x + S0(t s)g(ξ)S(s)xds, t 0, Z0 − ≥ where S0(t) is the semigroup generated by A with g = 0. In the sequel, in order to make the computation easier, we restrict our self to the mono-dimension case, thus we consider Ω =(0,1). In this case the semigroup (S0(t)) is is given by

αjt 2 S0(t)x = ∑ e− x, φj X φj, x L (Ω) j 1 h i ∀ ∈ ≥ with α = j2π2, j 1 is the set of eigenvalues of ∆ with the corresponding orthonormal basis of L2(Ω): j ≥ − φj(x) = √2 sin(jπx). Moreover, the semigroup S(t) is a contraction if in addition

2 2 1 Ω g(ξ)y (ξ)dξ y 1 Ω , y H0 ( ). Ω H0 ( ) Z ≤ k k ∀ ∈ Thus, in the sequel we suppose this condition satisﬁed. Then the operator B can be expressed as

1 2 2 Ω Bx = ∑ αj x, φj X φj, x L ( ). j 1 h i ∈ ≥

10 2 2 Here, B is unbounded on L (Ω) and it is bounded from L (Ω) onto the space X 1 defined as the completion 1 − 2 1 2 2 2 of L (Ω) for the norm y = ∑ y, φj , y L (Ω), which can be also interpreted as the dual k k j 1 αj h i ∀ ∈ 1 ≥ space of D(( ∆) 2 ) with respect to the L2(Ω) topology (the space L2(Ω) being the pivot space). Note − − 1 2 1 2 also that the space D(( ∆) 2 ) can be normed with x 1 = ∑ αj x, φj X . − k kD(( ∆) 2 ) |h i | − j 1 ≥ Let us first note that with the feedback control ν(t) = ρ1 , the system (27) admits a unique − t 0 / x(t)=0 global mild solution x(t) (see e.g. [3, 13]). { ≥ 6 } p 1 Let p > 1, T > 0 and let u L (0, T; X). It follows from the fact that D(A 1) = X D(( ∆) 2 ) 1 ∈ T − 1 ⊂ − and ( ∆) 2 (X, X 1), that the X 1 valued integral S 1(T s)( ∆) 2 u(s)ds is well-defined − ∈ L − − − 0 − − − (see [6], Theorem 5.34). Moreover, since the semigroup (S(t))t 0 is analytic, then so is ((S 1(t))t 0. R ≥ − ≥ 1 T T s ∆ 2 This implies that S 1( −2 )( ) u(s) X, s [0, T) (see [6], p. 101). Then we have S 1(T − − ∈ ∀ ∈ 0 − − 1 Z s)( ∆) 2 u(s)ds X, which gives the admissibility of B (see [14], Prop. 3.3 and [17], Lemma. 4.3.9). − ∈ On the other hand, using again the analyticity of the semigroup S(t), we deduce that for u Lp(0, T; X) ∈ we have

t 1 t 1 B S 1(T s)Bu(s)ds = ( ∆) 2 S 1(T s)( ∆) 2 u(s)ds Z0 − − − Z0 − − − 1 t 1 = ( ∆) 2 ( ∆) 2 S(T s)u(s)ds − Z0 − − t = S(T s)( ∆)u(s)ds X Z0 − − ∈ which gives (i). Let us check (ii). For all x L2(Ω), t 0 and j 1, we have ∈ ≥ ≥ t t αj(t s) S0(t s)gS(s)xds, φj X = gS(s)x, e− − φj Xds |hZ0 − i | | Z0 h i | α t 1 e− j g L∞(Ω) x X − . ≤ k k k k αj We deduce that

αjt αjt 1 e− S(t)x, φj X e− x X + g L∞(Ω) − x X, t 0, j 1. (28) |h i | ≤ k k k k αj k k ≥ ≥ Now for any x X, we have ∈ 1 ( ) = 2 ( ) BS t x ∑ αj S t x, φj X φj. j 1 ≥ This combined with (28) implies that BS(t)x X for all x X and for any t > 0, and gives (13). Thus ∈ ∈ the assumption (ii) of Theorem 2 is satisﬁed. Now, using again the series expansion of BS(t)y for y D(A), we get after integrating: ∈ 1 ( ) ( ) = 2 ( ) 2 BS t x, S t x X ∑ αj S t x, φj X h i j 1 ≥ S(t)x 2 ≥ k kX S(T)x 2 , t [0, T] ≥ k kX ∀ ∈

11 hence (iii) is fulﬁlled. According to Theorem II.4, we conclude that for ρ > 0 small enough, the control ν(t) = ρ1 − t 0, x(t)=0 guarantees the uniform exponential stability of the system (27). { ≥ 6 }

Example 2 Consider the following system

∂ (ζ, t) = ∂ x(ζ, t) αx(ζ, t) + ν(t)h(ζ)x(ζ, t) in (0,1) (0, ∞), ∂t ∂ζ − × (S0) x(1, t) = 0 in (0, ∞). x( ,0) = x L2(0,1) · 0 ∈ where X = L2(0,1), α > 0 and h L∞(0,1) is such that h c > 0, for some positive constant c. Here ∈ ≥ we can take A = d αid with domain D(A) := x H1(0,1) : x(1) = 0 . The operator A is the dζ − ∈ generator of contractions semigroup (S(t)t 0) given by ≥ e αtx(ζ + t), if ζ + t 1 − ≤ S(t)x (ζ) =  0, else. 

According to previous theorems, the system (S0) is exponentially stablilizable as here, the semigroup S(t) is a contraction (so that S(t) is decreasing) and the linear bounded operator B := h id satisﬁes the k k 1 observation condition (since h c > 0). Let us now consider the following system ≥

x˙(t) = xζ(t) αx(ζ, t) + ν(t)h(ζ)x(t) in (0,1) (0, ∞) (S1) − × (x(1, t) + ǫψ(x(t)) = 0 in (0, ∞) where ψ : X R is a non null linear functional of X. This may be seen as a perturbed version → of (S0) on its boundary conditions. According to Riesz representation, one can assume that ψ(x) = 1 f (s)x(s)ds, x X for some f X (0). 0 ∀ ∈ ∈ − We aim to show that under small valuers of ǫ > 0, this system is still exponentially stabilizable. R The latter system can be reformulated as:

x˙(t) = x(t) + ν(t)h(ζ)x(t) in (0,1) (0, ∞) (S2) A × (x(0) = x0 in (0,1) where : D( ) X X is deﬁned by: A A ⊂ → x := Ax ǫhx, x D( ) := x H1(0,1), x(1) + ǫψ(x) = 0 . A − ∀ ∈ A ∈ n o We claim that is the generator of a strongly continuous semigroup on X. In order to verify this assertion, A we will consider as a perturbation of the generator A. A In order to write the system (S2) in the form (1), let us consider the function θ(ζ) = 1(ζ) := 1, ζ X, d 1 ∈ which is such that Amθ = 0, θ(1) = 1, where Am := dζ with domain D(Am) := H (0,1). Let us introduce the following operator

Bx = hx ψ(x)A 1θ, x X − − ∀ ∈ which is a one to one operator since we have θ D(A). 6∈ In the sequel, we will verify the assumptions of Theorem II.5 and then conclude the stabilization of the perturbed system (S1). From the boundary conditions of (S ), we can see that • 2 x X, x D( ) x H1(0,1) and x + ǫψ(x)θ D(A) ∀ ∈ ∈ A ⇔ ∈ ∈ ·

12 This together with the deﬁnition of θ implies that for x D( ), we have ∈ A X x = A x ǫhx ∋ A m − = A x + ǫψ(x)θ ǫhx m − = A x + ǫψ(x)θ ǫhx − = A 1 x + ǫψ(x)θ ǫhx − − = A 1x ǫBx − − = A 1 ǫB Xx − − | Moreover, for all x D((A 1 ǫB) X), we have A 1 x + ǫψ(x)θ X, and hence x + ǫψ(x)θ 1 ∈ − − | 1 − ∈ ∈ D(A) H (0,1) which implies that x H (0,1). Then we have (A 1 ǫB Xx = x. In other words, ⊂ ∈ − − | A we have , D( ) = (A 1 ǫB) X, D(A 1 ǫB) X) . A A − − | − − | The operator (A 1 ǫB)X is a generator if we can show that • − − 1 S 1(1 r)ψ(u(r))A 1θdr X, Z0 − − − ∈ or, equivalently 1 1 2 S 1(1 r) (.)ψ(u(r))dr D(A), u L (0,1; X). Z0 − − ∈ ∀ ∈ We have 1 1 1 1 S 1(1 r) (.)ψ(u(r))dr = ψ(u(r))S(1 r) ( )dr Z0 − − Z0 − · 1 α(1 r) = e− − ψ(u(r))dr := g(.). Z· Since ψou L2(0,1), this implies that g H1(0,1) and g(1) = 0. In other words, g D(A). Hence, ∈ ∈ ∈ for ǫ > 0 small enough, the system (S1) is well-posed. Here we can take X 1 = span(A 1θ), so we obtain an admissible decomposition for the pair (A 1, ǫB). • − − − − Indeed, it is clear that Bx = hx, x X, so B is a bounded operator from X to X. X ∈ X Moreover, for all x D((A 1 ǫB) X) = D (A 1 + ǫψ(.)A 1θ) X , we have ∈ − − | − − | A 1x = A 1(x + ǫψ(x)θ) ǫψ(x)A 1θ = A(x + ǫψ(x)θ) ǫψ(x)A 1θ, − − − − − − from which it comes that

(A 1)x = A(x + ǫψ(x)θ), x D((A 1 ǫB) X), X − ∀ ∈ − − | where 2 D((A 1 ǫB) X) = x L (0,1) / x + ǫψ(x)θ D(A) − − | { ∈ ∈ } Then for x D (A 1 ǫB X , we have (A 1 ǫB)x X or equivalently x + ǫψ(x)θ D(A), and ∈ − − | − − ∈ ∈ (A 1)x, x = A(x + ǫψ(x)θ), x hX − i h i = A (x + ǫψ(x)θ), x h m i = A x, x h m i 1 2 = x′(s)x(s)ds α x Z0 − k k ǫ2 f 2 1 k k α x 2 x2(0) ≤ 2 − k k − 2 · 13 1/2 2α Thus the operator (A 1) is dissipative in D((A 1 ǫB) X) for every 0 < ǫ . X − − − | ≤ f Finally, the observation estimate follows from the fact that h c > 0 and that k k for any x X, the • ≥ ∈ mapping t S(t)x is decreasing. 7→ k k > We conclude by Theorem II.5 that for ǫ 0 small enough, the control ν(t) = ǫ1 t 0: x(t)=0 − { ≥ 6 } guarantees the exponentially stabilization of the system (S1).

IV. Conclusion

This paper provided sufﬁcient conditions for exponential stability of a linear system under a Desch-Schapacher perturbation of the dynamic. The main assumptions of sufﬁciency are formu- lated in terms of admissibility and observability assumptions of unbounded linear operators. An explicit decay rate of the stabilized state is provided. The previous research on this problem con- cerned either bounded or Miyadera’s type perturbations [11, 15]. The main stabilization result is further applied to show the uniform exponential stabilization of unbounded bilinear reaction diffusion and transport equations using a bang bang controller.

Conflict of interest

The authors declare that they have no conﬂict of interest.

References

[1] Adler, M., Bombieri, M., & Engel, K. J. (2014). On Perturbations of Generators of-Semigroups. In Abstract and Applied Analysis (Vol. 2014). Hindawi.

[2] Ammari, K., El Alaoui, S., & Ouzahra, M. (2020). Feedback stabilization of linear and bilinear unbounded systems in Banach space. arXiv e-prints, arXiv-2006.

[3] Berrahmoune, L. (2009). A note on admissibility for unbounded bilinear control systems. Bulletin of the Belgian Mathematical Society-Simon Stevin, 16(2), 193-204.

[4] Boulouz, A., Bounit, H., & Hadd, S. (2021). Well-posedness and exponential stability of boundary control systems with dynamic boundary conditions. Systems & Control Letters, 147, p. 104825.

[5] Desch, W., & Schappacher, W. (1989). Some generation results for pertubed semigroup, Semi- group Theory and Applications (Clémnet, Invernizzi, Mitidieri, and Vrabie, eds.). Lect. Notes Pure Appl. Math, 116, 125-152.

[6] Engel, K. J., & Nagel, R. (2001, June). One-parameter semigroups for linear evolution equations. In Semigroup forum (Vol. 63, No. 2). Springer-Verlag.

[7] G. Greiner, Perturbing the boundary conditions of a generator, Houston Journal of Mathemat- ics, 13 (1987), 213–229.

[8] Hadd, S., Manzo, R., & Rhandi, A. (2015). Unbounded perturbations of the generator domain. Discrete & Continuous Dynamical Systems-A, 35(2), 703.

14 [9] Idrissi, A. (2003). On the unboundedness of control operators for bilinear systems. Quaes- tiones Mathematicae, 26(1), 105-123.

[10] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2003). Fractional differential equations: A emergent ﬁeld in applied and mathematical sciences. In Factorization, singular operators and related problems (pp. 151-173). Springer, Dordrecht.

[11] Liu, K., Liu, Z., & Rao, B. (2001). Exponential stability of an abstract nondissipative linear system. SIAM journal on control and optimization, 40(1), 149-165.

[12] Metzler, R., & Klafter, J. (2000). The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1), 1-77.

[13] Maragh, F., Bounit, H., Fadili, A., & Hammouri, H. (2014). On the admissible control operators for linear and bilinear systems and the Favard spaces. Bulletin of the Belgian Mathematical Society-Simon Stevin, 21(4), 711-732.

[14] Nagel, R., and E. Sinestrari. Inhomogeneous Volterra integrodifferential equations for Hille- Yosida operators, Functional Analysis Proc. Essen Conference (KD Bierstedt and A. Pietsch and WM Ruess and D. Vogt, eds.). (1994): 51-70.

[15] Ouzahra, M. (2017). Exponential stability of nondissipative linear system in Banach space and application to unbounded bilinear systems. Systems & Control Letters, 109, 53-62.

[16] A. Pazy, Semi-groups of linear operators and applications to partial differential equations, Springer Verlag, New York, 1983.

[17] Van Neerven, J. (1992). The adjoint semigroup. In The Adjoint of a Semigroup of Linear Operators (pp. 1-18). Springer, Berlin, Heidelberg.

[18] Weiss, G. (1989). Admissibility of unbounded control operators. SIAM Journal on Control and Optimization, 27(3), 527-545.