Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Abstract

Let 𝐴 be a generator of an exponentially stable operator semigroup in a Banach space, and let 𝐶(𝑡) (𝑡≥0) be a linear bounded variable operator. Assuming that ∫^𝑡₀𝐶(𝑠)𝑑𝑠 is sufficiently small in a certain sense for the equation 𝑑𝑥/𝑑𝑡=𝐴𝑥+𝐶(𝑡)𝑥, we derive exponential stability conditions. Besides, we do not require that for each 𝑡₀≥0, the “frozen” autonomous equation 𝑑𝑥/𝑑𝑡=𝐴𝑥+𝐶(𝑡₀)𝑥 is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.

1. Introduction and Statement of the Main Result

In this paper, we investigate stability of linear nonautonomous equations in a Banach space, which can be considered as integrally small perturbations of autonomous equations. The stability theory of evolution equations in a Banach space is well developed, compare and confare with [1] and references therein, but the problem of stability analysis of evolution equations continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution. One of the basic methods for the stability analysis is the direct Lyapunov method. By that method, many strong results were established, compare and confare with [2, 3]. But finding the Lyapunov functionals is usually a difficult mathematical problem. A fundamental approach to the stability of diffusion parabolic equations is the method of upper and lower solutions. A systematical treatment of that approach is given in [4]. In [5], stability conditions are established by a normalizing mapping. Note that a normalizing mapping enables us to use more complete information about the equation than a usual (number) norm. In [6], the “freezing” method for ordinary differential equations is extended to equations in a Banach space. About the recent results, see the interesting papers [7–11]. In particular, in [7] the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces is investigated. In the paper [8], a Rolewicz’s type theorem of in-solid function spaces is proved. Dragan and Morozan [9] established criteria for exponential stability of linear differential equations on ordered Banach spaces. Paper [10] deals with the stability and controllability of hyperbolic type abstract evolution equations. Pucci and Serrin [11] investigated the asymptotic stability for nonautonomous wave equations.

Certainly, we could not survey the whole subject here and refer the reader to the previously listed publications and references given therein.

Let 𝑋 be a complex Banach space with a norm ‖⋅‖_𝑋 and the unit operator 𝐼. For a bounded operator 𝐾, ‖𝐾‖ is the operator norm.

Everywhere below 𝐴 is a linear operator in 𝑋 with a domain Dom(𝐴), generating a strongly continuous semigroup 𝑇(𝑡); that is, 𝐴=lim_ℎ↓0(1/ℎ)(𝑇(ℎ)−𝐼) in the strong topology, and 𝐶(𝑡)(𝑡≥0) is a linear bounded variable operator mapping Dom(𝐴) into itself. Put 𝐵(𝑡)=𝐴+𝐶(𝑡). In the present paper, we establish stability conditions for the equation as follows: 𝑑𝑢(𝑡)𝑑𝑡=𝐵𝑢,𝑡≥0. (1) It should be noted that in the previously pointed papers it is assumed that for each 𝑡₀≥0, the “frozen” autonomous equation 𝑑𝑥/𝑑𝑡=𝐵(𝑡₀)𝑥 is stable. We do not require that condition. The aim of this paper is to generalize the main result from the paper [12], which deals with finite dimensional equations.

A solution of (1), for a given 𝑢₀∈Dom(𝐴) is a function 𝑢∶[0.∞)→Dom(𝐴) having a strong derivative, satisfying (1) and 𝑢(0)=𝑢₀. We will investigate (1) as a perturbation of the following equation: 𝑑𝑣𝑑𝑡=𝐴𝑣,𝑡≥0. (2) Put 𝐽(𝑡)∶=^𝑡₀𝐶(𝑠)𝑚(𝑡)‖(𝑡)(𝑡)𝐵(𝑡,().𝑑𝑠,∶=𝐴𝐽−𝐽)‖𝑡≥0 (3) We say that (1) is exponentially stable if there is an 𝛼=const>0, such that ‖𝑢(𝑡)‖_𝑋≤𝑒^−𝛼𝑡‖𝑢(0)‖_𝑋  (𝑡≥0) for any solution 𝑢(𝑡) with 𝑢(0)∈Dom(𝐴). Now we are in a position to formulate the main result of the paper.

Theorem 1. Let ‖𝑇‖_𝐿¹∶=^∞₀‖𝑇(𝑡)‖𝑑𝑡<∞,sup_𝑡≥0‖𝐽(𝑡)‖+^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑑𝑠<1. (4)(5) Then, (1) is exponentially stable.

The proof of this theorem is divided into a series of lemmas which are presented in the next section. To the best of our knowledge, Theorem 1 is new even in the case of bounded operators. In Section 3 we consider particular cases of Theorem 1. In Section 4, the previously pointed results are applied to a partial differential equation. For the brevity, we restrict ourselves by a scalar equation with the periodic boundary condition, but our results enable us to consider coupled systems of equations and other boundary conditions, for example, the Dirichlet condition.

2. Proofs

We need the following simple result.

Lemma 2. Let 𝑤(𝑡),𝑓(𝑡), and 𝑣(𝑡)(0≤𝑡≤𝑎≤∞) be functions whose values are bounded linear operators. Assume that 𝑤(𝑡) is integrable and 𝑓(𝑡) and 𝑣(𝑡) have integrable derivatives on [0,𝑎]. Then, with the notation 𝑗_𝑤(𝑡)=∫^𝑡₀𝑤(𝑠)𝑑𝑠, one has ^𝑡₀𝑓(𝑠)𝑤(𝑠)𝑣(𝑠)(𝑡)𝑗𝑑𝑠=𝑓_𝑤(𝑡)𝑣(𝑡)−^𝑡₀𝑓^′(𝑠)𝑗_𝑤(𝑠)𝑣(𝑠)+𝑓(𝑠)𝑗_𝑤(𝑠)𝑣^′(𝑠)𝑑𝑠,(𝑡≤𝑎). (6)

Proof. Clearly, 𝑑𝑓(𝑡)𝑗𝑑𝑡^𝑤(𝑡)𝑣(𝑡)=𝑓^′(𝑡)𝑗_𝑤(𝑡)𝑣(𝑡)+𝑓(𝑡)𝑤(𝑡)𝑣(𝑡)+𝑓(𝑡)𝑗_𝑤(𝑡)𝑣^′(𝑡). (7) Integrating this equality and taking into account that 𝑗_𝑤(0)=0, we arrive at the required result.

Let 𝑉(𝑡) be the Cauchy operator to (1); that is, 𝑉(𝑡)𝑢(0)=𝑢(𝑡) for a solution 𝑢(𝑡) of (1).

Lemma 3. One has ((𝑡𝑉(𝑡)(𝑡)+𝐼−𝐽))=𝑇^𝑡₀𝑇(𝑡−𝑠)[𝐴𝐽(𝑠)−𝐽(𝑠)𝐵(𝑠)]𝑉(𝑠)𝑑𝑠. (8)

Proof. As it is well known, 𝑉(𝑡)−𝑇(𝑡)=^𝑡₀𝑇(𝑡−𝑠)𝐶(𝑠)𝑉(𝑠)𝑑𝑠, (9) compare and confare with [13]. Thanks to the previous lemma, one has ^𝑡₀𝑇()𝐶(𝑠)𝑉(𝑠)(0)𝐽(𝑡)𝑉(𝑡)−𝑡−𝑠𝑑𝑠=𝑇^𝑡₀𝑑𝑇(𝑡−𝑠)𝐽(𝑠)𝑉(𝑠)()𝐽(𝑠)𝑉𝑑𝑠+𝑇𝑡−𝑠^′(𝑠)𝑑𝑠. (10) But 𝑑𝑇(𝑡−𝑠)/𝑑𝑠=−𝐴𝑇(𝑡−𝑠). In addition, 𝑉^′(𝑠)=𝐵(𝑠)𝑉(𝑠). Thus, ^𝑡₀𝑇()𝐶(𝑠)𝑉(𝑠)(𝑡)𝑉(𝑡)+𝑡−𝑠𝑑𝑠=𝐽^𝑡₀𝑇(𝑡−𝑠)[𝐴𝐽(𝑠)−𝐽(𝑠)𝐵(𝑠)]𝑉(𝑠)𝑑𝑠. (11) Now, (9) implies the required result.

Let, 𝜁(𝑡)∶=inf_{‖ℎ‖ℎ∈𝑋;=1}‖(𝐽(𝑡)−𝐼)ℎ‖. (12)

Lemma 4. Let condition inf_𝑡≥0𝜁(𝑡)>0 (13) hold. Then, ‖𝑉(𝑡)‖≤𝑧(𝑡), 𝑡≥0, where 𝑧(𝑡) is a solution of the following equation: 𝑧(𝑡)=1𝜁(𝑡)×‖𝑇(𝑡)‖+^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑧(𝑠)𝑑𝑠,𝑡≥0. (14)

Proof. Thanks to the previous lemma, (𝑡𝑉(𝑡≤‖𝑇(𝑡+‖(𝐼−𝐽)))‖)‖^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)‖𝑉(𝑠)‖𝑑𝑠. (15) Hence 𝜁(𝑡𝑉(𝑡≤‖𝑇(𝑡+)‖)‖)‖^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)‖𝑉(𝑠)‖𝑑𝑠. (16) Then by the well-known (comparison) Lemma 3.2.1 from [14] we have the required result.

Let 𝜂₀∶=sup_𝑡≥01𝜁(𝑡)^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑑𝑠<1. (17) Then (14) implies sup_𝑡𝑧(𝑡)≤sup_𝑡≥0‖𝑇(𝑡)‖𝜁(𝑡)+sup_𝑡𝑧(𝑡)𝜂₀. (18) Due to the previous lemma we get the following.

Lemma 5. Let conditions (13) and (17) hold. Then sup_𝑡≥0‖𝑉(𝑡)‖≤sup_𝑡≥0‖𝑇(𝑡)‖1−𝜂₀𝜁(𝑡). (19)

Proof of Theorem 1. Assume that 𝑗(𝑡)∶=‖𝐽(𝑡)‖≤𝑞<1,𝑞=const;𝑡≥0, (20) then 𝜁(𝑡)≥1−𝑗(𝑡). If 𝜂₁∶=sup_𝑡≥011−𝑗(𝑡)^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑑𝑠<1, (21) then 𝜂₀≤𝜂₁<1 and thanks to the previous lemma, (1) is stable. But condition (5) implies that 𝑗(𝑡)+^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑑𝑠<1 (22) or 11−𝑗(𝑡)^𝑡₀‖𝑇(𝑡−𝑠)‖𝑚(𝑠)𝑑𝑠<1,(𝑡≥0). (23) Thus (5) implies the inequality 𝜂₁<1, and therefore, from (5), condition (21) follows. This proves the stability. To prove the exponential stability we use the well-known Theorem 4.1 [13, p. 116] (see also Theorem 2.44 [1, p. 49]). It asserts that the finiteness of the 𝐿¹-norm of 𝑇 implies the inequality ‖𝑇(𝑡)‖≤𝑀𝑒^−𝛼𝑡,𝑡≥0, (24) where 𝑀=const≥1, 𝛼=const>0. So for 0<𝜖<𝛼, the semigroup 𝑇_𝜖(𝑡) generated by 𝐴+𝐼𝜖 satisfies the inequality ‖𝑇_𝜖(𝑡)‖≤𝑀𝑒^{−(𝛼−𝜖)𝑡}, 𝑡≥0, and therefore it also has a finite 𝐿¹-norm. Substitute the equality 𝑢(𝑡)=𝑦(𝑡)𝑒^−𝜖𝑡 (25) into (1). Then we obtain the equation 𝑦̇=(𝐵(𝑡)+𝐼𝜖)𝑦. (26) Denote the Cauchy operator of (26) by 𝑉_𝜖(𝑡). Repeating our above arguments with 𝑉_𝜖(𝑡) instead of 𝑉(𝑡) and the equation 𝑥̇=(𝐴+𝐼𝜖)𝑥 instead of (2), due to Lemma 5 we can assert that 𝑉_𝜖(𝑡) is bounded. Now (25) implies ‖𝑉(𝑡)‖≤𝑒^−𝜖𝑡sup_𝑡≥0𝑉_𝜖(𝑡),𝑡≥0. (27) This proves the theorem.

3. A particular Case of Theorem 1

To illustrate Theorem 1, consider the following equation: 𝑑𝑢(𝑡)𝐶𝑑𝑡=𝐴𝑢+𝑐₀𝑢, (28) where 𝐶₀ is a constant operator and 𝑐(𝑡) is a scalar real piece-wise continuous function bounded on [0,∞). So, 𝐶(𝑡)=𝑐(𝑡)𝐶₀. Without any loss of generality, assume that sup_𝑡|𝑐(𝑡)|=1, (29) and with the notation 𝑖_𝑐(𝑡)=^𝑡₀𝑐(𝑠)𝑑𝑠, (30) we obtain 𝑚(𝑡)=‖𝐴𝐽(𝑡)−𝐽(𝑡)𝐵(𝑡)‖≤𝑖_𝑐(𝑡)𝐴𝐶₀−𝐶₀𝐴+𝑐(𝑡)𝐶₀≤𝑖_𝑐(𝑡)𝐴𝐶₀−𝐶₀𝐴+|𝑐(𝑡)|𝐶²₀≤𝑖_𝑐(𝑡)𝐴𝐶₀−𝐶₀𝐴+𝐶²₀. (31) Due to (24), ^𝑡₀‖𝑇(𝑡−𝑠)‖𝑑𝑠≤𝑀^𝑡₀𝑒^−𝛼𝑠𝑀𝑑𝑠≤𝛼,(𝑡≥0). (32) Thus, denoting 𝜃₀=sup_𝑡^𝑡₀𝑐(𝑠)𝑑𝑠, (33) due to Theorem 1, we arrive at the following result.

Corollary 6. If the inequality 𝜃₀𝐶₀+𝑀𝛼𝐴𝐶₀−𝐶₀𝐴+𝐶²₀<1 (34) holds, then (28) is exponentially stable.

For example, let 𝑐(𝑡)=sin(𝜔𝑡)  (𝜔>0). Then, 𝑖_𝑐(𝑡)≤2/𝜔 and 𝑚(𝑡)≤2𝜔𝐴𝐶₀−𝐶₀𝐴+𝐶²₀. (35) Thus, (34) takes the following form: 𝐶₀+𝑀𝛼𝐴𝐶₀−𝐶₀𝐴+𝐶²₀<𝜔2. (36)

4. Equations with Periodic Boundary Conditions

Consider the problem 𝜕𝑢(𝑥,𝑡)=()𝜕𝑡𝜕𝑢𝑥,𝑡+𝜕𝑥−𝑏₀(𝑡)𝑎(𝑥𝑢(),(),𝑢()(),(),+𝑐)𝑥,𝑡0≤𝑥≤10,𝑡=𝑢1,𝑡𝑡≥0 (37)(38) with a positive constant 𝑏₀ and a real differentiable function 𝑎(𝑥); 𝑐(𝑡) is the same as in the previous section.

Take 𝑋=𝐿²(0,1), where 𝐿²=𝐿²(0,1) is the Hilbert space of real functions 𝑓,ℎ defined on [0,1] with the scalar product 𝑓,ℎ=¹₀𝑓(𝑥)ℎ(𝑥)𝑑𝑥, (39) and the norm ‖ℎ‖=(ℎ,ℎ). Set Dom(𝐴)=𝑓∈𝐿²∶𝑓^′∈𝐿²(0)(1(𝑥)=(𝑥);𝑓=𝑓),𝐴𝑢𝑑𝑢𝑑𝑥−𝑏₀𝑢(𝑥)(𝑢∈Dom(𝐴)), (40)(41) then we have (𝐴𝑢,𝑢)=¹₀𝑑𝑢(𝑥)𝑑𝑥−𝑏₀𝑢(𝑥)𝑢(𝑥)=𝑑𝑥₁₀𝑑𝑢(𝑥)𝑑𝑥−𝑏₀𝑢(𝑥)𝑢(𝑥)=𝑑𝑥₁₀12𝑑𝑢²(𝑥)𝑑𝑥−𝑏₀𝑢²(𝑥)=1𝑑𝑥2𝑢²(1)−𝑢²(0)−𝑏₀¹₀𝑢²(𝑥)𝑑𝑥=−𝑏₀¹₀𝑢²(𝑥)𝑑𝑥. (42) Let 𝑣=𝑣(𝑡,𝑥) be a solution of (2) with 𝐴 defined by (41). Then, we obtain 𝑑()=(̇)+(𝑣)̇=()+()()𝑑𝑡𝑣,𝑣𝑣,𝑣𝑣,𝐴𝑣,𝑣𝑣,𝐴𝑣=2𝐴𝑣,𝑣≤−2𝑏₀(𝑣,𝑣). (43) Hence, (𝑑/𝑑𝑡)‖𝑣‖≤−𝑏₀‖𝑣‖. Thus, ‖𝑇(𝑡)‖≤𝑒^−𝑏0𝑡. In addition, 𝐶₀𝑢(𝑥)=𝑎(𝑥)𝑢(𝑥), 𝐴𝐶₀−𝐶₀𝐴𝑢(𝑥)=𝑑(𝑎(𝑥)𝑢(𝑥))(𝑥)(𝑥)𝑑𝑥−𝑎𝑑𝑢𝑑𝑥=𝑎^′(𝑥)𝑢(𝑥),(𝑢∈Dom(𝐴)). (44) Due to condition (34), we obtain the following.

Corollary 7. If the inequality 𝜃₀|𝑎(𝑥+1)|𝑏₀𝑎^′(𝑥)+|𝑎(𝑥)|²<1,(0≤𝑥≤1) (45) holds, then (37) is exponentially stable.

For example, let 𝑐(𝑡)=sin(𝜔𝑡)(𝜔>0). Then 𝑖_𝑐(𝑡)≤2/𝜔 and (45) takes the form |𝑎(𝑥+1)|𝑏₀𝑎^′(𝑥)+|𝑎(𝑥)|²<𝜔2,(0≤𝑥≤1). (46)