Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 76, pp. 1–8. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.76 DELAY-DEPENDENT STABILITY CONDITIONS FOR DELAY DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATORS IN BANACH SPACES MICHAEL GIL’ Abstract. We consider the equation du(t)/dt = Au(t) + Bu(t − h) where t > 0, h is a positive constant, and A is a linear unbounded and B is a linear bounded operators. We establish explicit delay-dependent conditions for exponential stability, and present applications to partial integro-differential equations with delay. 1. Introduction and statement of the main result In this article we suggest delay-dependent stability conditions for delay differen- tial equations with unbound operators in a Banach space. The basic method for the stability analysis of functional differential equations is the Lyapunov-Krasovskij method [4, 13]. By that method, many results have been obtained. Recently, that method has been extended to functional differential equations in a Hilbert space, see [1, 6, 14, 15] and references given therein. In [8, 10] the delay-dependent stability conditions for equations in a Banach space with bounded operators have been derived. To the best of our knowledge, the delay- dependent stability conditions for equations in a Banach space with unbounded operators are not investigated in the available literature. It should be noted that finding the Lyapunov-Krasovskij type functionals or solving the corresponding operator inequalities are often connected with serious mathematical difficulties, To the contrary, the stability conditions presented in this paper are explicitly formulated in terms of the coefficients and delays. The literature on the delay- dependent stability criteria is rather rich, but mainly equations in a finite dimen- sional space are considered, see [2, 3, 13]. Everywhere below, X is a complex Banach space with a norm ∥ · ∥X = ∥ · ∥ and the unit operator IX = I. By B(X ), we denote the set of all bounded linear operators in X . For a linear operator T , σ(T ) is the spectrum and ∥T∥X = ∥T∥ is the operator norm of T if it is bounded. 2020 Mathematics Subject Classification. 34K30, 34K06, 34K20. Key words and phrases. Banach space; delay differential equation; stability; integro-differential equation. ©2024. This work is licensed under a CC BY 4.0 license. Submitted May 16, 2024. Published November 26, 2024. 1 2 M. GIL’ EJDE-2024/76 Furthermore, C(J,X ) is the space of X -valued functions f defined and continuous on a finite or infinite real segment J, and equipped with the finite norm. ∥f∥C(J) = ∥f∥C(J,X ) := sup t∈J ∥f(t)∥X . In addition, W (J,X ) is the space of X -valued functions f defined and strongly continuously differentiable on J, and equipped with the norm ∥f∥W (J) = ∥f∥W (J,X ) := max{sup t∈J ∥f ′(t)∥X , sup t∈J ∥f(t)∥X }. Denote also R+ = [0,∞) and Rη = [−η,∞) for a finite η > 0. Throughout this article A is a closed linear operator with a dense domainD(A) ⊆ X , generating a strongly continuous semigroup eAt on X , and B ∈ B(X ) maps X into D(A). Our main object is to study the equation y′(t) = Ay(t) +By(t− h) (t > 0; 0 < h = const.∞) (1.1) with the initial condition y(t) = ϕ(t) (−h ≤ t ≤ 0), (1.2) where ϕ ∈W ([−h, 0],X ) ∩D(A) is given. Various integro-differential equations with differential operators A and integral operators B are examples of (1.1). A solution of problem (1.1), (1.2) is defined as a continuous function y(t) defined on Rη with values in D(A), having a continuous derivative for all t > 0 and the right derivative at zero, and satisfying (1.1), and (1.2). Let ∫ ∞ 0 ∥eAs∥Xds <∞. (1.3) Since AB is defined on the whole X , due to the Banach theorem [12, Section 2] AB is bounded, and consequently, ψA := ∫ ∞ 0 ∥eAsAB∥Xds <∞. In addition, put M = A+B and assume that∫ ∞ 0 ∥eMs∥Xds <∞. (1.4) Therefore ψM := ∫ ∞ 0 ∥eMsB∥Xds <∞. Now we are in a position to formulate the main result of the paper. Theorem 1.1. Let conditions (1.3),(1.4) and hψM (ψA + ∥B∥X ) < 1 (1.5) hold. Then problem (1.1), (1.2) with ϕ ∈ W (−h, 0) ∩D(A) has a unique solution y(t), which satisfies the inequality ∥y∥C(R+) ≤ c0∥ϕ∥W (−h,0), where the constant c0 ≥ 1 does not depend on ϕ. EJDE-2024/76 DELAY-DEPENDENT STABILITY CONDITIONS 3 The proof of this theorem is presented in the next section. Theorem 1.1 gives us the conditions for the Lyapunov stability with respect to W (−h, 0). We will say that (1.1) is exponentially stable with respect to W (−h, 0), if there are constants α > 0 and c1 ≥ 1 independent of ϕ ∈W (−h, 0), such that ∥y(t)∥X ≤ c1e −αt∥ϕ∥W (−h,0) (t ≥ 0) for any solution of (1.1), (1.2). Assume that the semigroups eAt and eMt are exponentially stable: ∥eAt∥X ≤ cAe −αAt and ∥eMt∥X ≤ cMe −αM t, (1.6) where t ≥ 0, αA > 0, αM > 0, cA ≥ 1, cM ≥ 1. Then ψA ≤ ∥AB∥X ∫ ∞ 0 cAe −αAtdt = cA∥AB∥X /αA, ψM ≤ cM∥B∥X /αM . So (1.5) is provided by the inequality hcM∥B∥X αM (cA∥AB∥X αA + ∥B∥X ) < 1. (1.7) Now Theorem 1.1 implies ∥y∥C(R+) ≤ c2∥ϕ∥W (−h,0), (1.8) where c2 does not depend on ϕ. In the next section we also show that Theorem 1.1 implies the following result. Corollary 1.2. Let conditions (1.6) and (1.7) hold. Then (1.1) is exponentially stable with respect to W (−h, 0). This corollary is sharp: if B = 0, then its conditions are necessary for the exponential stability. Moreover, its conditions are necessary if h = 0 and A = 0. 2. Proofs of Theorem 1.1 and Corollary 1.2 Proof of Theorem 1.1. According to [9, Theorem 1], problem (1.1), (1.2) has a unique differentiable solution y(t). Since y(t) ∈ D(A), by the variation of con- stants formula [5, Sect. III.1], (1.1) is equivalent to the equation y(t) = eAtϕ(0) + ∫ t 0 eA(t−s)By(s− h)ds. Consequently, in view of (1.1), dy(t)/dt = Ay(t) +By(t− h) = A(eAtϕ(0) + ∫ t 0 eA(t−s)By(s− h)ds) +By(t− h). Since AB is bounded the integral ∫ t 0 eA(t−s)ABy(s − h)ds (0 < t < ∞) converges and A ∫ t 0 eA(t−s)By(s− h)ds = ∫ t 0 eA(t−s)ABy(s− h)ds. Thus, (1.1) can be written as dy dt = AeAtϕ(0) + ∫ t 0 eA(t−s)ABy(s− h)ds+By(t− h). Hence, with the notation |y|t := sup 0≤s≤t ∥y(s)∥X (0 < t <∞) and a0 := sup t≥0 ∥eAt∥X , 4 M. GIL’ EJDE-2024/76 we can write |y′|t ≤ a0∥Aϕ(0)∥X + ∫ t 0 ∥eA(t−s)ABy(s− h)∥Xds+ ∥B∥X sup 0≤s≤t ∥y(s− h)∥X , and therefore |y′|t ≤ a0∥Aϕ(0)∥X + (ψA + ∥B∥X ) sup 0≤s≤t ∥y(s− h)∥X , i. e. |y′|t ≤ a0∥Aϕ(0)∥X + (ψA + ∥B∥X )(∥ϕ∥C(−h,0) + |y|t). (2.1) From (1.1) and (1.2) it follows that ϕ′(0) = Aϕ(0) +Bϕ(−h). Hence, ∥Aϕ(0)∥X ≤ (1 + ∥B∥X )∥ϕ∥W (−h,0). Now (2.1) yields |y′|t ≤ a0(1 + ∥B∥X )∥ϕ∥W (−h,0) + (ψA + ∥B∥X )∥ϕ∥C(−h,0) + (ψA + ∥B∥X )|y|t and thus |y′|t ≤ ĉ∥ϕ∥W (−h,0) + (ψA + ∥B∥X )|y|t, (2.2) where ĉ = a0(1 + ∥B∥X ) + ψA + ∥B∥X . Furthermore, we rewrite (1.1) as y′(t) =My(t) +B(y(t− h)− y(t)) (t > 0). (2.3) Recall that M = A+B. from the above mentioned variation of constants formula, y(t) = eMtϕ(0) + ∫ t 0 eM(t−s)B(y(s− h)− y(s))ds. Hence, |y|t ≤ m0∥ϕ(0)∥X + ∫ t 0 ∥eM(t−s)B∥Xds sup s≤t ∥y(s− h)− y(s)∥X , (2.4) where m0 := supt≥0 ∥eMt∥X , and therefore |y|t ≤ m0∥ϕ(0)∥X + ψM sup 0≤s≤t ∥y(s− h)− y(s)∥X . (2.5) Note that ∥y(s− h)− y(s)∥X = ∥ ∫ s s−h y′(s1)ds1∥X ≤ h∥y′∥C(−h,t) ≤ h∥ϕ′∥C(−h,0) + h|y′|t (s ≤ t). Using (2.5), we arrive at the inequality |y|t ≤ m0∥ϕ(0)∥X + ψMh(∥ϕ′∥C(−h,0) + |y′|t). Now (2.2) implies |y|t ≤ ∥ϕ∥W (−h,0)(m0 + ψMh+ hψM ĉ) + hψM (ψA + ∥B∥X )|y|t, or |y|t ≤ ĉ2∥ϕ∥W (−h,0) + hψM (ψA + ∥B∥X )|y|t, EJDE-2024/76 DELAY-DEPENDENT STABILITY CONDITIONS 5 where ĉ2 = m0 + ψMh+ hψM ĉ. According (1.5) we obtain |y|t ≤ ĉ2∥ϕ∥W (−h,0)(1− hψM (ψA + ∥B∥X ))−1. Hence, letting t→ ∞, we obtain ∥y∥C(R+) ≤ (1− hψM (ψA + ∥B∥X ))−1ĉ2∥ϕ∥W (−h,0). This proves the required result. □ Proof of Corollary 1.2. Substitute y(t) = e−ϵtyϵ(t) (2.6) with ϵ > 0 into (1.1). We obtain the equation y′ϵ(t) = (A+ ϵI)yϵ(t) +Beϵhyϵ(t− h) (t > 0). (2.7) Put M(ϵ) = A+ ϵI +Beϵh. We have eM(ϵ)t − eMt = ∫ t 0 eM(t−s)(M(ϵ)−M)eM(ϵ)sds = − ∫ t 0 eM(t−s)(ϵI +B(eϵh − 1))eM(ϵ)sds. Hence, ∥eM(ϵ)t∥ ≤ ∥eMt∥+ ∫ t 0 ∥eM(t−s)∥δ(ϵ)∥eM(ϵ)s∥ds ≤ e−αM t + δ(ϵ) ∫ t 0 e−αM (t−s)∥eM(ϵ)s∥ds, where δ(ϵ) = ∥ϵI +B(eϵh − 1)∥ → 0 as ϵ→ 0. Thus we obtain ∥e(M(ϵ)+αMI)t∥ ≤ 1 + δ(ϵ) ∫ t 0 ∥e(M(ϵ)+αMI)s∥ds. Now the Gronwall lemma yields ∥eM(ϵ)t∥ ≤ e−αM (ϵ)t, where αM (ϵ) = αM − δ(ϵ). So αM (0) = αM . If (1.6), (1.7) hold, then for small enough ϵ > 0, hcMe ϵh∥B∥ αM (ϵ) (cAeϵh∥AB + ϵB∥ αA − ϵ + eϵh∥B∥ ) < 1. From inequality (1.8), which follows from Theorem 1.1, a solution of (2.7) with the initial function ϕ ∈ W (−h, 0) satisfies the inequality ∥yϵ∥C(R+) ≤ cϵ∥ϕ∥W (−h,0), where cϵ does not depend on ϕ. Hence, (2.6) yields ∥y(t)∥C(R+) ≤ cϵe −ϵt∥ϕ∥W (−h,0) (t ≥ 0). This proves the exponential stability. □ 6 M. GIL’ EJDE-2024/76 3. Example In this section X = L2(0, 1), where L2(0, 1) = L2 is the traditional Hilbert space of complex-valued functions defined on [0, 1] with the scalar product (f, f1) = ∫ 1 0 f(x)f1(x)dx (f, f1 ∈ L2). We consider the equation ∂u(t, x) ∂t = ∂2u(t, x) ∂x2 + b(x)u(t, x) + ∫ 1 0 K(x, s)u(t− h, s)ds (3.1) for 0 ≤ x ≤ 1 and t ≥ 0, where b(x) is a complex valued function defined and bounded on [0, 1]; K(x, s) is defined on [0, 1]×[0, 1], twice continuously differentiable in x ∈ [0, 1], and bounded and measurable in s, and K(0, s) = K(1, s) = 0 (s ∈ [0, 1]). We consider the boundary conditions u(0, t) = u(1, t) = 0 (t ≥ 0). (3.2) We will consider problem (3.1), (3.2) in L2(0, 1) with D(A) = {f ∈ L2(0, 1) : f ′′ ∈ L2(0, 1), f(0) = f(1) = 0}, A and B are defined by (Af)(x) = d2f(x) dx2 + b(x)f(x) (f ∈ D(A)), (Bf)(x) = ∫ 1 0 K(x, s)f(s)ds (f ∈ L2). Thus B maps L2(0, 1) into D(A). Also (ABf)(x) = ∫ 1 0 [K ′′(x, s) + b(x)K(x, s)]f(s)ds (f ∈ L2(0, 1)). Simple calculations show that the largest eigenvalue of the self-adjoint operator d2 dx2 on D(A) is −π2 and sup f∈D(A) Re(Af, f)/(f, f) ≤ ν̂A := −π2 + sup x Re b(x). Note that the function w(t) = eAtw(0) with w(0) ∈ D(A) satisfies d dt (w(t), w(t)) = (w′(t), w(t)) + (w(t), w′(t)) = (Aw(t), w(t)) + (w(t), Aw(t)) ≤ 2ν̂A(w(t), w(t)). Hence d dt ∥w(t)∥ ≤ ν̂A∥w(t)∥ and therefore ∥eAt∥ ≤ eν̂At (t ≥ 0). (3.3) With ν̂A = −π2 + sup x Re b(x) < 0 EJDE-2024/76 DELAY-DEPENDENT STABILITY CONDITIONS 7 we have ψA = ∫ ∞ 0 ∥eAtAB∥dt ≤ ∥AB∥/|ν̂A|. (3.4) Furthermore, with M = A+B, we obtain sup f∈D(A) Re(Mf, f)/(f, f) ≤ ν̂M := ν̂A + ν̂B where ν̂B := 1 2 sup f∈L2(0,1) ((B +B∗)f, f)/(f, f) <∞, where B∗ is the adjoint of B, i.e. ν̂B is the largest eigenvalue of the self-adjoint operator (B +B∗)/2. With ν̂M < 0 similarly to (3.3) and (3.4) we have ∥eMt∥L2 ≤ eν̂M t (t ≥ 0), ψM = ∫ ∞ 0 ∥eMtB∥dt ≤ ∥B∥L2/|ν̂M |. (3.5) According to (3.3) and (3.5) cA = cM = 1. Using Corollary 1.2, we arrive at the following result. Theorem 3.1. Let ν̂A < 0, ν̂M < 0 and h∥B∥L2 ν̂M (∥AB∥L2 ν̂A + ∥B∥L2 ) < 1 . Then (3.1), (3.2) is exponentially stable with respect to W (−h, 0). Acknowledgments. The author is very grateful to the anonymous referee for his/her helpful remarks. References [1] O. Arino, M.L. Hbid, E.H. Dads (eds.); Delay Differential Equations and Applications, Springer, Dordrecht, The Netherlands, 2006. [2] L. Berezansky, E.Braverman; On exponential stability of linear delay equations with oscilla- tory coefficients and kernels. Differential and Integral Equations, 35, no. 9-10 (2022), 559–580 . [3] L. Berezansky, J. Diblik, Z. Svoboda, Z. Šmarda; Simple tests for uniform exponential sta- bility of a linear delayed vector differential equation. IEEE Trans. Automat. Control 67, no. 3 (2022), 1537–1542. [4] C. Corduneanu, Yizeng Li, M. Mahdavi; Functional Differential Equations. Advances and applications, Pure and Applied Mathematics (Hoboken). 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Michael Gil’ Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer- Sheva 84105, Israel Email address: gilmi@bezeqint.net 1. Introduction and statement of the main result 2. Proofs of Theorem ?? and Corollary 1.2 3. Example Acknowledgments References