Robust Stability and Stability Radius for Variational Control Systems

Abstract

We consider an integral variational control system on a Banach space 𝑋 and we study the connections between its uniform exponential stability and the (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stability, where 𝐼 and 𝑂 are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.

1. Introduction

In the past few years a significant progress was made in the asymptotic theory of dynamical systems and their applications in control theory (see [1–41]). It is well known that if 𝑋 is a Banach space, Θ is a compact metric space, 𝜎 is a flow on Θ, and 𝐴∶Θ→ℬ(𝑋) is a continuous mapping, then Φ(𝜃,𝑡), the solution operator of the linear differential equation𝑥̇(𝑡)=𝐴(𝜎(𝜃,𝑡))𝑥(𝑡),𝑡≥0, (1.1)is a cocycle and the pair 𝜋=(Φ,𝜎) is a linear skew-product flow. Often, (1.1) arises from the linearization of nonlinear equations (see [9, 31] and the references therein). Equation (1.1) is the starting point of our paper. We consider an integral model for the above type of systems, defined in terms of linear skew-product flows and we study the exponential stability of solutions.

Input-output conditions in the study of the asymptotic properties of evolution equations have a long and impressive history that goes back to the work of Perron. In recent years, new ideas were developed in this theory, providing input-output theorems for stability, expansiveness, and dichotomy and also a number of applications in control theory (see [2, 7, 9, 11, 13, 15, 19–27, 29, 30, 32–41] and the references therein). The aim of our paper is to establish input-output conditions for uniform exponential stability of variational control systems, identifying the viable classes for input spaces and output spaces, as well. We provide a new approach based on the fundamental properties of Banach function spaces. Our attention is devoted to the relationship between the uniform exponential stability of a variational system and the stability of a pair of spaces which are translations invariant. We establish the connections between (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stability and uniform exponential stability of a variational control system on a Banach space 𝑋, when the input space 𝐼(ℝ₊,𝑋) and the output space 𝑂(ℝ₊,𝑋) belong to specific classes of Banach function spaces. By examples we motivate our techniques and also discuss some optimization methods with respect to the input space.

In what follows, we consider a generalization of systems described by differential equations of the form𝑥̇(𝑡)=𝐴(𝜎(𝜃,𝑡))𝑥(𝑡)+𝐵(𝜎(𝜃,𝑡))𝑢(𝑡),𝑦(𝑡)=𝐶(𝜎(𝜃,𝑡))𝑥(𝑡), (1.2)where 𝜎 is a flow on a locally compact metric space Θ, 𝐴(𝜃) are unbounded operators on a Banach space 𝑋, and the operators 𝐵(𝜃)∈ℬ(𝑈,𝑋), 𝐶(𝜃)∈ℬ(𝑋,𝑌), where 𝑈, 𝑌 are Banach spaces. Roughly speaking, the family {𝐴(𝜃)}_𝜃∈Θ will be subjected to additive structured perturbations, so that the perturbed system is 𝑥̇(𝑡)=[𝐴(𝜎(𝜃,𝑡))+𝐵(𝜎(𝜃,𝑡))Δ(𝜎(𝜃,𝑡))𝐶(𝜎(𝜃,𝑡))]𝑥(𝑡), which may be interpreted as a system obtained by applying the feedback 𝑢(𝑡)=Δ(𝜎(𝜃,𝑡))𝑦(𝑡) to the time-varying system (1.2). The main question is how general may be the perturbation structure such that the property of exponential stability is preserved. We will answer this question by determining a general lower bound for the stability radius of such systems.

In the last decades, stability radius became a subject of large interest and various estimations for the lower bound of the stability radius of systems were obtained (see [11, 17–20, 32, 33, 41]). This concept was introduced by Hinrichsen, Ilchmann, and Pritchard (see [18]) and led to a systematic study of the stability of linear infinite-dimensional systems under structured time-varying perturbations. The main idea was to estimate the size of the smallest disturbance operator under which the additively perturbed system loses exponential stability. A significant theorem which gives a lower bound for the stability radius of a system associated to a mild evolution operator in terms of the norm of the input-output operator of the system has been obtained by Hinrichsen and Pritchard (see [19, Theorem 3.2]). A distinct approach of this result was given by Clark et al. in [11], employing an evolution semigroup technique. The variational case was firstly treated in [32], where we obtained a lower bound for the stability radius in terms of the Perron operators associated with a linear skew-product semiflow. In [41], Wirth and Hinrichsen introduced a concept of stability radius for the case of discrete time-varying systems under structured perturbations of multi-output feedback type. In the spirit of this theory, we have obtained in [33] an estimation for the lower bound of the stability radius of a control system of difference equations on a Banach space 𝑋, under structured infinite multiperturbations, in terms of the norm of the input-output operator associated with a system on 𝑙^𝑝(ℕ,𝑋) with 𝑝∈[1,∞]. An interesting approach for systems in finite-dimensional spaces was presented by Jacob in [20]. There the author deduces a formula for the stability radius of time-varying systems with coefficients in 𝐿^loc_∞(0,∞;𝐊^𝑛×𝑛), where 𝐊∈{ℝ,𝐂}, in terms of the norms of a family of input-output operators.

In the fourth section of the present paper we propose a new approach, deducing a lower bound for the stability radius of variational systems in terms of input-output operators acting on rearrangement invariant Banach function spaces. Until now, the most common function spaces used for estimating the stability radius are the 𝐿^𝑝-spaces with 𝑝∈[1,∞), which satisfy in particular the requirements of the classes introduced in this paper. Thus, our results provide a unified treatment in a large class of function spaces, extending the above-mentioned contributions. As particular cases, we obtain a lower bound for the stability radius of a variational system in terms of 𝐿^𝑝-spaces, with 𝑝∈[1,∞], and generalize the main result in [32].

In the last section, the central results of the paper are applied at the study of the uniform exponential stability of nonautonomous systems. We provide a complete analysis concerning the implications of the variational case for nonautonomous systems. We point out several interesting situations and deduce as particular cases the stability results due to Datko, Clark, Latushkin, Montgomery-Smith, Randolph, Neerven, Megan, and many others. Moreover, as an application we obtain a lower bound for the stability radius of nonautonomous systems in terms of the norm of the input-output operators acting on translations invariant spaces. Our results extend the existing contributions in the literature on this topic.

2. Preliminary results on Banach function spaces

Let ℳ(ℝ₊,ℝ) be the linear space of all Lebesgue measurable functions 𝑢∶ℝ₊→ℝ, identifying the functions equal a.e.Definition 2.1. A linear subspace 𝐵 of ℳ(ℝ₊,ℝ) is called a normed function space if there is a mapping |⋅|_𝐵∶𝐵→ℝ₊ such that (i)|𝑢|_𝐵=0 if and only if 𝑢=0 a.e.;(ii)|𝛼𝑢|_𝐵=|𝛼||𝑢|_𝐵, for all (𝛼,𝑢)∈ℝ×𝐵;(iii)|𝑢+𝑣|_𝐵≤|𝑢|_𝐵+|𝑣|_𝐵, for all 𝑢,𝑣∈𝐵;(iv) if |𝑢(𝑡)|≤|𝑣(𝑡)| a.e. 𝑡∈ℝ₊ and 𝑣∈𝐵, then 𝑢∈𝐵 and |𝑢|_𝐵≤|𝑣|_𝐵.
If (𝐵,|⋅|_𝐵) is complete, then 𝐵 is called a Banach function space.

Remark 2.2. (i) If (𝐵,|⋅|_𝐵) is a Banach function space and 𝑢_𝑛→𝑢 in 𝐵, then there is a subsequence (𝑢_𝑘𝑛) such that 𝑢_𝑘𝑛→𝑢 a.e. (see, e.g., [28]).
(ii) If (𝐵,|⋅|_𝐵) is a Banach function space and 𝑢∈𝐵, then |𝑢(⋅)|∈𝐵.

Definition 2.3. A Banach function space (𝐵,|⋅|_𝐵) is said to be invariant to translations if for every 𝑢∶ℝ₊→ℝ and every 𝑡>0, 𝑢∈𝐵 if and only if the function𝑢_𝑡∶ℝ₊→ℝ,𝑢_𝑡(𝑠)=𝑢(𝑠−𝑡),𝑠≥𝑡0,𝑠∈[0,𝑡) (2.1)belongs to 𝐵 and |𝑢_𝑡|_𝐵=|𝑢|_𝐵.

Let 𝐶_𝑐(ℝ₊,ℝ) be the linear space of all continuous functions 𝑣∶ℝ₊→ℝ with compact support contained in ℝ₊ and let 𝐶_0𝑐(ℝ₊,ℝ)∶={𝑣∈𝐶_𝑐(ℝ₊,ℝ)∶𝑣(0)=0}. Let 𝐿¹_loc(ℝ₊,ℝ) be the linear space of all locally integrable functions 𝑢∶ℝ₊→ℝ.

We denote by 𝒯(ℝ₊) the class of all Banach function spaces 𝐵 which are invariant to translations and with the following properties:(i)𝐶_𝑐(ℝ₊,ℝ)⊂𝐵⊂𝐿¹_loc(ℝ₊,ℝ);(ii)if 𝐵⧵𝐿¹(ℝ₊,ℝ)≠∅, then there is a continuous function 𝛿∈𝐵⧵𝐿¹(ℝ₊,ℝ).

For every 𝐴⊂ℝ₊ we denote by 𝜒_𝐴 the characteristic function of the set 𝐴.

Remark 2.4. If 𝐵∈𝒯(ℝ₊), then 𝜒_[0,𝑡)∈𝐵, for all 𝑡>0.

Indeed, let 𝑡>0. If 𝛼∶ℝ₊→[0,1] is a continuous function with compact support such that 𝛼(𝑠)=1, for 𝑠∈[0,𝑡] and 𝛼(𝑠)=0, for 𝑠≥𝑡+1, then we have that 𝜒_[0,𝑡)(𝑠)≤𝛼(𝑠), for all 𝑠≥0. Since 𝛼∈𝐵, using (iv) from Definition 2.1, we deduce that 𝜒_[0,𝑡)∈𝐵.

Definition 2.5. If 𝐵∈𝒯(ℝ₊), then the function 𝐹_𝐵∶(0,∞)→ℝ₊, 𝐹_𝐵(𝑡)=|𝜒_[0,𝑡)|_𝐵 is called the fundamental function of the space 𝐵.

Remark 2.6. The function 𝐹_𝐵 is nondecreasing.

Example 2.7. Let 𝑝∈[1,∞). Then the space 𝐿^𝑝(ℝ₊,ℝ)={𝑢∈ℳ(ℝ₊,ℝ)∶∫^∞₀|𝑢(𝜏)|^𝑝𝑑𝜏<∞} with respect to the norm ‖𝑢‖_𝑝=(∫^∞₀|𝑢(𝜏)|^𝑝𝑑𝜏)^1/𝑝 is a Banach function space which belongs to 𝒯(ℝ₊).Example 2.8. Let 𝐿^∞(ℝ₊,ℝ) be the linear space of all essentially bounded functions 𝑢∈ℳ(ℝ₊,ℝ). With respect to the norm ‖𝑢‖_∞∶=esssup_𝑡≥0|𝑢(𝑡)|, 𝐿^∞(ℝ₊,ℝ) is a Banach function space which belongs to 𝒯(ℝ₊).Example 2.9 (Orlicz spaces). Let 𝜑∶ℝ₊→ℝ₊ be a nondecreasing left-continuous function, which is not identically zero on (0,∞). The Young function associated with 𝜑 is defined by 𝑌_𝜑(𝑡)=∫^𝑡₀𝜑(𝑠)𝑑𝑠. For every 𝑢∈ℳ(ℝ₊,ℝ) we define 𝑀_𝜑(𝑢)∶=∫^∞₀𝑌_𝜑(|𝑢(𝑠)|)𝑑𝑠. The set 𝑂_𝜑 of all 𝑢∈ℳ(ℝ₊,ℝ) with the property that there is 𝑘>0 such that 𝑀_𝜑(𝑘𝑢)<∞ is a linear space. With respect to the norm |𝑢|_𝜑∶=inf{𝑘>0∶𝑀_𝜑(𝑢/𝑘)≤1}, 𝑂_𝜑 is a Banach space called the Orlicz space associated with 𝜑.

Remark 2.10. It is easy to verify that the Orlicz spaces introduced in Example 2.9 belong to 𝒯(ℝ₊).

Remark 2.11. The spaces 𝐿^𝑝(ℝ₊,ℝ), 𝑝∈[1,∞] are Orlicz spaces, which may be obtained for 𝜑(𝑡)=𝑝𝑡^𝑝−1, if 𝑝∈[1,∞) and for𝜑(𝑡)=0,𝑡∈[0,1]1,𝑡>1if𝑝=∞. (2.2)

Lemma 2.12. Let 𝐵∈𝒯(ℝ₊) and 𝜈>0. Then the function 𝑒_𝜈∶ℝ₊→ℝ₊, 𝑒_𝜈(𝑡)=𝑒^−𝜈𝑡, belongs to 𝐵.

Proof. We have that𝑒_𝜈(𝑡)=^∞_𝑛=0𝑒^−𝜈𝑡𝜒_{[𝑛,𝑛+1)}(𝑡)≤^∞_𝑛=0𝑒^−𝜈𝑛𝜒_{[𝑛,𝑛+1)}(𝑡),∀𝑡≥0. (2.3)This shows that 𝑒_𝜈∈𝐵 and |𝑒_𝜈|_𝐵≤|𝜒_[0,1)|_𝐵/(1−𝑒^−𝜈).

Definition 2.13. Let 𝑢,𝑣∈ℳ(ℝ₊,ℝ). Say that 𝑢 and 𝑣 are equimeasurable if for every 𝑡>0 the sets {𝑠∈ℝ₊∶|𝑢(𝑠)|>𝑡} and {𝑠∈ℝ₊∶|𝑣(𝑠)|>𝑡} have the same measure.Definition 2.14. A Banach function space (𝐵,|⋅|_𝐵) is rearrangement invariant if for every equimeasurable functions 𝑢,𝑣∶ℝ₊→ℝ₊ with 𝑢∈𝐵, one has 𝑣∈𝐵 and |𝑢|_𝐵=|𝑣|_𝐵.

Remark 2.15. The Orlicz spaces are rearrangement invariant (see [4, Theorem 8.9]).

We denote by ℛ(ℝ₊) the class of all Banach function spaces 𝐵∈𝒯(ℝ₊) which are rearrangement invariant.

Remark 2.16. If 𝐵∈ℛ(ℝ₊), then 𝐵 is an interpolation space between 𝐿¹(ℝ₊,ℝ) and 𝐿^∞(ℝ₊,ℝ) (see [4, Theorem 2.2, page 106]).

Lemma 2.17. Let 𝐵∈𝒯(ℝ₊) and 𝜈>0. Then for every 𝑢∈𝐵 the function𝜓_𝑢∶ℝ₊→ℝ,𝜓_𝑢(𝑡)=^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}𝑢(𝑠)𝑑𝑠 (2.4)belongs to 𝐵. Moreover, there is 𝜆_𝐵>0 such that |𝜓_𝑢|_𝐵≤𝜆_𝐵|𝑢|_𝐵, for all 𝑢∈𝐵.

Proof. It is easy to see that𝐷_𝜈∶𝐿^∞(ℝ₊,ℝ)→𝐿^∞(ℝ₊,ℝ),(𝐷_𝜈(𝑢))(𝑡)=^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}𝑢(𝑠)𝑑𝑠 (2.5)is a correctly defined bounded linear operator. Moreover, the restriction 𝐷_𝜈|∶𝐿¹(ℝ₊,ℝ)→𝐿¹(ℝ₊,ℝ) is correctly defined and is a bounded linear operator. Then from Remark 2.16 it follows that 𝐷_𝜈|𝐵∶𝐵→𝐵 is a correctly defined and bounded linear operator. Taking 𝜆_𝐵=‖𝐷_𝜈|𝐵‖, the proof is complete.

We denote by 𝒬(ℝ₊) the class of all Banach function spaces 𝐵∈𝒯(ℝ₊) with the property that sup_𝑡>0𝐹_𝐵(𝑡)=∞.

Remark 2.18. If 𝜑(𝑡)∈(0,∞), for all 𝑡>0, then the Orlicz space 𝑂_𝜑∈𝒬(ℝ₊) (see, e.g., [24, Proposition 2.1]).

Let 𝐶₀(ℝ₊,ℝ) be the space of all continuous functions 𝑢∶ℝ₊→ℝ with lim_𝑡→∞𝑢(𝑡)=0. Lemma 2.19. If 𝐵∈𝒯(ℝ₊)⧵𝒬(ℝ₊), then 𝐶₀(ℝ₊,ℝ)⊂𝐵.

Proof. Let 𝑢∈𝐶₀(ℝ₊,ℝ). Then there is an unbounded increasing sequence (𝑡_𝑛)⊂(0,∞) such that |𝑢(𝑡)|≤1/(𝑛+1), for all 𝑡≥𝑡_𝑛 and 𝑛∈ℕ. We set 𝑢_𝑛=𝑢𝜒_[0,𝑡𝑛). According to our hypothesis, 𝑀∶=sup_𝑡>0𝐹_𝐵(𝑡)<∞. Then, we observe that|𝑢_𝑛+𝑝−𝑢_𝑛|_𝐵≤1𝑛+1|𝜒_{[𝑡𝑛,𝑡𝑛+𝑝)}|_𝐵≤𝑀𝑛+1,∀𝑝∈ℕ^∗,∀𝑛∈ℕ. (2.6)Hence, the sequence (𝑢_𝑛) is fundamental in 𝐵, so there is 𝑣∈𝐵 such that 𝑢_𝑛→𝑣 in 𝐵. From Remark 2.2 there exists a subsequence (𝑢_𝑘𝑛) such that 𝑢_𝑘𝑛→𝑣 a.e. This implies that 𝑢=𝑣 a.e., so 𝑢=𝑣 in 𝐵. It follows that 𝑢∈𝐵 and the proof is complete.

We denote by ℒ(ℝ₊) the class of all Banach function spaces 𝐵∈𝒯(ℝ₊) with the property that 𝐵⧵𝐿¹(ℝ₊,ℝ)≠∅.

Remark 2.20. Using Remark 2.2(ii) we have that if 𝐵∈ℒ(ℝ₊), then there is a continuous function 𝛿∶ℝ₊→ℝ₊ with 𝛿∈𝐵⧵𝐿¹(ℝ₊,ℝ).

Lemma 2.21. If 𝐵∈𝒯(ℝ₊), then 𝐵∈𝒬(ℝ₊) or 𝐵∈ℒ(ℝ₊).

Proof. Suppose by contrary that there exists 𝐵∈𝒯(ℝ₊) such that 𝐵∉𝒬(ℝ₊) and 𝐵∉ℒ(ℝ₊). Then 𝑀∶=sup_𝑡>0𝐹_𝐵(𝑡)<∞ and 𝐵⊂𝐿¹(ℝ₊,ℝ). It follows that there is 𝑐>0 such that ‖𝑢‖₁≤𝑐|𝑢|_𝐵, for all 𝑢∈𝐵. In particular, for 𝑢=𝜒_[0,𝑡) we deduce that 𝑡≤𝑐|𝜒_[0,𝑡)|_𝐵=𝑐𝐹_𝐵(𝑡)≤𝑐𝑀, for all 𝑡>0, which is absurd.

Let (𝑋,‖⋅‖) be a real or complex Banach space. We denote by 𝐶_𝑐(ℝ₊,𝑋) the linear space of all continuous functions 𝑣∶ℝ₊→𝑋 with compact support, let 𝐶_0𝑐={𝑣∈𝐶_𝑐(ℝ₊,𝑋)∶𝑣(0)=0} and let 𝐿¹_loc(ℝ₊,𝑋) be the linear space of all locally integrable functions 𝑢∶ℝ₊→𝑋.

For every 𝐵∈𝒯(ℝ₊) we denote by 𝐵(ℝ₊,𝑋) the linear space of all Bochner measurable functions 𝑢∶ℝ₊→𝑋 with the property that the mapping 𝑁_𝑢∶ℝ₊→ℝ₊, 𝑁_𝑢(𝑡)=‖𝑢(𝑡)‖ lies in 𝐵. Endowed with the norm ‖𝑢‖_{𝐵(ℝ+,𝑋)}∶=|𝑁_𝑢|_𝐵, 𝐵(ℝ₊,𝑋) is a Banach space.

3. Stability of variational integral control systems

In the stability theory of evolution operators in Banach spaces, an evolution family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} is said to be (𝐸(ℝ₊,𝑋),𝐹(ℝ₊,𝑋)) stable if for every 𝑓∈𝐸(ℝ₊,𝑋) the mapping 𝑃_𝑓 belongs to 𝐹(ℝ₊,𝑋) where𝑃_𝑓(𝑡)=^𝑡₀𝑈(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. (3.1)The classical input-output results concerning exponential stability of evolution families in Banach spaces may be stated as follows.

Let𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0}be an evolution operator on a Banach space X and𝑝,𝑞∈[1,∞)with𝑝≤𝑞. Then the following assertions are equivalent:(i)𝒰is uniformly exponentially stable;(ii)𝒰is(𝐶₀(ℝ₊,𝑋),𝐶₀(ℝ₊,𝑋))stable;(iii)𝒰is(𝐶₀(ℝ₊,𝑋),𝐶_𝑏(ℝ₊,𝑋))stable;(iv)𝒰is(𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋))stable.

The implication (iv)⇒(i) was obtained by Datko in 1973 (see [16]), using one of his stability results contained in the same paper. In the last few years, there were pointed out new and interesting methods of proving the above theorem. The equivalences (i)⇔(ii)⇔(iv) were proved by Neerven for the special case of 𝐶₀ semigroups of linear operators (see [30]) and 𝑝=𝑞. In [11], Clark et al. have provided an inedit technique of proving the equivalences (i)⇔(ii)⇔(iv) for the case 𝑝=𝑞. There the authors established important connections between the asymptotic properties of evolution families and those of the associated evolution semigroups with a large number of applications in control theory. Roughly speaking the authors described the behavior of a nonautonomous system in terms of the properties of an associated autonomous system. Therefore, in their approach the input space must coincide with the output space.

The equivalence (i)⇔(ii)⇔(iii) has been also proved by van Minh et al. (see [29]), employing an evolution semigroup technique. The equivalence (i)⇔(ii) was also treated by Buşe in [7]. The case 𝑝≠𝑞 was completely treated in [23], where several new stability results of Perron type were established for exponential stability of evolution families, generalizing the above equivalences and obtaining that uniform exponential stability of an evolution family can be expressed using boundedly locally dense subsets of 𝐶_𝑏(ℝ₊,𝑋) and 𝐿^𝑝(ℝ₊,𝑋), respectively. The main results in [23] were extended in [25] for the case of linear skew-product flows, where the input-output operator was replaced with a family of operators acting between the function spaces of the admissible pair. The stability results obtained in [25] generalized some theorems from [7, 16, 23, 29, 30]. Moreover, the stability theorems in [25] led to several interesting consequences in control theory (see [26, 32]).

In what follows, the input-output techniques in the stability theory of variational equations will be treated from the perspective of Banach function spaces arising from the interpolation theory. In this section, our main purpose is to give a complete and unified study of the exponential stability of variational systems via input-output methods, providing the “structure” of the classes of input and output spaces, respectively.

Let 𝑋 be a Banach space, let (Θ,𝑑) be a metric space, and let ℰ=𝑋×Θ. We denote by ℬ(𝑋) the Banach algebra of all bounded linear operators on 𝑋. The norm on 𝑋 and on ℬ(𝑋) will be denoted by ‖⋅‖. Definition 3.1. Let 𝐽∈{ℝ₊,ℝ}. A continuous mapping 𝜎∶Θ×𝐽→Θ is called a flow on Θ if 𝜎(𝜃,0)=𝜃 and 𝜎(𝜃,𝑠+𝑡)=𝜎(𝜎(𝜃,𝑠),𝑡), for all (𝜃,𝑠,𝑡)∈Θ×𝐽².Definition 3.2. A pair 𝜋=(Φ,𝜎) is called a linear skew-product flow on ℰ=𝑋×Θ if 𝜎 is a flow on Θ and Φ∶Θ×ℝ₊→ℬ(𝑋) satisfies the following conditions:(i)Φ(𝜃,0)=𝐼_𝑑, the identity operator on 𝑋, for all 𝜃∈Θ;(ii)Φ(𝜃,𝑡+𝑠)=Φ(𝜎(𝜃,𝑡),𝑠)Φ(𝜃,𝑡), for all (𝜃,𝑡,𝑠)∈Θ×ℝ²₊ (the cocycle identity);(iii)(𝜃,𝑡)↦Φ(𝜃,𝑡)𝑥 is continuous, for every 𝑥∈𝑋;(iv) there are 𝑀≥1 and 𝜔>0 such that ‖Φ(𝜃,𝑡)‖≤𝑀𝑒^𝜔𝑡, for all (𝜃,𝑡)∈Θ×ℝ₊.

The mapping Φ given by Definition 3.2 is called the cocycle associated to the linear skew-product flow 𝜋=(Φ,𝜎).

Let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ. In what follows, we consider the variational integral control system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ (𝑆)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋. Definition 3.3. The system (𝑆) is said to be(i)uniformly stable if there is 𝐿>0 such that‖𝑥_𝜃(𝑡;𝑥₀,0)‖≤𝐿‖𝑥₀‖,∀(𝜃,𝑡)∈Θ×ℝ₊,∀𝑥₀∈𝑋; (3.2)(ii)uniformly exponentially stable if there are 𝐾,𝜈>0 such that‖𝑥_𝜃(𝑡;𝑥₀,0)‖≤𝐾𝑒^−𝜈𝑡‖𝑥₀‖,∀(𝜃,𝑡)∈Θ×ℝ₊,∀𝑥₀∈𝑋. (3.3)

Remark 3.4. (i) The system (𝑆) is uniformly stable if and only if there is 𝐿>0 such that ‖Φ(𝜃,𝑡)‖≤𝐿, for all (𝜃,𝑡)∈Θ×ℝ₊.

(ii) The system (𝑆) is uniformly exponentially stable if and only if there are 𝐾,𝜈>0 such that ‖Φ(𝜃,𝑡)‖≤𝐾𝑒^−𝜈𝑡, for all (𝜃,𝑡)∈Θ×ℝ₊.

Definition 3.5. Let 𝐼, 𝑂 be two Banach function spaces with 𝐼,𝑂∈𝒯(ℝ₊). The system (𝑆) is said to be (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋))stable if the following assertions hold:(i)for every 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) and every 𝜃∈Θ the solution 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋);(ii)there is 𝜆>0 such that ‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}, for all (𝑢,𝜃)∈𝐶_0𝑐(ℝ₊,𝑋)×Θ.𝐼(ℝ₊,𝑋) is called the input space and 𝑂(ℝ₊,𝑋) is called the output space.

We begin with a sufficient condition for uniform stability. Theorem 3.6. Let 𝐼,𝑂∈𝒯(ℝ₊). If the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then the system (𝑆) is uniformly stable.

Proof. Let 𝛼∶ℝ₊→[0,2] be a continuous function with supp𝛼⊂(0,1) and ∫¹₀𝛼(𝜏)𝑑𝜏=1.
Let 𝑀,𝜔∈(0,∞) be given by Definition 3.2 and let 𝜆>0 be given by Definition 3.5.
Let (𝑥,𝜃)∈𝑋×Θ. We consider the function 𝑢∶ℝ₊→𝑋, 𝑢(𝑡)=𝛼(𝑡)Φ(𝜃,𝑡)𝑥. We have that 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) and ‖𝑢(𝑡)‖=𝛼(𝑡)‖Φ(𝜃,𝑡)𝑥‖≤𝑀𝑒^𝜔‖𝑥‖𝛼(𝑡), for all 𝑡≥0. This implies that‖𝑢‖_{𝐼(ℝ+,𝑋)}≤𝑀𝑒^𝜔‖𝑥‖|𝛼|_𝐼. (3.4)From hypothesis we have that 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋) and‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}. (3.5)We observe that𝑥_𝜃(𝑡;0,𝑢)=^𝑡₀=Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠_𝑡0𝛼(𝑠)𝑑𝑠Φ(𝜃,𝑡)𝑥=Φ(𝜃,𝑡)𝑥,∀𝑡≥1. (3.6) Then, for 𝑡≥2 we deduce that‖Φ(𝜃,𝑡)𝑥‖𝜒_{[𝑡−1,𝑡)}(𝑠)≤𝑀𝑒^𝜔‖Φ(𝜃,𝑠)𝑥‖=𝑀𝑒^𝜔‖𝑥_𝜃(𝑠;0,𝑢)‖,∀𝑠≥0 (3.7)which implies that‖Φ(𝜃,𝑡)𝑥‖|𝜒_{[𝑡−1,𝑡)}|_𝑂≤𝑀𝑒^𝜔‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}. (3.8)Since 𝑂 is invariant to translations we have that |𝜒_{[𝑡−1,𝑡)}|_𝑂=|𝜒_[0,1)|_𝑂=𝐹_𝑂(1). Then, from relations (3.4)–(3.8) we obtain that‖Φ(𝜃,𝑡)𝑥‖≤𝜆𝑀𝑒^𝜔𝐹_𝑂(1)‖𝑢‖_{𝐼(ℝ+,𝑋)}≤𝜆𝑀²𝑒^2𝜔|𝛼|_𝐼𝐹_𝑂(1)‖𝑥‖,∀𝑡≥2. (3.9)For 𝑡∈[0,2] we have that ‖Φ(𝜃,𝑡)‖≤𝑀𝑒^2𝜔. Then, for 𝐿=max{𝑀𝑒^2𝜔,(𝜆𝑀²𝑒^2𝜔|𝛼|_𝐼)/𝐹_𝑂(1)} we deduce that ‖Φ(𝜃,𝑡)𝑥‖≤𝐿‖𝑥‖, for all 𝑡≥0. Taking into account that 𝐿 does not depend on 𝜃 or 𝑥, it follows that ‖Φ(𝜃,𝑡)𝑥‖≤𝐿‖𝑥‖, for all (𝜃,𝑡)∈Θ×ℝ₊ and all 𝑥∈𝑋 and the proof is complete.

The first main result of this section is the following.

Theorem 3.7. Let 𝐼,𝑂∈𝒯(ℝ₊). If 𝐼∈ℒ(ℝ₊) and the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then the system (𝑆) is uniformly exponentially stable.

Proof. Let 𝜆>0 be given by Definition 3.5. From Theorem 3.6 we have that there is 𝐿>0 such that ‖Φ(𝜃,𝑡)‖≤𝐿, for all (𝜃,𝑡)∈Θ×ℝ₊.
Since 𝐼∈ℒ(ℝ₊) from Remark 2.20 it follows that there is a continuous function 𝛿∶ℝ₊→ℝ₊ such that 𝛿∈𝐼⧵𝐿¹(ℝ₊,ℝ). Let ℎ>0 be such that^ℎ₀𝛿(𝑠)𝑑𝑠≥2𝑒𝜆𝐿²|𝛿|_𝐼𝐹_𝑂.(1) (3.10)
For every 𝑛∈ℕ^∗ let 𝛼_𝑛∶ℝ₊→[0,1] be a continuous function with 𝛼_𝑛(𝑡)=1, for 𝑡∈[1/𝑛,ℎ], 𝛼_𝑛(0)=0, and 𝛼_𝑛(𝑡)=0, for 𝑡≥ℎ+1. Since^ℎ₀𝛼_𝑛(𝑠)𝛿(𝑠)𝑑𝑠→^ℎ₀𝛿(𝑠)𝑑𝑠,as𝑛→∞ (3.11)there is 𝑗∈ℕ^∗ such that𝑎∶=^ℎ₀𝛼_𝑗1(𝑠)𝛿(𝑠)𝑑𝑠≥2^ℎ₀𝛿(𝑠)𝑑𝑠≥𝑒𝜆𝐿²|𝛿|_𝐼𝐹_𝑂.(1) (3.12)
Let (𝑥,𝜃)∈ℰ=𝑋×Θ. We consider the function𝑢∶ℝ₊→𝑋,𝑢(𝑡)=𝛼_𝑗(𝑡)𝛿(𝑡)Φ(𝜃,𝑡)𝑥. (3.13)We have that 𝑢∈𝐶_0𝑐(ℝ₊,𝑋), so 𝑢∈𝐼(ℝ₊,𝑋). In addition, ‖𝑢(𝑡)‖≤𝐿‖𝑥‖𝛿(𝑡), for all 𝑡≥0. This implies that ‖𝑢‖_{𝐼(ℝ+,𝑋)}≤𝐿‖𝑥‖|𝛿|_𝐼. We observe that‖𝑥_𝜃(𝑡;0,𝑢)‖=‖^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠‖=^𝑡₀𝛼_𝑗(𝑠)𝛿(𝑠)𝑑𝑠‖Φ(𝜃,𝑡)𝑥‖≥𝑎‖Φ(𝜃,𝑡)𝑥‖,∀𝑡≥ℎ. (3.14) Then, we have that𝐿‖Φ(𝜃,ℎ+1)𝑥‖≤𝐿‖Φ(𝜃,𝑡)𝑥‖≤𝑎‖𝑥_𝜃(𝑡;0,𝑢)‖,∀𝑡∈[ℎ,ℎ+1) (3.15)which implies that‖Φ(𝜃,ℎ+1)𝑥‖𝜒_[ℎ,ℎ+1)𝐿(𝑡)≤𝑎‖𝑥_𝜃(𝑡;0,𝑢)‖,∀𝑡≥0. (3.16)Since 𝑂 is invariant to translations we have that |𝜒_[ℎ,ℎ+1)|_𝑂=|𝜒_[0,1)|_𝑂=𝐹_𝑂(1). Then from (3.16) we deduce that‖Φ(𝜃,ℎ+1)𝑥‖𝐹_𝑂𝐿(1)≤𝑎‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}. (3.17)According to our hypothesis‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}. (3.18)Using relations (3.17) and (3.18) it follows that𝑎‖Φ(𝜃,ℎ+1)𝑥‖≤𝜆𝐿𝐹_𝑂(1)‖𝑢‖_{𝐼(ℝ+,𝑋)}≤𝜆𝐿²|𝛿|_𝐼𝐹_𝑂(1)‖𝑥‖. (3.19)We set 𝑟=ℎ+1. From (3.19) and (3.12) we obtain that ‖Φ(𝜃,𝑟)𝑥‖≤(1/𝑒)‖𝑥‖. Since 𝑟 does not depend on 𝜃 or 𝑥 we deduce that ‖Φ(𝜃,𝑟)𝑥‖≤(1/𝑒)‖𝑥‖, for all (𝑥,𝜃)∈ℰ.
Let 𝜈=1/𝑟 and 𝐾=𝐿𝑒. Let 𝑡≥0. Then there is 𝑛∈ℕ and 𝑠∈[0,𝑟) such that 𝑡=𝑛𝑟+𝑠. It follows that ‖Φ(𝜃,𝑡)‖≤𝐿‖Φ(𝜃,𝑛𝑟)‖≤𝐿𝑒^−𝑛≤𝐾𝑒^−𝜈𝑡. This implies that the system (𝑆) is uniformly exponentially stable.

The second main result of this section is the following.

Theorem 3.8. Let 𝐼,𝑂∈𝒯(ℝ₊). If 𝑂∈𝒬(ℝ₊) and the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆) is uniformly exponentially stable.

Proof. Let 𝜆>0 be given by Definition 3.5 and let 𝐿>0 be given by Theorem 3.6. Let ℎ>0 be such that𝐹_𝑂(ℎ)≥2𝑒𝜆𝐿²𝐹_𝐼(1). (3.20)Let 𝛼∶ℝ₊→[0,2] be a continuous function with supp𝛼⊂(0,1) and ∫¹₀𝛼(𝜏)𝑑𝜏=1.
Let (𝑥,𝜃)∈ℰ=𝑋×Θ. We consider the function𝑢∶ℝ₊→𝑋,𝑢(𝑡)=𝛼(𝑡)Φ(𝜃,𝑡)𝑥. (3.21)Since 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) we have that 𝑢∈𝐼(ℝ₊,𝑋). Moreover, ‖𝑢(𝑡)‖≤𝐿‖𝑥‖𝛼(𝑡)≤2𝐿‖𝑥‖𝜒_[0,1)(𝑡), for all 𝑡≥0, which implies that‖𝑢‖_{𝐼(ℝ+,𝑋)}≤2𝐿‖𝑥‖𝐹_𝐼(1). (3.22)We observe that 𝑥_𝜃(𝑡,0,𝑢)=Φ(𝜃,𝑡)𝑥, for all 𝑡≥1. Then, we have that‖Φ(𝜃,ℎ+1)𝑥‖𝜒_[1,ℎ+1)(𝑡)≤𝐿‖Φ(𝜃,𝑡)𝑥‖𝜒_[1,ℎ+1)(𝑡)≤𝐿‖𝑥_𝜃(𝑡;0,𝑢)‖,∀𝑡≥0 (3.23) Since 𝑂∈𝒯(ℝ₊) we have that |𝜒_[1,ℎ+1)|_𝑂=|𝜒_[0,ℎ)|_𝑂=𝐹_𝑂(ℎ). Then from (3.23) and (3.22) we deduce that‖Φ(𝜃,ℎ+1)𝑥‖𝐹_𝑂(ℎ)≤𝐿‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆𝐿‖𝑢‖_{𝐼(ℝ+,𝑋)}≤2𝜆𝐿²𝐹_𝐼(1)‖𝑥‖. (3.24) From relations (3.20) and (3.24) it follows that ‖Φ(𝜃,ℎ+1)𝑥‖≤(1/𝑒)‖𝑥‖. Taking into account that ℎ does not depend on 𝜃 or 𝑥 we obtain that ‖Φ(𝜃,ℎ+1)𝑥‖≤(1/𝑒)‖𝑥‖, for all (𝑥,𝜃)∈ℰ=𝑋×Θ. Using similar arguments as in Theorem 3.7 we obtain that the system (𝑆) is uniformly exponentially stable.

The central result of this section is the following.

Theorem 3.9. Let 𝐼,𝑂∈𝒯(ℝ₊) be such that 𝐼∈ℒ(ℝ₊) or 𝑂∈𝒬(ℝ₊). Then, the following assertions hold:(i)if the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆) is uniformly exponentially stable;(ii)if 𝐼⊂𝑂 and one of the spaces 𝐼, 𝑂 belongs to the class ℛ(ℝ₊), then the system (𝑆) is uniformly exponentially stable if and only if the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.

Proof. (i) This follows from Theorems 3.7 and 3.8.
(ii) Necessity. Let 𝐾,𝜈>0 be such that ‖Φ(𝜃,𝑡)𝑥‖≤𝐾𝑒^−𝜈𝑡‖𝑥‖, for all (𝑥,𝜃)∈ℰ and all 𝑡≥0. Since 𝐼⊂𝑂, there is 𝛾>0 such that|𝑤|_𝑂≤𝛾|𝑤|_𝐼,∀𝑤∈𝐼. (3.25)
Let 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) and let𝜓_𝑢∶ℝ₊→ℝ,𝜓_𝑢(𝑡)=^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}‖𝑢(𝑠)‖𝑑𝑠. (3.26)Then for every 𝜃∈Θ we have that‖𝑥_𝜃(𝑡;0,𝑢)‖≤𝐾𝜓_𝑢(𝑡),𝑡≥0. (3.27)Case 1. 𝐼∈ℛ(ℝ₊) Let 𝜆_𝐼>0 be given by Lemma 2.17. Since 𝑢∈𝐼(ℝ₊,𝑋), from Lemma 2.17 we have that 𝜓_𝑢∈𝐼 and |𝜓_𝑢|_𝐼≤𝜆_𝐼‖𝑢‖_{𝐼(ℝ+,𝑋)}. Then from (3.27) it follows that 𝑥_𝜃(⋅;0,𝑢)∈𝐼(ℝ₊,𝑋), so 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋). Moreover, from (3.25) and (3.27) we have that‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝛾‖𝑥_𝜃(⋅;0,𝑢)‖_{𝐼(ℝ+,𝑋)}≤𝛾𝐾𝜆_𝐼‖𝑢‖_{𝐼(ℝ+,𝑋)}. (3.28)Taking 𝜆=𝛾𝐾𝜆_𝐼 and observing that 𝜆 does not depend on 𝑢 or 𝜃 we deduce that the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋))stable.Case 2. 𝑂∈ℛ(ℝ₊) Let 𝜆_𝑂>0 be given by Lemma 2.17. From 𝑢∈𝑂(ℝ₊,𝑋), using Lemma 2.17 we have that 𝜓_𝑢∈𝑂 and |𝜓_𝑢|_𝑂≤𝜆_𝑂‖𝑢‖_{𝑂(ℝ+,𝑋)}. Then, from (3.25) and (3.27) we deduce that 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋) and‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝐾𝜆_𝑂‖𝑢‖_{𝑂(ℝ+,𝑋)}≤𝛾𝐾𝜆_𝑂‖𝑢‖_{𝐼(ℝ+,𝑋)}. (3.29)Taking 𝜆=𝛾𝐾𝜆_𝑂 we conclude that the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.Sufficiency. It follows from (i).
In what follows, we prove that the main result given by Theorem 3.9 is the most general in this topic. Specifically, we will show that if 𝐼∉ℒ(ℝ₊) and 𝑂∉𝒬(ℝ₊), then the (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stability of the system (𝑆) does not imply the uniform exponential stability of (𝑆).

Example 3.10. Let Θ=ℝ₊ and let 𝜎∶Θ×ℝ₊→Θ, 𝜎(𝜃,𝑡)=𝜃+𝑡. Let 𝑋=𝐶₀(ℝ₊,ℝ) and for every (𝜃,𝑡)∈Θ×ℝ₊, letΦ(𝜃,𝑡)∶𝑋→𝑋,(Φ(𝜃,𝑡)𝑥)(𝑠)=𝑥(𝑠+𝑡). (3.30)Then 𝜋=(Φ,𝜎) is a linear skew-product flow on ℰ=𝑋×Θ. We consider the system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ (𝑆)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋. Observing that ‖Φ(𝜃,𝑡)‖=1, for all (𝜃,𝑡)∈Θ×ℝ₊, we deduce that the system (𝑆) is not uniformly exponentially stable.
Let 𝐼,𝑂∈𝒯(ℝ₊) with 𝐼∉ℒ(ℝ₊) and 𝑂∉𝒬(ℝ₊). Then 𝐼⊂𝐿¹(ℝ₊,ℝ) and from Lemma 2.19 we have that 𝐶₀(ℝ₊,ℝ)⊂𝑂. In what follows, we prove that the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.
Let 𝑢∈𝐿¹(ℝ₊,𝑋) and 𝜃∈Θ. Let 𝜀>0. Since 𝑢∈𝐿¹(ℝ₊,𝑋), there is ℎ>0 such that ∫^∞_ℎ‖𝑢(𝑠)‖𝑑𝑠<𝜀/2. Let 𝑣∶=∫^ℎ₀Φ(𝜎(𝜃,𝜏),ℎ−𝜏)𝑢(𝜏)𝑑𝜏. From 𝑣∈𝑋 we have that there is 𝛿>0 such that |𝑣(𝜏)|<𝜀/2, for all 𝑡≥𝛿. Then, for 𝑡≥𝛿+ℎ, we deduce that^ℎ₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠=‖Φ(𝜎(𝜃,ℎ),𝑡−ℎ)𝑣‖=sup_𝑟≥0𝜀|𝑣(𝑡−ℎ+𝑟)|<2. (3.31)This implies that‖𝑥_𝜃(𝑡;0,𝑢)‖≤^ℎ₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠+^𝑡_ℎ<𝜀‖Φ(𝜎(𝜃,𝑠),𝑡−𝑠)‖‖𝑢(𝑠)‖𝑑𝑠2+^𝑡_ℎ‖𝑢(𝑠)‖𝑑𝑠<𝜀,∀𝑡≥𝛿+ℎ (3.32) so 𝑥_𝜃(⋅;0,𝑢)∈𝐶₀(ℝ₊,𝑋). It follows that for every (𝑢,𝜃)∈𝐿¹(ℝ₊,𝑋)×Θ we have that 𝑥_𝜃(⋅;0,𝑢)∈𝐶₀(ℝ₊,𝑋).
It makes sense to define the linear operator𝑃∶𝐿¹(ℝ₊,𝑋)→𝐶₀(ℝ₊,𝑋),𝑃(𝑢)=𝑥₀(⋅;0,𝑢). (3.33)It is easy to verify that 𝑃 is closed, so from the closed graph theorem we obtain that 𝑃 is bounded. Setting 𝛾=‖𝑃‖ we have that‖𝑥₀(⋅;0,𝑢)‖_{𝐶0(ℝ+,𝑋)}≤𝛾‖𝑢‖_𝐿¹_(ℝ+,𝑋),∀𝑢∈𝐿¹(ℝ₊,𝑋). (3.34)Since 𝐼⊂𝐿¹(ℝ₊,ℝ) there is 𝛼>0 such that‖𝑢‖₁≤𝛼|𝑢|_𝐼,∀𝑢∈𝐼. (3.35)Since 𝐶₀(ℝ₊,ℝ)⊂𝑂 there is 𝛽>0 such that|𝑤|_𝑂≤𝛽‖𝑤‖_{𝐶0(ℝ+,ℝ)},∀𝑤∈𝐶₀(ℝ₊,ℝ). (3.36)
Let (𝑢,𝜃)∈𝐶_0𝑐(ℝ₊,𝑋)×Θ. Since 𝑥_𝜃(⋅;0,𝑢)∈𝐶₀(ℝ₊,𝑋) and 𝐶₀(ℝ₊,𝑋)⊂𝑂(ℝ₊,𝑋) it follows that 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋). It is easy to observe that 𝑥_𝜃(⋅;0,𝑢)=𝑥₀(⋅;0,𝑢). Setting 𝜆=𝛼𝛽𝛾, from relations (3.34)–(3.36) we deduce that‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}=‖𝑥₀(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝛽‖𝑥₀(⋅;0,𝑢)‖_{𝐶0(ℝ+,𝑋)}≤𝛽𝛾‖𝑢‖_𝐿¹_(ℝ+,𝑋)≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)} (3.37) Taking into account that 𝜆 does not depend on 𝑢 or 𝜃 we conclude that the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable. But, for all that (𝑆) is not uniformly exponentially stable.

Remark 3.11. The input-output characterizations for asymptotic properties of systems have a wide applicability area if the input space is as small as possible and the output space is very general. We note that in the main result given by Theorem 3.9 the input functions belong to 𝐶_0𝑐(ℝ₊,𝑋), while the output space is a general function space.
Moreover, the class 𝒯(ℝ₊) is closed to finite intersections. If 𝐼₁,𝐼₂,…,𝐼_𝑛∈𝒯(ℝ₊), then we may consider the space 𝐼∶=𝐼₁∩𝐼₂∩⋯∩𝐼_𝑛 with respect to the norm|𝑢|_𝐼∶=max{|𝑢|_𝐼1,|𝑢|_𝐼2,…,|𝑢|_𝐼𝑛} (3.38)which is a Banach function space in 𝒯(ℝ₊). Now, if we analyze condition (ii) in Definition 3.5, it is obvious that if the input space is 𝐼(ℝ₊,𝑋), then in the right member of the inequality we have a “larger” norm and thus the estimation is more permissive.

Let 𝑞∈[1,∞], 𝑛∈ℕ^∗, and 𝑝₁,…,𝑝_𝑛∈(1,∞). We consider the space𝐼^{𝑝1,…,𝑝𝑛,𝑞}(ℝ₊,ℝ)=𝐿^𝑝1(ℝ₊,ℝ)∩⋯∩𝐿^𝑝𝑛(ℝ₊,ℝ)∩𝐿^𝑞(ℝ₊,ℝ) (3.39)which is a Banach space with respect the norm‖𝑢‖_{𝑝1,…,𝑝𝑛,𝑞}=max{‖𝑢‖_𝑝1,…,‖𝑢‖_𝑝𝑛,‖𝑢‖_𝑞}. (3.40)Corollary 3.12. The system (𝑆) is uniformly exponentially stable if and only if the system (𝑆) is (𝐼^{𝑝1,…,𝑝𝑛,𝑞}(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable.

Proof. If 𝑞∈[1,∞), then 𝐿^𝑞(ℝ₊,ℝ)∈𝒬(ℝ₊). If 𝑞=∞, then 𝐼∈ℒ(ℝ₊). By applying Theorem 3.9 we obtain the conclusion.

Corollary 3.13. Let 𝑊∈ℛ(ℝ₊). The system (𝑆) is uniformly exponentially stable if and only if (𝑆) is (𝑊(ℝ₊,𝑋),𝑊(ℝ₊,𝑋)) stable.

Proof. This follows from Theorem 3.9 and Lemma 2.21.

Lemma 3.14. Let 𝜈>0 and 𝑝,𝑞∈[1,∞] with 𝑝≤𝑞. For every 𝑢∈𝐿^𝑝(ℝ₊,ℝ), the function𝑓_𝑢∶ℝ₊→ℝ,𝑓_𝑢(𝑡)=^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}𝑢(𝑠)𝑑𝑠 (3.41)belongs to 𝐿^𝑞(ℝ₊,ℝ). In addition, there is 𝛾>0 such that ‖𝑓_𝑢‖_𝑞≤𝛾‖𝑢‖_𝑝, for all 𝑢∈𝐿^𝑝(ℝ₊,𝑋).

Proof. Using Hölder's inequality it follows that for every 𝑢∈𝐿^𝑝(ℝ₊,ℝ₊) the function 𝑓_𝑢∈𝐿^𝑞(ℝ₊,ℝ₊). Then, from|𝑓_𝑢(𝑡)|≤^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}|𝑢(𝑠)|𝑑𝑠,∀𝑡≥0,∀𝑢∈𝐿^𝑝(ℝ₊,ℝ) (3.42)we deduce that for every 𝑢∈𝐿^𝑝(ℝ₊,ℝ) the function 𝑓_𝑢 belongs to 𝐿^𝑞(ℝ₊,ℝ). Hence, it makes sense to define the linear operator Γ∶𝐿^𝑝(ℝ₊,ℝ)→𝐿^𝑞(ℝ₊,ℝ), Γ(𝑢)=𝑓_𝑢. It is easy to see that Γ is closed, so it is bounded. Setting 𝛾=‖Γ‖ we obtain the conclusion.

Corollary 3.15. Let 𝑝,𝑞∈[1,∞] with (𝑝,𝑞)≠(1,∞). The following assertions hold:(i)if the system (𝑆) is (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable, then (𝑆) is uniformly exponentially stable;(ii) if 𝑝≤𝑞, then the system (𝑆) is uniformly exponentially stable if and only if it is (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable.

Proof. (i) It follows from Theorem 3.9(i).
(ii) Necessity. Let 𝛾>0 be given by Lemma 3.14. Let 𝐾,𝜈>0 be such that ‖Φ(𝜃,𝑡)𝑥‖≤𝐾𝑒^−𝜈𝑡‖𝑥‖, for all (𝑥,𝜃)∈ℰ and all 𝑡≥0. Let (𝑢,𝜃)∈𝐶_0𝑐(ℝ₊,𝑋)×Θ. Then‖𝑥_𝜃(𝑡;0,𝑢)‖≤𝐾^𝑡₀𝑒^{−𝜈(𝑡−𝑠)}‖𝑢(𝑠)‖𝑑𝑠,∀𝑡≥0. (3.43)Using Lemma 3.14 we obtain that 𝑥_𝜃(⋅;0,𝑢)∈𝐿^𝑞(ℝ₊,𝑋) and‖𝑥_𝜃(⋅;0,𝑢)‖_𝐿^𝑞_(ℝ+,𝑋)≤𝐾𝛾‖𝑢‖_𝐿^𝑝_(ℝ+,𝑋). (3.44)Taking into account that 𝛾 and 𝐾 do not depend on 𝑢 or 𝜃, it follows that the system (𝑆) is (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable.
Sufficiency. This follows from (i).

Definition 3.16. Let 𝐼, 𝑂 be two Banach function spaces with 𝐼,𝑂∈𝒯(ℝ₊). The system (𝑆) is said to be completely(𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋))stable if the following assertions hold:(i)for every 𝑢∈𝐼(ℝ₊,𝑋) and every 𝜃∈Θ the solution 𝑥_𝜃(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋);(ii)there is 𝜆>0 such that ‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}, for all (𝑢,𝜃)∈𝐼(ℝ₊,𝑋)×Θ.Theorem 3.17. Let 𝐼,𝑂∈𝒯(ℝ₊) be such that 𝐼∈ℒ(ℝ₊) or 𝑂∈𝒬(ℝ₊). Then, the following assertions hold:(i) if the system (𝑆) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆) is uniformly exponentially stable;(ii) if 𝐼⊂𝑂 and one of the spaces 𝐼, 𝑂 belongs to the class ℛ(ℝ₊), then the system (𝑆) is uniformly exponentially stable if and only if the system (𝑆) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.

Proof. (i) This follows from Theorem 3.9(i).
(ii) Necessity follows using similar arguments as in the necessity part of Theorem 3.9(ii).
Sufficiency is given by (i).

Corollary 3.18. Let 𝑞∈[1,∞], 𝑛∈ℕ^∗ and 𝑝₁,…,𝑝_𝑛∈(1,∞). The system (𝑆) is uniformly exponentially stable if and only if the system (𝑆) is completely (𝐼^{𝑝1,…,𝑝𝑛,𝑞}(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable.

Proof. This follows from Corollary 3.12.

Corollary 3.19. Let 𝑊∈ℛ(ℝ₊). The system (𝑆) is uniformly exponentially stable if and only if (𝑆) is completely (𝑊(ℝ₊,𝑋),𝑊(ℝ₊,𝑋)) stable.

Remark 3.20. Let 𝑊∈ℛ(ℝ₊). If the system (𝑆) is uniformly exponentially stable, then for every 𝜃∈Θ the linear operator𝑃^𝜃_𝑊∶𝑊(ℝ₊,𝑋)→𝑊(ℝ₊,𝑋),(𝑃^𝜃_𝑊𝑢)(𝑡)=^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠 (3.45)is correctly defined and bounded. Moreover, if 𝜆>0 is given by Definition 3.16, then we have that sup_𝜃∈Θ‖𝑃^𝜃_𝑊‖≤𝜆.

Corollary 3.21. Let 𝑝,𝑞∈[1,∞] with (𝑝,𝑞)≠(1,∞). The following assertions hold:(i) if the system (𝑆) is completely (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable, then (𝑆) is uniformly exponentially stable;(ii) if 𝑝≤𝑞, then the system (𝑆) is uniformly exponentially stable if and only if it is completely (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋))stable.

Proof. (i) This follows from Corollary 3.15(i).
(ii) Necessity follows using similar arguments as in the necessity of Corollary 3.15(ii). Sufficiency follows from (i).

Remark 3.22. A distinct proof for Corollary 3.21(i) was given in [27] (see Theorem 5.2).

Let 𝐶_𝑏(ℝ₊,ℝ) be the space of all bounded continuous functions 𝑢∶ℝ₊→ℝ and 𝐶₀₀(ℝ₊,ℝ)={𝑢∈𝐶₀(ℝ₊,ℝ)∶𝑢(0)=0}. Definition 3.23. Let 𝑈,𝑌∈{𝐶_𝑏(ℝ₊,ℝ),𝐶₀(ℝ₊,ℝ),𝐶₀₀(ℝ₊,ℝ)}. The system (𝑆) is said to be completely(𝑈(ℝ₊,𝑋),𝑌(ℝ₊,𝑋))stable if the following assertions hold:(i)for every 𝑢∈𝑈(ℝ₊,𝑋) and every 𝜃∈Θ the solution 𝑥_𝜃(⋅;0,𝑢)∈𝑌(ℝ₊,𝑋);(i) there is 𝜆>0 such that ‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑌(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝑈(ℝ+,𝑋)}, for all (𝑢,𝜃)∈𝑈(ℝ₊,𝑋)×Θ.Corollary 3.24. Let 𝑈,𝑌∈{𝐶_𝑏(ℝ₊,ℝ),𝐶₀(ℝ₊,ℝ),𝐶₀₀(ℝ₊,ℝ)} with 𝑈⊂𝑌. The system (𝑆) is uniformly exponentially stable if and only if (𝑆) is completely (𝑈(ℝ₊,𝑋),𝑌(ℝ₊,𝑋)) stable.

Proof. Necessity is a simple exercise.
Sufficiency. If the system (𝑆) is completely (𝑈(ℝ₊,𝑋),𝑌(ℝ₊,𝑋)) stable, then it is (𝐿^∞(ℝ₊,𝑋),𝐿^∞(ℝ₊,𝑋)) stable. By applying Corollary 3.15 we deduce that (𝑆) is uniformly exponentially stable.

Remark 3.25. A different proof for Corollary 3.24 for the cases (𝑈,𝑌)=(𝐶₀₀(ℝ₊,ℝ),𝐶₀₀(ℝ₊,ℝ)) and (𝑈,𝑌)=(𝐶₀₀(ℝ₊,ℝ),𝐶_𝑏(ℝ₊,ℝ)) was given in [27] (see Theorem 5.1).

4. Stability radius

In this section, we will obtain a lower bound for the stability radius of linear skew-product flows in terms of input-output operators acting on Banach function spaces which belong to a class of rearrangement invariant spaces.

Let 𝑈, 𝑌 be Banach spaces and let Θ be a locally compact metric space. We denote by ℬ(𝑈,𝑌) the space of all bounded linear operators from 𝑈 into 𝑌 and by 𝒞_𝑠(Θ,ℬ(𝑈,𝑌)) the space of all continuous bounded mappings 𝐻∶Θ→ℬ(𝑈,𝑌). With respect to the norm ‖|𝐻‖|∶=sup_𝜃∈Θ‖𝐻(𝜃)‖, 𝒞_𝑠(Θ,ℬ(𝑈,𝑌)) is a Banach space.

Remark 4.1. If 𝜋=(Φ,𝜎) is a linear skew-product flow on ℰ=𝑋×Θ and 𝑃∈𝒞_𝑠(Θ,ℬ(𝑋)), then there exists a unique linear skew-product flow denoted 𝜋_𝑃=(Φ_𝑃,𝜎) on 𝑋×Θ such thatΦ_𝑃(𝜃,𝑡)𝑥=Φ(𝜃,𝑡)𝑥+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑃(𝜎(𝜃,𝑠))Φ_𝑃(𝜃,𝑠)𝑥𝑑𝑠 (4.1)for all (𝑥,𝜃,𝑡)∈𝑋×Θ×ℝ₊ (see [26, Theorem 2.1]).

Let 𝑋 be a Banach space, let Θ be a locally compact metric space, and let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ. We consider the variational integral control system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ (𝑆)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋.

For every 𝑃∈𝒞_𝑠(Θ,ℬ(𝑋)) we consider the perturbed system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ_𝑃(𝜃,𝑡)𝑥₀+^𝑡₀Φ_𝑃(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ (𝑆,𝑃)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋.

Let 𝑈,𝑌 be Banach spaces, let 𝐵∈𝒞_𝑠(Θ,ℬ(𝑈,𝑋)) and let 𝐶∈𝒞_𝑠(Θ,ℬ(𝑋,𝑌)). We consider the system (𝑆,𝐵,𝐶) described by the following integral model:𝑥_𝜃(𝑡,𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+∫^𝑡₀𝑦Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝐵(𝜎(𝜃,𝑠))𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ,_𝜃(𝑡,𝑥₀,𝑢)=𝐶(𝜎(𝜃,𝑡))𝑥_𝜃(𝑡,𝑥₀,𝑢),𝑡≥0,𝜃∈Θ, (4.2)where 𝑥₀∈𝑋 and 𝑢∈𝐿¹_loc(ℝ₊,𝑈).

Throughout this section, we suppose that (𝑆) is uniformly exponentially stable. The stability radius of (𝑆) with respect to the perturbation structure (𝐵,𝐶) is defined by𝑟_stab(𝑆,𝐵,𝐶)=sup{𝑟≥0∶∀Δ∈𝒞_𝑠(Θ,ℬ(𝑌,𝑈))with‖|Δ‖|≤𝑟⇒thesystem(𝑆,𝐵Δ𝐶)isuniformlyexponentiallystable}. (4.3)

Let 𝒱(ℝ₊) be the class of all Banach function spaces 𝐵∈ℛ(ℝ₊) with the property that for every 𝑢∈𝐵, |𝑢𝜒_[0,𝑛)|_𝐵→|𝑢|_𝐵 as 𝑛→∞.

Remark 4.2. It is easy to verify that the Orlicz spaces belong to the class 𝒱(ℝ₊).

Let 𝑉∈𝒱(ℝ₊). Since the system (𝑆) is uniformly exponentially stable from Remark 3.20 we have that for every 𝜃∈Θ, the linear operators𝑃^𝜃_𝑉∶𝑉(ℝ₊,𝑋)→𝑉(ℝ₊,𝑋),(𝑃^𝜃_𝑉𝑢)(𝑡)=^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠 (4.4)are bounded and sup_𝜃∈Θ‖𝑃^𝜃_𝑉‖<∞. Lemma 4.3. For every 𝜃∈Θ, the linear operators,𝐵^𝜃_𝑉∶𝑉(ℝ₊,𝑈)→𝑉(ℝ₊,𝑋),(𝐵^𝜃_𝑉𝐶𝑢)(𝑡)=𝐵(𝜎(𝜃,𝑡))𝑢(𝑡),_𝜃𝑉∶𝑉(ℝ₊,𝑋)→𝑉(ℝ₊,𝑌),(𝐶^𝜃_𝑉𝑢)(𝑡)=𝐶(𝜎(𝜃,𝑡))𝑢(𝑡), (4.5) are bounded. Moreover, sup_𝜃∈Θ‖𝐵^𝜃_𝑉‖≤‖|𝐵‖| and sup_𝜃∈Θ‖𝐶^𝜃_𝑉‖≤‖|𝐶‖|.

Proof. Let 𝜃∈Θ. For every 𝑢∈𝑉(ℝ₊,𝑈) we have that‖(𝐵^𝜃_𝑉𝑢)(𝑡)‖≤‖𝐵(𝜎(𝜃,𝑡))‖‖𝑢(𝑡)‖≤‖|𝐵‖|‖𝑢(𝑡)‖,∀𝑡≥0 (4.6)which implies that ‖𝐵^𝜃_𝑉𝑢‖_{𝑉(ℝ+,𝑋)}≤‖|𝐵‖|‖𝑢‖_{𝑉(ℝ+,𝑈)}. This shows that ‖𝐵^𝜃_𝑉‖≤‖|𝐵‖|, for all 𝜃∈Θ. Similarly, we obtain that ‖𝐶^𝜃_𝑉‖≤‖|𝐶‖|, for all 𝜃∈Θ.

Definition 4.4. For every 𝜃∈Θ, we consider the bounded linear operators𝐿^𝜃_𝑉∶𝑉(ℝ₊,𝑈)→𝑉(ℝ₊,𝑌),𝐿^𝜃_𝑉=𝐶^𝜃_𝑉𝑃^𝜃_𝑉𝐵^𝜃_𝑉. (4.7)The family {𝐿^𝜃_𝑉}_𝜃∈Θ is called the family of input-output operators associated to the system (𝑆,𝐵,𝐶). In all what follows we denote𝛼_𝑉(𝑆,𝐵,𝐶)∶=sup_𝜃∈Θ‖𝐿^𝜃_𝑉‖. (4.8)

Remark 4.5. Using Remark 3.20 and Lemma 4.3 we have that 𝛼_𝑉(𝑆,𝐵,𝐶)<∞.

For every Δ∈𝒞_𝑠(Θ,ℬ(𝑌,𝑈)) and every 𝜃∈Θ, we consider the bounded linear operatorΔ^𝜃_𝑉∶𝑉(ℝ₊,𝑌)→𝑉(ℝ₊,𝑈),(Δ^𝜃_𝑉𝑢)(𝑡)=Δ(𝜎(𝜃,𝑡))𝑢(𝑡). (4.9)Then we have that ‖Δ^𝜃_𝑉‖≤‖|Δ‖|, for all 𝜃∈Θ.

In what follows we suppose that there is 𝑚>0 such that ‖𝐶(𝜃)𝑥‖_𝑌≥𝑚‖𝑥‖_𝑋, for all (𝑥,𝜃)∈𝑋×Θ. Theorem 4.6. Let Δ∈𝒞_𝑠(Θ,ℬ(𝑌,𝑈)) with ‖|Δ‖|<1/𝛼_𝑉(𝑆,𝐵,𝐶). Then, for every (𝑥,𝜃)∈𝑋×Θ, the function𝑓_𝑥,𝜃∶ℝ₊→𝑋,𝑓_𝑥,𝜃(𝑡)=Φ_𝐵Δ𝐶(𝜃,𝑡)𝑥 (4.10)belongs to 𝑉(ℝ₊,𝑋).

Proof. Let 𝛾_𝑉=𝛼_𝑉(𝑆,𝐵,𝐶)‖|Δ‖|. Let 𝐾,𝜈>0 be such that ‖Φ(𝜃,𝑡)‖≤𝐾𝑒^−𝜈𝑡, for all (𝜃,𝑡)∈Θ×ℝ₊.
Let (𝑥,𝜃)∈ℰ. For every 𝑛∈ℕ^∗ let𝑓_𝑛∶ℝ₊→𝑋,𝑓_𝑛(𝑡)=𝜒_[0,𝑛)(𝑡)𝑓_𝑥,𝜃𝑔(𝑡),_𝑛∶ℝ₊→𝑋,𝑔_𝑛(𝑡)=𝐶(𝜎(𝜃,𝑡))𝑓_𝑛(𝑡). (4.11) Using Remark 4.1, we have that 𝑓_𝑛(𝑡)=𝜒_[0,𝑛)(𝑡)Φ(𝜃,𝑡)𝑥+(𝑃^𝜃_𝑉𝐵^𝜃_𝑉Δ^𝜃_𝑉𝑔_𝑛)(𝑡), for all 𝑡∈[0,𝑛), which implies that𝑔_𝑛(𝑡)=𝜒_[0,𝑛)(𝑡)𝐶(𝜎(𝜃,𝑡))Φ(𝜃,𝑡)𝑥+(𝐿^𝜃_𝑉Δ^𝜃_𝑉𝑔_𝑛)(𝑡),∀𝑡∈[0,𝑛). (4.12)Then, for 𝑡∈[0,𝑛) we have that‖𝑔_𝑛(𝑡)‖_𝑌≤‖|𝐶‖|𝐾‖𝑥‖𝑒^−𝜈𝑡+‖𝐿^𝜃_𝑉‖‖Δ^𝜃_𝑉‖‖𝑔_𝑛(𝑡)‖_𝑌≤‖|𝐶‖|𝐾‖𝑥‖𝑒^−𝜈𝑡+𝛾_𝑉‖𝑔_𝑛(𝑡)‖_𝑌 (4.13) Since 𝑔_𝑛(𝑡)=0 for 𝑡≥𝑛, we deduce that‖𝑔_𝑛(𝑡)‖_𝑌≤‖|𝐶‖|𝐾‖𝑥‖𝑒^−𝜈𝑡+𝛾_𝑉‖𝑔_𝑛(𝑡)‖_𝑌,∀𝑡≥0. (4.14)
Let 𝑒_𝜈∶ℝ₊→ℝ₊, 𝑒_𝜈(𝑡)=𝑒^−𝜈𝑡. From 𝑉∈𝒱(ℝ₊) and using Lemma 2.12 we have that 𝑒_𝜈∈𝑉. Then, from (4.14) it follows that‖𝑔_𝑛‖_{𝑉(ℝ+,𝑌)}≤𝛿_𝑥|𝑒_𝜈|_𝑉+𝛾_𝑉‖𝑔_𝑛‖_{𝑉(ℝ+,𝑌)}, (4.15)where 𝛿_𝑥=‖|𝐶‖|𝐾‖𝑥‖. The above inequality shows that‖𝑔_𝑛‖_{𝑉(ℝ+,𝑌)}≤𝛿_𝑥|𝑒_𝜈|_𝑉1−𝛾_𝑉,∀𝑛∈ℕ^∗. (4.16)Since 𝑉∈𝒱(ℝ₊) from (4.16) we deduce that the function 𝑔_𝑥,𝜃∶ℝ₊→𝑌, 𝑔_𝑥,𝜃(𝑡)=𝐶(𝜎(𝜃,𝑡))𝑓_𝑥,𝜃(𝑡) belongs to 𝑉(ℝ₊,𝑌). Then, from 𝑚‖𝑓_𝑥,𝜃(𝑡)‖_𝑋≤‖𝑔_𝑥,𝜃(𝑡)‖_𝑌, for all 𝑡≥0, we obtain that 𝑓_𝑥,𝜃∈𝑉(ℝ₊,𝑋).

Theorem 4.7. The following estimation holds:𝑟_stab1(𝑆,𝐵,𝐶)≥𝛼_𝑉.(𝑆,𝐵,𝐶) (4.17)

Proof. Let Δ∈𝒞_𝑠(Θ,ℬ(𝑌,𝑈)) be such that ‖|Δ‖|<1/𝛼_𝑉(𝑆,𝐵,𝐶). Let 𝛾_𝑉=𝛼_𝑉(𝑆,𝐵,𝐶)‖|Δ‖|.
We prove that the perturbed system (𝑆,𝐵Δ𝐶) is completely (𝑉(ℝ₊,𝑋),𝑉(ℝ₊,𝑋)) stable. Let 𝑣∈𝑉(ℝ₊,𝑋) and 𝜃∈Θ. The corresponding solution of the system (𝑆,𝐵Δ𝐶) is𝑥_𝜃(⋅;0,𝑣)∶ℝ₊→𝑋,𝑥_𝜃(𝑡;0,𝑣)=^𝑡₀Φ_𝐵Δ𝐶(𝜎(𝜃,𝑠),𝑡−𝑠)𝑣(𝑠)𝑑𝑠. (4.18)Let 𝑛∈ℕ^∗. We set 𝑣_𝑛=𝑣𝜒_[0,𝑛) and 𝑥_𝑛=𝑥_𝜃(𝑛;0,𝑣). Then, we have that𝑥_𝜃(𝑡;0,𝑣_𝑛)=Φ_𝐵Δ𝐶(𝜎(𝜃,𝑛),𝑡−𝑛)𝑥_𝑛,∀𝑡≥𝑛. (4.19)From Theorem 4.6 we have that the function𝜑∶ℝ₊→𝑋,𝜑(𝑡)=Φ(𝜎(𝜃,𝑛),𝑡)𝑥_𝑛 (4.20)belongs to 𝑉(ℝ₊,𝑋). Since 𝑉 is invariant to translations, it follows that𝜓∶ℝ₊→𝑋,𝜓(𝑡)=Φ(𝜎(𝜃,𝑛),𝑡−𝑛)𝑥_𝑛,𝑡≥𝑛,0,𝑡∈[0,𝑛), (4.21)belongs to 𝑉(ℝ₊,𝑋).
Since 𝑥_𝜃(⋅;0,𝑣_𝑛) is continuous, setting 𝑀_𝑛=sup_{𝑡∈[0,𝑛]}‖𝑥_𝜃(𝑡;0,𝑣_𝑛)‖ and using (4.19) we deduce that‖𝑥_𝜃(𝑡;0,𝑣_𝑛)‖≤𝑀_𝑛𝜒_[0,𝑛)(𝑡)+‖𝜓(𝑡)‖,∀𝑡≥0 (4.22)which implies that 𝑥_𝜃(⋅;0,𝑣_𝑛)∈𝑉(ℝ₊,𝑋).
Using Remark 4.1 and Fubini's theorem we deduce that(𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛))(𝑡)=(𝐶^𝜃_𝑉𝑃^𝜃_𝑉𝑣_𝑛)(𝑡)+(𝐿^𝜃_𝑉Δ^𝜃_𝑉𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛))(𝑡),∀𝑡≥0. (4.23)This implies that‖𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑌)}≤‖|𝐶‖|sup_𝜃∈Θ‖𝑃^𝜃_𝑉‖‖𝑣‖_{𝑉(ℝ+,𝑋)}+𝛾_𝑉‖𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑌)}. (4.24) Setting 𝛿=‖|𝐶‖|sup_𝜃∈Θ‖𝑃^𝜃_𝑉‖ it follows that‖𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑌)}≤𝛿1−𝛾_𝑉‖𝑣‖_{𝑉(ℝ+,𝑋)}. (4.25)From ‖𝐶^𝜃_𝑉𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑌)}≥𝑚‖𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑋)}, using (4.25) we deduce that‖𝑥_𝜃(⋅;0,𝑣_𝑛)‖_{𝑉(ℝ+,𝑋)}≤𝛿𝑚(1−𝛾_𝑉)‖𝑣‖_{𝑉(ℝ+,𝑋)}. (4.26)Taking into account that 𝑥_𝜃(𝑡;0,𝑣)=𝑥_𝜃(𝑡;0,𝑣_𝑛), for all 𝑡∈[0,𝑛), from (4.26) it follows that‖𝑥_𝜃(⋅;0,𝑣)𝜒_[0,𝑛)‖_{𝑉(ℝ+,𝑋)}≤𝛿𝑚(1−𝛾_𝑉)‖𝑣‖_{𝑉(ℝ+,𝑋)},∀𝑛∈ℕ^∗. (4.27)Since 𝑉∈𝒱(ℝ₊) as 𝑛→∞ in (4.27), we obtain that‖𝑥_𝜃(⋅;0,𝑣)‖_{𝑉(ℝ+,𝑋)}≤𝛿𝑚(1−𝛾_𝑉)‖𝑣‖_{𝑉(ℝ+,𝑋)}. (4.28)Since 𝛿,𝑚,𝛾_𝑉 do not depend on 𝜃 or 𝑣, we deduce that the system (𝑆,𝐵Δ𝐶) is completely (𝑉(ℝ₊,𝑋),𝑉(ℝ₊,𝑋)) stable. By applying Corollary 3.19, we have that the system (𝑆,𝐵Δ𝐶) is uniformly exponentially stable and the proof is complete.

The main result of this section is the following. Theorem 4.8. The following estimation holds:𝑟_stab(𝑆,𝐵,𝐶)≥sup_{𝑉∈𝒱(ℝ+)}1𝛼_𝑉.(𝑆,𝐵,𝐶) (4.29)

Proof. This follows from Theorem 4.7.

A lower bound for the stability radius of (𝑆) with respect to the perturbation structure (𝐵,𝐶) in terms of 𝐿^𝑝-spaces is given by the following.

Corollary 4.9. The following estimation holds:𝑟_stab(𝑆,𝐵,𝐶)≥sup_{𝑝∈[1,∞]}1𝛼_𝐿^𝑝.(𝑆,𝐵,𝐶) (4.30)

5. Applications for nonautonomous systems

In this section, we apply the main results of this paper in order to obtain characterizations for uniform exponential stability of nonautonomous systems and deduce lower bound for the stability radius of such systems.

Let 𝑋 be a Banach space and let 𝐼_𝑑 be the identity operator on 𝑋. Definition 5.1. A family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0}⊂ℬ(𝑋) is called an evolution family if the following properties hold:(i)𝑈(𝑡,𝑡)=𝐼_𝑑 and 𝑈(𝑡,𝑠)𝑈(𝑠,𝑡₀)=𝑈(𝑡,𝑡₀), for all 𝑡≥𝑠≥𝑡₀≥0;(ii)there are 𝑀≥1 and 𝜔>0 such that ‖𝑈(𝑡,𝑠)‖≤𝑀𝑒^{𝜔(𝑡−𝑠)}, for all 𝑡≥𝑠≥0;(iii) for every 𝑥∈𝑋 the mapping (𝑡,𝑠)↦𝑈(𝑡,𝑠)𝑥 is continuous.

Let 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} be an evolution family on 𝑋. We consider the nonautonomous integral control system𝑥_𝑠(𝑡;𝑥₀,𝑢)=𝑈(𝑡,𝑠)𝑥₀+^𝑡_𝑠𝑈(𝑡,𝜏)𝑢(𝜏)𝑑𝜏,𝑡≥𝑠,𝑠≥0 (𝑆_𝒰)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋. Definition 5.2. The system (𝑆_𝒰) is said to be uniformly exponentially stable if there are 𝐾,𝜈>0 such that ‖𝑥_𝑠(𝑡;𝑥₀,0)‖≤𝐾𝑒^{−𝜈(𝑡−𝑠)}‖𝑥₀‖, for all 𝑡≥𝑠≥0 and all 𝑥₀∈𝑋.Definition 5.3. Let 𝐼, 𝑂 be two Banach function spaces with 𝐼,𝑂∈𝒯(ℝ₊). The system (𝑆_𝒰) is said to be (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋))stable if the following properties hold:(i)for every 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) the solution 𝑥₀(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋);(ii) there is 𝜆>0 such that ‖𝑥₀(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}, for all 𝑢∈𝐶_0𝑐(ℝ₊,𝑋).Lemma 5.4. Let 𝐵∈𝒯(ℝ₊), 𝑢∈𝐵, and 𝑡>0. Then, the function 𝑢_𝑡∶ℝ₊→ℝ, 𝑢_𝑡(𝑠)=𝑢(𝑠+𝑡) belongs to 𝐵 and |𝑢_𝑡|_𝐵≤|𝑢|_𝐵.

Proof. Let𝑣∶ℝ₊→ℝ,𝑣(𝑠)=𝑢(𝑠),𝑠≥𝑡,0,𝑠∈[0,𝑡). (5.1)Then 𝑣∈𝐵 and |𝑣|_𝐵≤|𝑢|_𝐵. Using the invariance to translations of 𝐵, we obtain that 𝑢_𝑡∈𝐵 and |𝑢_𝑡|_𝐵=|𝑣|_𝐵, which concludes the proof.

The first main result of this section is as follows. Theorem 5.5. Let 𝐼,𝑂∈𝒯(ℝ₊) be such that 𝐼∈ℒ(ℝ₊) and 𝑂∈𝒬(ℝ₊). Then, the following assertions hold:(i) if the system (𝑆_𝒰) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆_𝒰) is uniformly exponentially stable;(ii) if 𝐼⊂𝑂 and one of the spaces 𝐼 or 𝑂 belongs to ℛ(ℝ₊), then the system (𝑆_𝒰) is uniformly exponentially stable if and only if the system (𝑆_𝒰) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.

Proof. (i) Let 𝜆>0 be given by Definition 5.3.
Let Θ=ℝ₊ and let 𝜎∶Θ×ℝ₊→Θ, 𝜎(𝜃,𝑡)=𝜃+𝑡. For every (𝜃,𝑡)∈Θ×ℝ₊, let Φ(𝜃,𝑡)=𝑈(𝑡+𝜃,𝜃). Then 𝜋=(Φ,𝜎) is a linear skew-product flow on 𝑋×Θ. We consider the variational integral control system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ. (𝑆)We prove that (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.
Indeed, let 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) and let 𝜃∈Θ. We consider the function𝑢_𝜃∶ℝ₊→𝑋,𝑢_𝜃(𝑠)=𝑢(𝑠−𝜃),𝑠≥𝜃,0,𝑠∈[0,𝜃). (5.2)Then 𝑢_𝜃∈𝐶_0𝑐(ℝ₊,𝑋) and from 𝐼∈𝒯(ℝ₊) we have that ‖𝑢_𝜃‖_{𝐼(ℝ+,𝑋)}=‖𝑢‖_{𝐼(ℝ+,𝑋)}. From hypothesis we have that the function𝑥₀(𝑡;0,𝑢_𝜃)=^𝑡₀𝑈(𝑡,𝜏)𝑢_𝜃(𝜏)𝑑𝜏,𝑡≥0 (5.3)belongs to 𝑂(ℝ₊,𝑋). Because 𝑂(ℝ₊,𝑋)∈𝒯(ℝ₊), using Lemma 5.4 we have that the function𝜑∶ℝ₊→𝑋,𝜑(𝑡)=𝑥₀(𝑡+𝜃;0,𝑢_𝜃) (5.4)belongs to 𝑂(ℝ₊,𝑋) and ‖𝜑‖_{𝑂(ℝ+,𝑋)}≤‖𝑥₀(⋅;0,𝑢_𝜃)‖_{𝑂(ℝ+,𝑋)}. Observing that𝜑(𝑡)=₀^𝑡+𝜃𝑈(𝑡+𝜃,𝜏)𝑢_𝜃(𝜏)𝑑𝜏=_𝜃^𝑡+𝜃𝑈(𝑡+𝜃,𝜏)𝑢(𝜏−𝜃)𝑑𝜏=^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠=𝑥_𝜃(𝑡;0,𝑢),∀𝑡≥0, (5.5) it follows that‖𝑥_𝜃(⋅;0,𝑢)‖_{𝑂(ℝ+,𝑋)}≤‖𝑥₀(⋅;0,𝑢_𝜃)‖_{𝑂(ℝ+,𝑋)}≤𝜆‖𝑢_𝜃‖_{𝐼(ℝ+,𝑋)}=𝜆‖𝑢‖_{𝐼(ℝ+,𝑋)}. (5.6)Since 𝑢∈𝐶_0𝑐(ℝ₊,𝑋) and 𝜃∈Θ were arbitrary, we deduce that the system (𝑆) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable. By applying Theorem 3.9(i) we deduce that the system (𝑆) is uniformly exponentially stable, so there are 𝐾,𝜈>0 such that‖Φ(𝜃,𝑡)𝑥₀‖≤𝐾𝑒^−𝜈𝑡‖𝑥₀‖,∀𝑡≥0,∀𝑥₀∈𝑋. (5.7)This implies that‖𝑥_𝑠(𝑡;𝑥₀,0)‖=‖𝑈(𝑡,𝑠)𝑥₀‖=‖Φ(𝑠,𝑡−𝑠)𝑥₀‖≤𝐾𝑒^{−𝜈(𝑡−𝑠)}‖𝑥₀‖,∀𝑡≥𝑠≥0,∀𝑥₀∈𝑋. (5.8) In conclusion, the system (𝑆_𝒰) is uniformly exponentially stable.
(ii) Necessity follows using similar arguments as those used in the proof of Theorem 3.9. Sufficiencyfollows from (i).

Remark 5.6. From Example 3.10 we deduce that the result given by Theorem 5.5 is the most general in this topic. Precisely, if 𝐼∉ℒ(ℝ₊) or 𝑂∉𝒬(ℝ₊), then the (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stability of the system (𝑆_𝒰) does not assure the uniform exponential stability of (𝑆_𝒰).Definition 5.7. Let 𝐼,𝑂∈𝒯(ℝ₊). The system (𝑆_𝒰) is said to be completely(𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋))stable if for every 𝑢∈𝐼(ℝ₊,𝑋), the solution 𝑥₀(⋅;0,𝑢)∈𝑂(ℝ₊,𝑋).

Remark 5.8. If the system (𝑆_𝒰) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then it makes sense to consider the linear operator𝒫∶𝐼(ℝ₊,𝑋)→𝑂(ℝ₊,𝑋),𝒫(𝑢)=𝑥₀(⋅;0,𝑢). (5.9)It is easy to see that 𝒫 is closed, so it is bounded.

Remark 5.9. If the system (𝑆_𝒰) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆_𝒰) is (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable (with 𝜆=‖𝒫‖).

As an application of Theorem 5.5 and using Remark 5.9 we obtain the following. Theorem 5.10. Let 𝐼,𝑂∈𝒯(ℝ₊) be such that 𝐼∈ℒ(ℝ₊) and 𝑂∈𝒬(ℝ₊). Then, the following assertions hold:(i) if the system (𝑆_𝒰) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable, then (𝑆_𝒰) is uniformly exponentially stable;(ii) if 𝐼⊂𝑂 and one of the spaces 𝐼 or 𝑂 belongs to ℛ(ℝ₊), then the system (𝑆_𝒰) is uniformly exponentially stable if and only if the system (𝑆_𝒰) is completely (𝐼(ℝ₊,𝑋),𝑂(ℝ₊,𝑋)) stable.Corollary 5.11. If 𝑊∈ℛ(ℝ₊), then the system (𝑆_𝒰) is uniformly exponentially stable if and only if the system (𝑆_𝒰) is completely (𝑊(ℝ₊,𝑋),𝑊(ℝ₊,𝑋)) stable.Corollary 5.12. Let 𝑝,𝑞∈[1,∞] with (𝑝,𝑞)≠(1,∞). The following assertions hold:(i) if the system (𝑆_𝒰) is completely (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable, then (𝑆_𝒰) is uniformly exponentially stable;(ii) if 𝑝≤𝑞, then the system (𝑆_𝒰) is uniformly exponentially stable if and only if it is completely (𝐿^𝑝(ℝ₊,𝑋),𝐿^𝑞(ℝ₊,𝑋)) stable.

Remark 5.13. For 𝑝,𝑞∈[1,∞), Corollary 5.12(i) was firstly proved by Datko in [16]. Recently, for 𝑝=𝑞∈[1,∞), Corollary 5.11 was proved by Clark et al. in [11], using evolution semigroups techniques. A different proof for this corollary was given by Megan et al. in [23]. The autonomous case was treated by van Neerven in [30], for 𝑝=𝑞∈[1,∞).

Corollary 5.14. Let 𝑈,𝑌∈{𝐶_𝑏(ℝ₊,ℝ),𝐶₀(ℝ₊,ℝ),𝐶₀₀(ℝ₊,ℝ)} with 𝑈⊂𝑌. The system (𝑆_𝒰) is uniformly exponentially stable if and only if the system (𝑆_𝒰) is completely (𝑈(ℝ₊,𝑋),𝑌(ℝ₊,𝑋)) stable.

Proof. Necessity is immediate.
Sufficiency follows from Theorem 5.10 observing that the complete (𝑈(ℝ₊,𝑋),𝑌(ℝ₊,𝑋)) stability implies the (𝐿^∞(ℝ₊,𝑋),𝐿^∞(ℝ₊,𝑋))stability of (𝑆_𝒰).

Remark 5.15. A distinct proof of Corollary 5.14 was given in [23]. The case 𝑈=𝑌=𝐶₀(ℝ₊,ℝ) was obtained in [11], reducing the nonautonomous case to the autonomous case and using evolution semigroups.

An interesting application of the results obtained in the fourth section is the study of the persistence of stability properties for the case of nonautonomous systems. We mention that the nonautonomous case was treated by Hinrichsen et al. in valuable papers (see [17–19, 41]), which represent the starting point for many research studies in this area.

In what follows, we will deduce a lower bound for the stability radius of nonautonomous systems, as a consequence of the behavior in the variational case, providing the connections between these cases.

Let 𝑋 be a Banach space and let 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} be an evolution family on 𝑋.

Remark 5.16. For every 𝑃∈𝒞_𝑠(ℝ₊,ℬ(𝑋)) (see, e.g., [14]) there is a unique evolution family 𝒰_𝑃={𝑈_𝑃(𝑡,𝑠)}_{𝑡≥𝑠≥0} such that𝑈_𝑃(𝑡,𝑠)𝑥=𝑈(𝑡,𝑠)𝑥+^𝑡_𝑠𝑈(𝑡,𝜏)𝑃(𝜏)𝑈_𝑃(𝜏,𝑠)𝑥𝑑𝜏,∀𝑡≥𝑠≥0,∀𝑥∈𝑋. (5.10)

We consider the nonautonomous integral control system𝑥_𝑠(𝑡;𝑥₀,𝑢)=𝑈(𝑡,𝑠)𝑥₀+^𝑡_𝑠𝑈(𝑡,𝜏)𝑢(𝜏)𝑑𝜏,𝑡≥𝑠,𝑠≥0 (𝑆_𝒰)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋), 𝑥₀∈𝑋, and respectively, for every 𝑃∈𝒞_𝑠(ℝ₊,𝐵(𝑋)) we consider the perturbed system𝑥_𝑠(𝑡;𝑥₀,𝑢)=𝑈_𝑃(𝑡,𝑠)𝑥₀+^𝑡_𝑠𝑈_𝑃(𝑡,𝜏)𝑢(𝜏)𝑑𝜏,𝑡≥𝑠,𝑠≥0 (𝑆_𝒰,𝑃)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋), 𝑥₀∈𝑋.

Let 𝑈, 𝑌 be Banach spaces, let 𝐵∈𝒞_𝑠(ℝ₊,ℬ(𝑈,𝑋)) and let 𝐶∈𝒞_𝑠(ℝ₊,ℬ(𝑋,𝑌)). We consider the system (𝑆_𝒰,𝐵,𝐶) described by the following integral model𝑥_𝑠(𝑡;𝑥₀,𝑢)=𝑈(𝑡,𝑠)𝑥₀+∫^𝑡_𝑠𝑦𝑈(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,𝑡≥𝑠,𝑠≥0,_𝑠(𝑡;𝑥₀,𝑢)=𝐶(𝑡)𝑥_𝑠(𝑡;𝑥₀,𝑢),𝑡≥𝑠,𝑠≥0, (5.11)with 𝑢∈𝐿¹_loc(ℝ₊,𝑈), 𝑥₀∈𝑋.

In what follows we suppose that (𝑆_𝒰) is uniformly exponentially stable. The stability radius of (𝑆_𝒰) with respect to the perturbation structure (𝐵,𝐶) is defined by𝑟_stab(𝑆_𝒰,𝐵,𝐶)=sup{𝑟≥0∶∀Δ∈𝒞_𝑠(ℝ₊,ℬ(𝑌,𝑈))with‖|Δ‖|≤𝑟⇒(𝑆_𝒰,𝐵Δ𝐶)isuniformlyexponentiallystable}. (5.12)

Let 𝑉∈𝒱(ℝ₊). Since the system (𝑆_𝒰) is uniformly exponentially stable, we have that (𝑆_𝒰) is completely (𝑉(ℝ₊,𝑋),𝑉(ℝ₊,𝑋)) stable, so it makes sense to consider the linear operator𝒫_𝑉∶𝑉(ℝ₊,𝑋)→𝑉(ℝ₊,𝑋),(𝒫_𝑉𝑢)(𝑡)=^𝑡₀𝑈(𝑡,𝜏)𝑢(𝜏)𝑑𝜏. (5.13)It is easy to verify that 𝒫_𝑉 is closed, so it is bounded.

We consider the operatorsℬ_𝑉∶𝑉(ℝ₊,𝑈)→𝑉(ℝ₊,𝑋),(ℬ_𝑉𝒞𝑢)(𝑡)=𝐵(𝑡)𝑢(𝑡),_𝑉∶𝑉(ℝ₊,𝑈)→𝑉(ℝ₊,𝑋),(𝒞_𝑉𝑢)(𝑡)=𝐶(𝑡)𝑢(𝑡), (5.14) and we note that ℬ_𝑉, 𝒞_𝑉 are bounded linear operators with ‖ℬ_𝑉‖≤‖|𝐵‖| and ‖𝒞_𝑉‖≤‖|𝐶‖|. Definition 5.17. The operatorℒ_𝑉∶𝑉(ℝ₊,𝑈)→𝑉(ℝ₊,𝑌),ℒ_𝑉(𝑢)=(𝒞_𝑉𝒫_𝑉ℬ_𝑉)(𝑢), (5.15)is called the input-output operator associated with the system (𝑆_𝒰,𝐵,𝐶).

In what follows, we suppose that there is 𝑚>0 such that ‖𝐶(𝑡)𝑥‖_𝑌≥𝑚‖𝑥‖, for all 𝑥∈𝑋 and all 𝑡≥0.

The second main result of this section is the following.Theorem 5.18. The following estimation holds:𝑟_stab(𝑆_𝒰1,𝐵,𝐶)≥‖ℒ_𝑉‖. (5.16)

Proof. Let Δ∈𝒞_𝑠(ℝ₊,ℬ(𝑌,𝑈)) be such that ‖|Δ‖|‖ℒ_𝑉‖<1.
Let Θ=ℝ₊,𝜎∶Θ×ℝ₊→Θ,𝜎(𝜃,𝑡)=𝜃+𝑡 and let Φ(𝜃,𝑡)=𝑈(𝑡+𝜃,𝜃), for all 𝜃,𝑡≥0. Then 𝜋=(Φ,𝜎) is a linear skew-product flow. We consider the variational integral control system𝑥_𝜃(𝑡;𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+^𝑡₀Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ (𝑆)with 𝑢∈𝐿¹_loc(ℝ₊,𝑋) and 𝑥₀∈𝑋 and, respectively, the system (𝑆,𝐵,𝐶) given by𝑥_𝜃(𝑡,𝑥₀,𝑢)=Φ(𝜃,𝑡)𝑥₀+∫^𝑡₀𝑦Φ(𝜎(𝜃,𝑠),𝑡−𝑠)𝐵(𝜎(𝜃,𝑠))𝑢(𝑠)𝑑𝑠,𝑡≥0,𝜃∈Θ,_𝜃(𝑡,𝑥₀,𝑢)=𝐶(𝜎(𝜃,𝑡))𝑥_𝜃(𝑡,𝑥₀,𝑢),𝑡≥0,𝜃∈Θ, (5.17)where 𝑥₀∈𝑋 and 𝑢∈𝐿¹_loc(ℝ₊,𝑈).
For every 𝜃∈Θ we associate with the system (𝑆,𝐵,𝐶) the operators 𝑃^𝜃_𝑉, 𝐵^𝜃_𝑉, 𝐶^𝜃_𝑉, and 𝐿^𝜃_𝑉 according to the definitions in Section 4. We denote by 𝛼_𝑉(𝑆,𝐵,𝐶)∶=sup_𝜃∈Θ‖𝐿^𝜃_𝑉‖.
We prove that 𝛼_𝑉(𝑆,𝐵,𝐶)≤‖ℒ_𝑉‖. Indeed, let 𝜃∈Θ and let 𝑢∈𝑉(ℝ₊,𝑈). Since 𝑉 is invariant to translations, the function𝑢_𝜃∶ℝ₊→𝑈,𝑢_𝜃(𝑡)=𝑢(𝑡−𝜃),𝑡≥𝜃,0,𝑡∈[0,𝜃), (5.18)belongs to 𝑉(ℝ₊,𝑈) and ‖𝑢_𝜃‖_{𝑉(ℝ+,𝑈)}=‖𝑢‖_{𝑉(ℝ+,𝑈)}. In addition, using Lemma 5.4 we have that the function𝜑∶ℝ₊→𝑌,𝜑(𝑡)=(ℒ_𝑉𝑢_𝜃)(𝑡+𝜃), (5.19)belongs to 𝑉(ℝ₊,𝑌) and ‖𝜑‖_{𝑉(ℝ+,𝑌)}≤‖ℒ_𝑉(𝑢_𝜃)‖_{𝑉(ℝ+,𝑌)}. We observe that𝐿^𝜃_𝑉(𝑢)(𝑡)=𝐶(𝜃+𝑡)^𝑡₀𝑈(𝜃+𝑡,𝜃+𝑠)𝐵(𝜃+𝑠)𝑢(𝑠)𝑑𝑠=𝐶(𝜃+𝑡)_𝜃^𝜃+𝑡𝑈(𝜃+𝑡,𝜏)𝐵(𝜏)𝑢(𝜏−𝜃)𝑑𝜏=𝐶(𝜃+𝑡)₀^𝜃+𝑡𝑈(𝜃+𝑡,𝜏)𝐵(𝜏)𝑢_𝜃(𝜏)𝑑𝜏=[ℒ_𝑉(𝑢_𝜃)](𝑡+𝜃)=𝜑(𝑡),∀𝑡≥0. (5.20) This implies that‖𝐿^𝜃_𝑉(𝑢)‖_{𝑉(ℝ+,𝑌)}=‖𝜑‖_{𝑉(ℝ+,𝑌)}≤‖ℒ_𝑉(𝑢_𝜃)‖_{𝑉(ℝ+,𝑌)}≤‖ℒ_𝑉‖‖𝑢_𝜃‖_{𝑉(ℝ+,𝑈)}=‖ℒ_𝑉‖‖𝑢‖_{𝑉(ℝ+,𝑈)} (5.21) Since 𝑢∈𝑉(ℝ₊,𝑈) and 𝜃∈Θ were arbitrary, from the above inequality we obtain that ‖𝐿^𝜃_𝑉‖≤‖ℒ_𝑉‖, for all 𝜃∈Θ, so 𝛼_𝑉(𝑆,𝐵,𝐶)≤‖ℒ_𝑉‖.
Since ‖|Δ‖|‖ℒ_𝑉‖<1, we have that ‖|Δ‖|𝛼_𝑉(𝑆,𝐵,𝐶)<1. Then, from Theorem 4.7 it follows that the perturbed system (𝑆,𝐵Δ𝐶) is uniformly exponentially stable, so there are 𝐾,𝜈>0 such that‖Φ_𝐵Δ𝐶(𝜃,𝑡)‖≤𝐾𝑒^−𝜈𝑡,∀𝑡≥0,∀𝜃∈Θ. (5.22)ButΦ_𝐵Δ𝐶(𝜃,𝑡)𝑥=𝑈(𝑡+𝜃,𝜃)𝑥+^𝑡₀𝑈(𝜃+𝑡,𝜃+𝑠)(𝐵Δ𝐶)(𝜃+𝑠)Φ_𝐵Δ𝐶(𝜃,𝑠)𝑥𝑑𝑠,∀𝜃,𝑡≥0,∀𝑥∈𝑋. (5.23) which is equivalent toΦ_𝐵Δ𝐶(𝜃,𝑡−𝜃)𝑥=𝑈(𝑡,𝜃)𝑥+^𝑡_𝜃𝑈(𝑡,𝜏)(𝐵Δ𝐶)(𝜏)Φ_𝐵Δ𝐶(𝜃,𝜏−𝜃)𝑥𝑑𝜏,∀𝑡≥𝜃≥0,∀𝑥∈𝑋. (5.24) On the other hand, we have that𝑈_𝐵Δ𝐶(𝑡,𝜃)𝑥=𝑈(𝑡,𝜃)𝑥+^𝑡_𝜃𝑈(𝑡,𝜏)(𝐵Δ𝐶)(𝜏)𝑈_𝐵Δ𝐶(𝜏,𝜃)𝑥𝑑𝜏,∀𝑡≥𝜃≥0,∀𝑥∈𝑋. (5.25)
Let 𝜃≥0. For every 𝑇>0 and 𝑥∈𝑋, let𝜑_𝑥,𝑇∶[𝜃,𝜃+𝑇]→𝑋,𝜑_𝑥,𝑇(𝜏)=Φ_𝐵Δ𝐶(𝜃,𝜏−𝜃)𝑥−𝑈_𝐵Δ𝐶(𝜏,𝜃)𝑥. (5.26)If 𝑀,𝜔>0 are such that ‖𝑈(𝑡,𝑠)‖≤𝑀𝑒^{𝜔(𝑡−𝑠)}, for all 𝑡≥𝑠≥0, then from (5.24) and (5.25) it follows that‖𝜑_𝑥,𝑇(𝑡)‖≤𝐾^𝑡_𝜃‖𝜑_𝑥,𝑇(𝜏)‖𝑑𝜏,∀𝑡∈[𝜃,𝜃+𝑇], (5.27)where 𝐾=𝑀𝑒^𝜔𝑇‖|𝐵Δ𝐶‖|. From (5.27), using Gronwall's lemma, we deduce that 𝜑_𝑥,𝑇(𝜏)=0, for all 𝜏∈[𝜃,𝜃+𝑇]. This implies thatΦ_𝐵Δ𝐶(𝜃,𝜏−𝜃)𝑥=𝑈_𝐵Δ𝐶(𝜏,𝜃)𝑥,∀𝜏∈[𝜃,𝜃+𝑇],∀𝑥∈𝑋. (5.28)Since 𝑇>0, 𝑥∈𝑋, and 𝜃∈Θ were arbitrary, we deduce thatΦ_𝐵Δ𝐶(𝜃,𝜏−𝜃)=𝑈_𝐵Δ𝐶(𝜏,𝜃),∀𝜏≥𝜃≥0. (5.29)Then, from (5.22) we obtain that‖𝑈_𝐵Δ𝐶(𝜏,𝜃)‖≤𝐾𝑒^{−𝜈(𝜏−𝜃)},∀𝜏≥𝜃≥0. (5.30)In conclusion, the perturbed system (𝑆_𝒰,𝐵Δ𝐶) is uniformly exponentially stable and the proof is complete.

Concluding the above estimations, we deduce the following. Theorem 5.19. The following estimation holds:𝑟_stab(𝑆_𝒰,𝐵,𝐶)≥sup_{𝑉∈𝒱(ℝ+)}1‖ℒ_𝑉‖. (5.31)

A lower bound for the stability radius of nonautonomous systems in terms of 𝐿^𝑝-spaces is given by the following. Corollary 5.20. The following estimation holds:𝑟_stab(𝑆_𝒰,𝐵,𝐶)≥sup_{𝑝∈[1,∞]}1‖ℒ_𝐿^𝑝‖. (5.32)

Remark 5.21. For the case 𝑝∈[1,∞), the result given by Corollary 5.20 was firstly proved in [19] (see [19, Theorem 3.2]) for a very general class of nonautonomous systems described by mild evolution families. A distinct proof for Corollary 5.20 was presented in [11], for 𝑝∈[1,∞), using evolution semigroups techniques (see [11, Theorem 4.2]).

Remark 5.22. A distinct approach for the estimation of the stability radius was given by Jacob in [20] for the case of time-varying systems in finite dimensional spaces. There, the author considered a time-varying system (𝐴,𝐵,𝐶) with 𝐴∈𝐿^∞_loc([0,∞),𝐊^𝑛×𝑛), 𝐵∈𝐿^∞([0,∞),𝐊^𝑛×𝑚), 𝐶∈𝐿^∞([0,∞),𝐊^𝑞×𝑛), where 𝐊∈{ℝ,𝐂}. The family of input-output operators {𝐿_𝑡0}_𝑡0≥0 is defined by(𝐿_𝑡0)𝑢(𝑡)∶=𝐶(𝑡)^𝑡_𝑡0Φ_𝐴(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,𝑡≥𝑡₀≥0,𝑢∈𝐿^𝑝([𝑡₀,∞),𝐊^𝑚), (5.33)where {Φ_𝐴(𝑡,𝑠)}_{𝑡≥𝑠≥0} denotes the evolution family generated by A. In this case, the author proved that𝑟_stab(𝐴,𝐵,𝐶)=sup_𝑡0≥01‖𝐿_𝑡0‖. (5.34)

Acknowledgments

The author is grateful to the referee for his suggestions and comments, which led to the improvement of the paper. This work is supported by the CEEX Research Grant ET 4/2006 and by a PN II type Research Grant from West University of Timişoara.