Integral input-to-state stability of unbounded bilinear control systems

We study input-to-state stability of bilinear control systems with possibly unbounded control operators. Natural sufficient conditions for integral input-to-state stability are given. The obtained results are applied to a bilinearly controlled Fokker-Planck equation.


Introduction
In this note we continue recent developments on input-to-state stability (ISS) for systems governed by evolution equations. This concept unifies both asymptotic stability with respect to the initial values and robustness with respect to the external inputs such as controls or disturbances. Loosely, if a system Σ is viewed as a mapping which sends initial values x 0 P X and inputs u : r0, 8q Ñ U to the time evolution x : r0, T q Ñ X for some maximal T ą 0, then Σ is ISS if T " 8 and for all t P r0, 8q, }xptq} X ď βp}x 0 } X , tq`γp sup sPr0,ts }upsq} U q, @x 0 , u, where the continuous functions β : R0ˆR0 Ñ R0 and γ : R0 Ñ R0 are of Lyapunov class KL and K respectively 1 . Here X is called the state space and U the input space equipped with norms }¨} X and }¨} U . For linear systems where A is the infinitesimal generator of a C 0 -semigroup pT ptqq tě0 on a Banach space X and B : U Ñ X is a bounded linear operator, ISS is equivalent to uniform exponential stability of the semigroup [4,10]. If B is not bounded as operator from U to X, which is typically the case for boundary controlled PDEs, the property of being ISS becomes non-trivial even for linear systems. In fact, this is closely related to suitable solution concepts see e.g. [10,20,28]. Along with the recent developments in ISS theory for infinite-dimensional systems [4,5,8,15,26], several partial results have been derived in the (semi)linear context, with a slight focus on parabolic equations, see e.g. [13,16,21,22,24,34]. We refer to recent surveys on ISS for infinite-dimensional systems [25,28] and the book [17]. The origin of ISS theory, introduced by Sontag in 1989 [29], are non-linear systems and we refer the reader to [30] for a survey on ISS for ODEs. Already seemingly harmless system classes such as bilinear systems 9 xptq " Axptq`Bpxptq, uptqq, which is called integral input-to-state stable (integral ISS), see also [31]. Note that the terms involving u in (1) and (3) cannot be compared for arbitrary t ą 0, general functions u, and fixed functions γ, γ 1 , γ 2 . Still integral ISS and ISS are equivalent for infinite-dimensional linear systems with a bounded linear operator B : U Ñ X, [10,23] as this reduces to uniform exponential stability of the uncontrolled system. The corresponding question for general infinite-dimensional systems seems to be much harder and notorious questions remain, see [10,27,13] and [12] for a negative result.
On the other hand in [23] the equivalence of integral ISS and uniform exponential stability is shown for a natural infinite-dimensional version of (2), with A generating a C 0 -semigroup and B : XˆU Ñ X satisfying a Lipschitz condition on bounded subsets of X uniformly in the second variable and being bounded in the sense that }Bpx, uq} À }x}γp}u}q for some K-function γ and all x and u. As indicated above, the property whether a system is ISS or integral ISS is more subtle when boundary controls are considered and consequently, the involved input operators become unbounded. This also applies for bilinear systems which -in the presence of boundary control -cannot be treated as in the references mentioned above.
In this article we establish the abstract theory to overcome such issues. More precisely, we study infinite-dimensional bilinear control systems of the abstract form 9 xptq " Axptq`B 1 F pxptq, u 1 ptqq`B 2 u 2 ptq, t ě 0, (ΣpA, rB 1 , B 2 s, F q) where A generates a C 0 -semigroup on a Banach space X and B 1 and B 2 are possibly unbounded linear operators defined on Banach spaces s X and U 2 , respectively. The nonlinearity F : XˆU 1 Ñ s X is assumed to satisfy a Lipschitz condition and to be bounded. In Section 2 we present the details of this abstract framework and derive the main result, which, in terms of integral ISS, see also Definition 2.1, reads as follows.
Main Result (Theorem 2.9). The bilinear system ΣpA, rB 1 , B 2 s, F q is integral ISS, if the linear systems ΣpA, r0, B 1 s, F q and ΣpA, r0, B 2 s, F q are integral ISS.
In order to prove this statement we show existence of global mild solutions to ΣpA, rB 1 , B 2 s, F q by classical fixed point arguments under the weak conditions on the operators B 1 , B 2 . We conclude by applying our abstract result to the example of a bilinearly controlled Fokker-Planck equation with reflective boundary conditions, which has recently appeared in [3,9].
2. Input-to-state stability for bilinear systems 2.1. System class and notions. In the following we study bilinear control systems of the form where ‚ X, s X and U 1 , U 2 are Banach spaces and x 0 P X, ‚ A generates a C 0 -semigroup pT ptqq tě0 on X, ‚ the input functions u 1 and u 2 are locally integrable function with values in U 1 and U 2 , respectively, that is, u 1 P L 1 loc p0, 8; U 1 q and u 2 P L 1 loc p0, 8; U 2 q, ‚ the operators B 1 and B 2 are defined on s X and U 2 respectively. Both operators map into a space (see below) in which X is densely embedded, ‚ the nonlinear operator F : XˆU 1 Ñ s X is bounded in the sense that there exists a constant m ą 0 such that }F px, uq}Ď X ď m}x} X }u} U1 @x P X, u P U 1 .
and Lipschitz continuous in the first variable on bounded subsets of X, where the Lipschitz constant depends on the U 1 -norm of the second argument, that is, for all bounded subsets ‚ s Þ Ñ F pf psq, gpsqq is measurable for any interval I and measurable functions f : I Ñ X, g : I Ñ U 1 , ‚ we write ΣpA, r0, B 2 sq if B 1 " 0 and thus System ΣpA, r0, B 2 sq is linear. Before explaining the details on the assumptions on B 1 and B 2 below, we list some examples for functions F and operators that fit our setting.
(a) s X " X, U " C and F px, uq " xu, Let X´1 be the completion of X with respect to the norm }x} X´1 " }pβÁ q´1x} X for some β in the resolvent set ρpAq of A. For a reflexive Banach space, X´1 can be identified with pDpA˚qq 1 , the continuous dual of DpA˚q with respect to the pivot space X. The operators B 1 and B 2 are assumed to map to X´1, more precisely, B 1 P Lp s X, X´1q and B 2 P LpU 2 , X´1q, where LpX, Y q refers to the bounded linear operators from X to Y . Only in the special case that B 1 or B 2 are in Lp s X, Xq or LpU 2 , Xq, we say that the respective operator is bounded. The C 0semigroup pT ptqq tě0 extends uniquely to a C 0 -semigroup pT´1ptqq tě0 on X´1 whose generator A´1 is the unique extension of A to an operator in LpX, X´1q, see e.g. [7]. Note that X´1 can be viewed as taking the role of a Sobolev space with negative index. With the above considerations we may consider System ΣpA, rB 1 , B 2 s, F q on the Banach space X´1. We want to emphasize that our interest is primarily in the situation where B 1 and B 2 are not bounded -something that typically happens if the control enters through point boundary actuation. Note, however, that the assumptions imply that "the unboundedness of B 1 and B 2 is not worse than the one of A" -which particularly means that if A P LpX, Xq then B 1 P Lp s X, Xq and B 2 P LpU 2 , Xq. For zero-inputs u 1 and u 2 , the solution theory for System ΣpA, rB 1 , B 2 s, F q is fully characterized by the property that A generates a C 0 -semigroup as this reduces to solving a linear, homogeneous equation. For non-trivial inputs, the solution concept is a bit more delicate.
More precisely, for given t 0 , t 1 P r0, 8q, t 0 ă t 1 , x 0 P X, u 1 P L 1 loc p0, 8; U 1 q and u 2 P L 1 loc p0, 8; U 2 q, a continuous function x : rt 0 , t 1 s Ñ X is called a mild solution of ΣpA, rB 1 , B 2 s, F q on rt 0 , t 1 s if for all t P rt 0 , t 1 s, We say that x : r0, 8q Ñ X is a global mild solution or a mild solution on r0, 8q of ΣpA, rB 1 , B 2 s, F q if x| r0,t1s is a mild solution on r0, t 1 s for every t 1 ą 0. We stress that existence of a mild solution is non-trivial, even when u 1 " 0. In this case, it is easy to see that x P Cpr0, 8q; X´1q, but not necessarily xptq P X, t ą 0, without further assumptions on B 2 . The existence of a mild solutions to the linear System ΣpA, r0, B 2 sq is closely related to the notion admissibility of the operator B 2 for the semigroup pT ptqq tě0 and various sufficient and necessary conditions are available, see e.g. Proposition 2.4 and [10].
We need the following well-known function classes from Lyapunov theory.
The following concept is central in this work. It originates from works by Sontag [29,31]. We refer e.g. to [25,26] for the infinite-dimensional setting.
Definition 2.1. The system ΣpA, rB 1 , B 2 s, F q is called (i) input-to-state stable (ISS), if there exist β P KL, µ 1 , µ 2 P K 8 such that for every x 0 P X, u 1 P L 8 p0, 8; U 1 q and u 2 P L 8 p0, 8; U 2 q there exists a unique global mild solution x of ΣpA, rB 1 , B 2 s, F q and for every t ě 0 }xptq} ď βp}x 0 }, tq`µ 1 p}u 1 } L 8 p0,t;U1q q`µ 2 p}u 2 } L 8 p0,t;U2q q; (ii) integral input-to-state stable (integral ISS), if there exist β P KL, θ 1 , θ 2 P K 8 and µ 1 , µ 2 P K such that for every x 0 P X, u 1 P L 8 p0, 8; U 1 q and u 2 P L 8 p0, 8; U 2 q System ΣpA, rB 1 , B 2 s, F q has a unique global mild solution x and for every t ě 0 One may define some mixed type of these definitions like (ISS,integral ISS) (and (integral ISS,ISS)), in the sense that one has an ISS-estimate for u 1 and some integral ISS-estimate for u 2 (and vice versa).
Although the terms involving u 1 and u 2 on the right-hand-side of the integral ISS estimate do not define norms in general. However, there are function spaces which are naturally linked to integral ISS [10]. In this context we briefly introduce the Orlicz space E Φ pI; Y q associated to a so-called Young function Φ for an interval I Ă R and a Banach space Y in the Appendix. Note that the Orlicz space E Φ corresponding to the Young function Φptq " t p , 1 ă p ă 8, is isomorphic to L p . Definition 2.2. Let pT ptqq tě0 be a C 0 -semigroup.
(i) We say that pT ptqq tě0 is of type pM, ωq if M ě 1 and ω P R are such that (ii) We say that pT ptqq tě0 is (uniformly) exponentially stable if pT ptqq tě0 is of type pM, ωq for some ω ą 0.
for pT ptqq tě0 , if for every t ą 0 and u P Zp0, t; U q it holds that ż t 0 T´1pt´sqBupsq ds P X.
We will neglect the reference to pT ptqq tě0 if this is clear from the context.
Recall that every C 0 -semigroup is of type pM, ωq for some M ě 1 and ω P R. Note that any bounded operator B is Z-admissible for all Z considered above.
This is a consequence of the closed graph theorem. Also note that B is Z-admissible for pe δt T ptqq tě0 for any δ P R. Furthermore, the function t Þ Ñ C t,B is nondecreasing and, if pT ptqq tě0 is exponentially stable, even bounded, that is, The following result clarifies the relation between admissibility and (integral) ISS. The interest to study admissibility with respect to Orlicz spaces follows by the natural connection to integral ISS for linear systems, see Proposition 2.4 piiiq. Note in particular that the existence of mild solutions for E Φ -admissible operators B 2 is based on the absolute continuity of the Orlicz norm with respect to the length of the interval and the strong continuity of the shift-semigroup on E Φ pI; Y q for any interval I and any Banach space Y . The latter can be proven by similar methods one uses to prove the strong continuity of the shift-semigroup on L p pI; Y q.  [13] in the case of boundary control.
The following proposition proves an estimate between Orlicz-norms and integral ISS estimates, which will be useful for the proof of the main result.
Proposition 2.5. Let Φ be a Young function. Then there exist K 8 -functions θ and µ such that for any Banach space U and t ą 0, Moreover, θ and µ can be chosen as where φ equals the right-derivative of Φ a.e. and, for α ą 0, If Φ satisfies the ∆ 2 -condition (c.f. Appendix), then µ " Φ can be chosen as well.
Proof. Note that we only need to show that µ and θ define K 8 -functions since (9) is immediate from the definition of θ. The proof is similar in spirit to an argument for all c ą 0, see [27,Lem. 1]. In the special case that Φ satisfies the ∆ 2 -condition (with s 0 " 0), the above properties also hold for µ " Φ, by the defining properties of the ∆ 2 -condition. This implies that whenever a sequence pf n q nPN with f n P L 8 p0, t n ; U q is such that it follows that lim nÑ8 }f n } EΦp0,tn;Uq " 0, see [27,Lem. 2]. Clearly, µ is a K 8function, since µ is a Young function. Therefore, it remains to consider θ. It is easy to see that θ is well-defined, non-decreasing and unbounded, whence we are left to show continuity. Moreover, since θpαq is of the form sup M α with nested sets pM α q αą0 , it follows that θ is right-continuous on p0, 8q. To see that θ is continuous at α " 0, let pα n q n be a decreasing sequence of positive numbers with lim nÑ8 α n " 0 and for every n P N let u n P L 8 p0, t n ; U q be such that ş tn 0 µp}u n psq}q ds ď α n and 0 ď θpα n q´}u n } EΦp0,tn;Uq ă 1 n . By the above mentioned argument, we can conclude that }u n } EΦp0,tn;Uq converges to 0 as n Ñ 8. Thus, lim nÑ8 θpα n q " 0. We finish the proof by showing that θ is left-continuous on p0, 8q. Now let α ą 0, For every n P N, we aim to findũ n P L 8 p0, t n ; U q such that ş tn 0 µp}ũ n psq}q ds ď α n and lim nÑ8 }u n´ũn } EΦp0,tn;Uq " 0. Indeed, then θpαq´θpα n q ď θpαq´}ũ n } EΦp0,tn;Uq ď θpαq´}u n } EΦp0,tn;Uq`} u n´ũn } EΦp0,tn;Uq tends to 0 as n Ñ 8, which shows left-continuity. We defineũ n :" u n χ Mn where the measurable set M n is chosen such that ż Mn µp}u n psq}q ds " α n , if ż tn 0 µp}u n psq}q ds ě α n , or M n " p0, t n q otherwise. It follows that ż tn 0 µp}u n psq´ũ n psq} U q ds " Thus, using the argument from the beginning of the proof again, we infer that }u n´ũn } EΦp0,tn;Uq Ñ 0 as n Ñ 8. This concludes the proof.
Combining Proposition 2.5 with [10, Prop 2.10] allows us to fomulate the following result: Corollary 2.6. If System ΣpA, r0, B 2 sq possesses a unique mild solution x for every x 0 P X and u 2 P L 8 p0, 8; U 2 q, then the following statements are equivalent.
(i) There exist functions β P KL and µ 2 P K 8 such that holds for all t ě 0 and u 2 P L 8 p0, 8; U 2 q. 6 (ii) There exist functions β P KL, θ 2 P K and µ 2 P K 8 such that (12) holds for all t ě 0 and u 2 P L 8 p0, 8; U 2 q.
Remark 2.7. Let us make the following comments on the construction of µ and θ in Proposition 2.5.
(1) If Φpsq " s p , s ą 0, then µpsq " s p and it is not hard to see that, up to a constant, θprq is given by Φ´1prq " r 1 p . This shows that the choice of θ is rather natural.
(2) With similar techniques as in the proof of Proposition 2.5, it has been shown in [10,27] that if a linear system ΣpA, r0, Bsq satisfies (11), then it is integral ISS with the estimate Proposition 2.5 shows that θ can actually be chosen independent of the semigroup pT ptqq tě0 and B provided the system is E Φ -admissible (which, however, depends on pT ptqq tě0 and B, of course). In some sense, this fact simplifies the proofs in [10,27]. On the other hand, the choice of θ based on (13) is more refined; in case the system was even E Ψ -admissible with some Ψ ď Φ, this would affect the choice of θ, even if µ is constructed from Φ only.
In contrast to linear systems, the existence of mild solutions is less clear for bilinear control systems of the form ΣpA, rB 1 , B 2 s, F q.
Sontag [31] showed that finite-dimensional bilinear systems are hardly ever ISS, but integral ISS if and only if the semigroup is exponentially stable. In [23] it was shown that this result generalizes to infinite-dimensional bilinear systems provided that B 1 and B 2 are bounded operators and s X " X. The following results give sufficient conditions for integral ISS and some combination of ISS and integral ISS of ΣpA, rB 1 , B 2 s, F q. We start with a result on existence of local solutions to ΣpA, rB 1 , B 2 s, F q. The proof involves typical arguments in the context of mild solutions for semilinear equations.
A similar result for the existence of the unique mild solution as in the following Lemma 2.8 were proved under slightly stronger conditions in [2] for L padmissible B 1 , scalar-valued inputs u 1 , F px, u 1 q " u 1 x and B 2 " 0. Our condition is more natural as the same condition guarantees the existence of continuous (and unique) global mild solutions of the linear systems ΣpA, r0, B 1 sq and ΣpA, r0, B 2 sq, see Proposition 2.4. Lemma 2.8. Let A generate a C 0 -semigroup pT ptqq tě0 on X. Suppose that B 1 P Lp s X, X´1q is E Φ -admissible and that B 2 P LpU 2 , X´1q is E Ψ -admissible. Then for every t 0 ě 0, x 0 P X, u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q there exists t 1 ą t 0 such that System ΣpA, rB 1 , B 2 s, F q possesses a unique mild solution x on rt 0 , t 1 s. Moreover, if t max ą t 0 denotes the supremum of all t 1 ą t 0 such that System 7 ΣpA, rB 1 , B 2 s, F q has a unique mild solution x on rt 0 , t 1 s, then t max ă 8 implies that lim tÑtmax }xptq} " 8.
Proof. We first show that for every t 0 ě 0, x 0 P X, u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q there exists t 1 ą t 0 such that System ΣpA, rB 1 , B 2 s, F q possesses a unique mild solution on rt 0 , t 1 s with initial condition x 0 and input functions u 1 and u 2 . Moreover, we show that t 1 " t 0`δ can be chosen such that δ is independent for any bounded sets of initial data x 0 and t 0 . Let T ą 0, r ą 0, u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q be arbitrarily. We first recall the following property of Orlicz spaces. For any ε ą 0 there exists δ ą 0 such that see e.g. [19,Thm. 3.15.6]. Let t 0 P r0, T s, t 1 ą t 0 and x 0 P K r p0q " tx P X : }x} ď ru and define the mapping T´1pt´sqrB 1 F pxpsq, u 1 psqq`B 2 u 2 psqs ds.
We will show next that t 1 can be chosen such that Φ t0,t1 maps M k pt 0 , t 1 q to M k pt 0 , t 1 q and is contractive on this set. Let C t,B1 and C t,B2 refer to the admissibility constants such that (8) holds for B 1 and B 2 which can be chosen non-decreasing in t. Let m be the boundedness constant of F from (4) and let L K k p0q be the Lipschitz constant of F such that (5) holds for the bounded set X b " txptq | x P M k pt 0 , t 1 q, t P rt 0 , t 1 su Ă X which is equal to K k p0q " tx P X : }x} ď ku. Now, let t 1 " t 0`δ with δ P p0, 1q be chosen such that for all t 0 P r0, T s, (i) e ωpt1´t0q " e ωδ ď 2, (ii) mC T`1,B1 }u 1 } EΦpt0,t0`δ;U1q ď 1 2 , (iii) C T,B2 }u 2 } EΨpt0,t0`δ;U2q ď M and (iv) C T`δ,B1 L K k p0q }u 1 } EΦpt0,t0`δ;U1q ă 1 holds, where we used (14) in (ii)-(iv). Note that apart from the parameters of the operators B 1 , B 2 , A, F , the choice of δ only depends on r and T , where the rdependence of δ arises from the r-dependence of k. It follows that for all t 0 P r0, T s, x P M k pt 0 , t 1 q and x 0 P K r p0q }Φ t0,t1 pxq} Cprt0,t1s;Xq ď M e ωpt1´t0q }x 0 }`C t1,B1 }F px, u 1 q} EΦpt0,t1; Ď Xq`Ct1,B2 }u 2 } EΨpt0,t1;U2q ď 2M }x 0 }`mC t1,B1 }u 1 } EΦpt0,t1;U1q }x} Cprt0,t1s;Xq`M ď k, where we used admissibility in the first inequality and (4) in the second inequality as well as the monotonicity of the Orlicz norm in both estimates. Hence, Φ t0,t1 maps M k pt 0 , t 1 q to M k pt 0 , t 1 q. The contractivity follows by pivq since }Φ t0,t1 pxq´Φ t0,t1 pxq} Cprt0,t1s;Xq ď sup tPrt0,t1s T pt´sqB 1 rF pxpsq, u 1 psqq´F pxpsq, u 1 psqqs ds where we used again admissibility, the Lipschitz property of F and the monotonicity of the Orlicz norm. By Banach's fixed-point theorem, we conclude that System ΣpA, rB 1 , B 2 s, F q possesses a unique mild solution on rt 0 , t 1 s with initial condition x 0 and input functions u 1 and u 2 . Now let t max be the supremum of all t 1 such that there exists a mild solution x of ΣpA, rB 1 , B 2 s, F q on rt 0 , t 1 s for every t 1 ă t max , where x 0 P X, u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q are given. Suppose that t max is finite. We will show, that then lim tÑtmax }xptq} " 8. If this is not the case, we have r " sup tPrt0,tmaxs }xptq} ă 8.
Let pt n q nPN be a sequence of positive real numbers converging to t max from below. Since t n P r0, t max s and }xpt n q} ď r for all n P N, there exists δ ą 0 independent of n P N such that the equation has a mild solution y on rt n , t n`δ s. Therefore, we can extend x by xptq " yptq, t P pt n , t n`δ s, to a solution of ΣpA, rB 1 , B 2 s, F q on rt 0 , t n`δ s. This contradicts the maximality of t max and hence, x has to be unbounded in t max . Theorem 2.9. Suppose that the linear systems ΣpA, r0, B 1 sq and ΣpA, r0, B 2 sq are integral ISS, then the bilinear system ΣpA, rB 1 , B 2 s, F q is integral ISS and (integral ISS,ISS).
The assumption that ΣpA, r0, B 2 sq is integral ISS is necessary.
Proof. The necessity of ΣpA, r0, B 2 sq being integral ISS follows by setting u 1 " 0 in the bilinear system. Proposition 2.4 says that integral ISS of the linear systems is equivalent to the exponential stability of the semigroup pT ptqq tě0 generated by A and the admissibility of the control operators B 1 and B 2 with respect to some Orlicz spaces E Φ and E Ψ , respectively. Using this characterization, we will give the proof in two steps. At first we prove the existence of a global mild solution x of ΣpA, rB 1 , B 2 s, F q (which does not need the exponential stability of pT ptqq tě0 ). Afterwards we prove the (integral) ISS properties. STEP I. Let pM, ωq denote the type of pT ptqq tě0 . By Remark 2.3 there exist C t,B1 , C t,B2 ą 0 such that for every t ě 0, Let x 0 P X, u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q and let t max be the supremum over all t 1 such that ΣpA, rB 1 , B 2 s, F q possesses a unique x mild solution on r0, t 1 s. Lemma 2.8 yields t max ą 0. For t P r0, t max q we have that where C ω,u2,t " C t,B2 e´ω 2 t › › e ω 2¨u 2 › › EΨp0,t;U2q . The }¨} EΦ -norm in the second term can be estimated by the boundedness of F , We pass over to the equivalent norm on E Φ given in the Appendix, (24). Therefore, for ε ą 0 there exists a function g P LΦp0, t; Cq with }g} LΦp0,t;Cq ď 1 such that Hence, by combining this with (15) gives where we used the generalized Hölder inequality for Orlicz spaces, see (25) in the Appendix. Thus, by letting ε tend to 0, multiplying with e´ω 2 t and using ab ď 1 2 a 2`1 2 b 2 for a, b P R, we obtain }xptq} ď M e´ω t }x 0 }`1 2 M 2 e´ω t sup rPr0,ts Thus, we have shown }xptq} ď βp}x 0 }, tq`γ 1 pC t,B1 }u 1 } EΦp0,tq q`γ 2 pC t,B2 e´ω 2 t }e ω 2¨u 2 } EΨp0,tq q (16) ď βp}x 0 }, tq`γ 1 pC t,B1 }u 1 } EΦp0,tq q`γ 2 pC t,B2 sup rPr0,ts e´ω 2 r }u 2 } EΨp0,tq q, for all u 1 P E Φ p0, 8; U 1 q, u 2 P E Ψ p0, 8; U 2 q and functions β P KL and γ 1 , γ 2 P K 8 , which can be choosen as βps, tq " M e´ω t s`1 2 M 2 e´ω t s 2 sup rPr0,ts e´ω r , γ 1 psq " 4m 2 s 2 e 4ms , γ 2 psq " s`1 2 s 2 .
Moreover, the mild solution exists on r0, 8q. Indeed, if this is not the case, we have t max ă 8 and Lemma 2.8 implies that x is unbounded in t max . This contradicts (16) since the right-hand-side is uniformly bounded in t on finite intervals r0, t max q. STEP II. Since we are dealing with an exponentially stable semigroup, Remark 2.3 implies that C t,B1 and C t,B2 are uniformly bounded in t and we can choose ω ą 0. Hence, (16) yields for all u 1 P E Φ p0, 8; U 1 q and u 2 P E Ψ p0, 8; U 2 q that }xptq} ď βp}x 0 }, tq`γ 1`CB1 }u 1 } EΦp0,t;U1q˘`γ2`CB2 }u 2 } EΨp0,t;U2qw ith C Bi " sup tě0 C t,Bi , i " 1, 2. Using Proposition 2.5 for u 1 and u 2 , we have shown that ΣpA, rB 1 , B 2 s, F q is integral ISS since L 8 is contained in any Orlicz space on compact intervals. If we apply Proposition 2.5 only for u 1 in (16), ΣpA, rB 1 , B 2 s, F q is (integral ISS,ISS) by realizing that there exists a constant C ą 0 such that for all u 2 P L 8 p0, 8; U 2 q and t ą 0. To see this, let ε ą 0 such that Ψpxq ď x for all x P p0, εq, which exists by the property that lim sÑ0 Ψpsq s " 0. Therefore, choosing This implies that The assumption in Theorem 2.9 that System ΣpA, r0, B 1 sq is integral ISS is not necessary as the choice F " 0 shows.

Remark 2.10.
(1) In Theorem 2.9 one cannot expect the bilinear systems to be ISS as the trivial finite-dimensional example 9 x "´x`u 1 x shows. (2) Using the definitions of γ 1 , γ 2 after (16) and the definitions of µ and θ from Proposition 2.5, up to constants the functions µ 1 , µ 2 , θ 1 and θ 2 in the integral ISS estimate for ΣpA, rB 1 , B 2 s, F q can be given explicitly. (3) The proof of Theorem 2.9 is easier in the case that the Orlicz spaces are L p spaces, since the L p -norm is already an integral of the form we are seeking for in the integral ISS estimate (c.f. Definition 2.1). (4) Note that the assumptions of Lemma 2.8 already yield that the unique mild solution is global. This is the first step of the proof of Theorem 2.9.
In order to investigate integral ISS, it is thus sufficient to check that the linear systems ΣpA, r0, B 1 sq and ΣpA, r0, B 2 sq are integral ISS, or by Proposition 2.4 equivalently, that A generates an exponential stable C 0 -semigroup and the control operators B 1 and B 2 are admissible. Note that there are control operators B which are E Φ -admissible for some Young function Φ but not L p -admissible for any p P r1, 8q. In the context of linear systems, such an example was already given in [10,Ex. 5.2] for an operator B defined on C using the connection between a Carleson-measure criterion and admissibility stated in [10], see also [11]. The next example extends this result to control operators defined on s X.
Example 2.11. Let X " ℓ 2 pNq and define F : XˆC Ñ X, by F px, uq :" ux and the diagonal operators Ae n "´2 n e n , Be n " 2 n n e n , n P N, where pe n q nN is the canonical basis of X and A is defined on its maximal domain. The general assumptions of Section 2.1 are satisfied with B 1 " B and B 2 " 0. Let x " p 1 n q nPN P X and p P r1, 8q. Following [10, Ex. 5.2], the operator b " Bx defined on C is not L p -admissible. Hence, B is not L p -admissible. Next, we show that B is E Φ -admissibile, where Φ is the complementary Young function toΦ psq " s lnplnps`eqq.
It is easy to check thatΦ is a Young function. Define the sequence k " pk n q nPN by k n " lnpCnq n , n P N, where C " lnp2q`lnp2eq ą 1. We choose n large enough, such that k n n " lnpCnq ě 1 holds. Similar to [33,Ex. 4.2.13] one can show Φˆ2 n k n n e´2 n t˙ď 2 n e´2 n t .
We deduce ż t 0Φ˜e´2 n pt´sq 2 n n k n¸d s ď 1´e´2 n t ă 1 and hence }e´2 n pt´¨q 2 n n } LΦp0,t;Cq ď k n for sufficiently large n. Using the generalized Hölder inequality (25), we get for u P E Φ p0, t; ℓ 2 pNqq and sufficiently large ňˇˇˇˆż Therefore, for some M ą 0, which shows that B is E Φ -admissible and thus ΣpA, rB, 0s, F q is integral ISS.
Here n refers to the outward normal vector on the boundary, ρ 0 denotes the initial probability distribution with ş Ω ρ 0 pxq dx " 1 and ν ą 0. Furthermore, the potential V is assumed to be of the form where W P W 2,8 pΩq and α P W 1,8 X H 2 pΩq satisfying the structural assumption ∇α¨ n " 0 on BΩ. Thus, the scalar-valued input function u enters via the spatial profile α in the potential. In order to cast the equations in an abstract framework, we introduce the following operators: Af " ν∆f`∇¨pf ∇W q, Bf " ∇¨pf ∇αq, where X " L 2 pΩq and H 1 pΩq, H 2 pΩq refer to the standard Sobolev spaces. By standard arguments, the operator A is seen to generate a bounded C 0 -semigroup on X, with discrete spectrum σpAq " σ p pAq Ď p´8, 0s and ρ 8 " ce´Φ is an eigenfunction to the simple eigenvalue 0, where Φ is given by Φ " ln ν`W ν and c ą 0 is such that ş Ω ρ 8 dx " 1, see [3]. Furthermore, we will identify B with its extension from X to X´1.
We now consider the system around the stationary distribution ρ 8 instead of the origin, see also [3], and decompose X according to the projections P : L 2 pΩq Ñ L 2 pΩq, y Þ Ñ y´ż Ω ypxq dxρ 8 and Q :" I´P.
In the remainder of the section we will lay out the proof of Theorem 3.1 based on Theorem 2.9 and Proposition 2.4. This includes to show that A, B 1 , B 2 satisfy the the assumptions of the abstract system class from Section 2.1 and considering the nonlinearity F : XˆC Ñ X , py, uq Þ Ñ yu.
Let M be the multiplication operator by e Φ 2 on L 2 pΩq. Clearly, M is bounded and invertible on L 2 pΩq, leaves H 1 pΩq invariant, and the inverse M´1 is the multiplication operator by e´Φ 2 . Hence,Ã given bỹ DpÃq "M DpAq is well-defined and self-adjoint.
To study admissibility of B we introduce the following well-known abstract interpolation and extrapolation spaces, see e.g. [32]. LetX 1 andX´1 be defined in the same way as X 1 and X´1, but usingÃ instead of A. We defineX 1 2 as the completion of DpÃq with respect to the norm given by }z} 2X 1 2 :" xpI´Ãqz, zy, x P DpÃq, and we denote byX´1 2 the dual space ofX 1 2 with respect to the pivot space X, i.e. the completion of X with respect to the norm sup }v}X 1 2 ď1 |xz, vy X |. The following embeddings are dense and continuous:X 1 ãÑX 1 2 ãÑ X ãÑX´1 2 ãÑX´1. We first prove that the operatorB :" M BM´1 defined on DpÃq has a unique extensionB P LpX,X´1 2 q which is L 2 -admissible forÃ. Integration by parts gives }v} 2X For f P DpÃq and v P DpÃq, }v}X where σ is the surface measure on BΩ. ThusB P LpX,X´1 2 q andB is L 2 -admissible forÃ by [32,Prop. 5.1.3]. We have for β P ρpAq " ρpÃq and f P X }M´1f } X´1 " }pβ´Aq´1M´1f } X " }M´1pβ´Ãq´1f } X ď }M´1}}f }X´1.
Thus, M´1 extends uniquely to an operator in LpX´1, X´1q. The same argument yields a unique extension M P LpX´1,X´1q. Note that these extensions are inverse to each other, so it is natural to denote the extensions again by M and M´1. We claim that M´1BM P LpX, X´1q extends B to an L 2 -admissible operator for A which we again denote by B. Indeed, if pT ptqq tě0 is the semigroup generated by A, then pSptqq tě0 with Sptq " M T ptqM´1 is the semigroup generated byÃ and for u P L 2 p0, t; Xq we have M u P L 2 p0, t; Xq and ż t 0 T´1pt´sqBupsq ds " M´1 ż t 0 Spt´sqBpM uqpsq ds.
As B 2 P LpC, X q, B 2 is clearly L 1 -admissible. The operator P commutes with the C 0 -semigroup generates by A [3, Eq. (3.12)], by [14,Lem. 4.4] the operator B 1 " B| X P LpX , X´1q is well-defined and L 2 -admissible for A.
Thus the bilinearly controlled Fokker-Planck system given by (18)- (20) can be written as a system ΣpA, rB 1 , B 2 s, F q.
Following the construction of the integral ISS estimate (c.f. (16)) we deduce an explicit integral ISS estimate: There exists constants C, ω ą 0 such that for any ρ 0 P L 2 pΩq with ş Ω ρ 0 pxqdx " 1 and u P L 2 p0, 8; U q, the global mild solution of the Fokker-Planck system (18) satisfies where γprq " Cre Cr

Conclusion
Bilinear systems appear naturally in control theory e.g. when considering multiplicative disturbances in feedback loops of linear systems. The results in this article draw a link between bilinear systems, which are a classical example class in (integral) ISS in finite-dimensions, and recent progress in ISS for infinite-dimensional systems. We emphasize that the most natural example in this context, 9 xptq " Axptq`uptqxptq, t ą 0, xp0q " x 0 , with A generating a C 0 -semigroup pT ptqq tě0 on X, is covered by the system class considered here. More precisely, by the results in Section 2, it follows that this system is integral ISS if and only if pT ptqq tě0 is exponentially stable. More precisely, the sufficiency follows since the identity is L 1 -admissible and hence the system is integral ISS. It seems that prior works on integral ISS [23, Sec. 4.2] did not cover this comparably simple class as the bilinearity x Þ Ñ xu fails to satisfy a Lipschitz condition uniform in u required there 2 .
Moreover, our results generalize to integral ISS assessment for bilinearities arising from boundary control (or lumped control).

Appendix
We briefly introduce Orlicz spaces of functions f : I Ñ Y for an interval I Ă R and a Banach space Y . For more details on Orlicz spaces we refer to [1,18,19]. Let Φ : R0 Ñ R0 be a Young function, i.e. Φ is continuous, increasing, convex with lim sÑ0 Φpsq s " 0 and lim sÑ8 Φpsq s " 8 and denote by L Φ pI; Y q the set of Bochner-measurable functions u : I Ñ Y for which there exists a constant k ą 0 such that Φpk}up¨q}q is integrable. We equip L Φ pI; Y q with the norm }u} LΦpI;Y q " inf " k ą 0ˇˇˇˇż Despite the fact that L Φ pI; Y q is typically referred to as "Orlicz space" in the literature, we prefer to call E Φ pI; Y q " tu P L 8 pI; Y q | ess supp u is boundedu the Orlicz space associated with the Young function Φ. We write }u} EΦpI;Y q " }u} LΦpI;Y q for u P E Φ . Note that u P E Φ pI; Y q implies that Φ˝}up¨q} is integrable. Typical examples of Orlicz spaces are L p -spaces; for Φptq " t p with p P p1, 8q it holds that E Φ pI; Y q is isomorphic to L p pI; Y q. A Young function Φ is said to satisfy the ∆ 2 -condition if there exist K ą 0 and s 0 ě 0 such that Φp2sq ď KΦpsq, s ě s 0 .
Note that E Φ pI; Y q " L Φ pI; Y q if and only if Φ satisfies the ∆ 2 -condition. Also note that Φpsq " s p , p P p1, 8q satisfies the ∆ 2 -condition. For a Young function Φ the complementary Young functionΦ is defined bỹ Again, this is a Young function and Φ can be recovered fromΦ in the same manner. The complementary Young function to Φpsq " s p p , 1 ă p ă 8, is given byΦpsq " s q q with 1 p`1 q " 1. As for L p spaces, an equivalent norm to }¨} LΦpI;Y q is given by Furthermore, for a Young functions Φ and its complementary Young functionΦ the following generalized Hölder inequality ż I }upsq}}vpsq}ds ď 2}u} LΦ }v} LΦ .
holds. This also implies the continuity of the embeddings if I is bounded. Although L 1 is not an Orlicz space, we will explicitly allow for Φptq " t in our notation referring to E Φ pI; Y q " L 1 pI; Y q. Note that the definition of the norm (23) is indeed consistent with the L 1 -norm and that Φ satisfies the ∆ 2 -condition. However, we will not define a "complementary Young function" for this particular Φ. An essential property of Orlicz spaces is the absolute continuity of the E Φ norm with respect to the length of the interval I (see e.g. [19,Thm. 3.15.6]), this is for u P E Φ pI; Y q and ε ą 0 there exists δ ą 0 such that for each interval I holds λpIq ă δ ùñ }u} EΦpI;Y q ă ε, where λ refers to the Lebesgue-measure on R.