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Funnel control for the Fokker-Planck equation corresponding to the Ornstein-Uhlenbeck process

In this paper the feasibility of funnel control techniques for the Fokker-Planck equation corresponding to the Ornstein-Uhlenbeck process on an unbounded spatial domain is explored. First, using weighted Lebesgue and Sobolev spaces, an auxiliary operator is defined via a suitable coercive sesquilinear form. This operator is then transformed to the desired Fokker-Planck operator. We show that any weak solution of the controlled Fokker-Planck equation (which is a probability density) has a variance that exponentially converges to a constant. After a simple feedforward control approach is discussed, we show feasibility of funnel control in the presence of disturbances by exploiting an energy estimate. We emphasize that the closed-loop system is a nonlinear and time-varying PDE. The results are illustrated by some simulations.


Introduction.
In this work we study output tracking control for the Fokker-Planck equation that corresponds to the Ornstein-Uhlenbeck process. The latter is a continuoustime stochastic process which was originally used to describe the motion of a massive Brownian particle under the influence of friction [39]. Although its investigation was mainly driven by physics and mathematics, several other important applications emerged, such as in neurobiology [35] and in finance [36]. The Ornstein-Uhlenbeck process is often considered in the context of optimal control, see e.g. [2,3,18,19]. The Fokker-Planck equation is a parabolic partial differential equation (PDE) which describes the evolution of the probability density function of the solution of a stochastic differential equation, see e.g. [30]. It will be the main tool to treat the output tracking control problem.
In this context, control means that we assume that the drift term of the stochastic differential equation can be manipulated by an external signal, which is called the control input. The resulting Fokker-Planck equation can be viewed as an abstract bilinear control system in terms of the state and the input, cf. [14,22]; see also the monograph [29] for several topics on bilinear control systems. The mean value (or expected value) of the Ornstein-Uhlenbeck process is chosen as the output and measurements of it are assumed to be available. For a given reference signal, we then seek to achieve that the difference between the mean value and the reference stays within a prescribed error margin for all times, thus allowing to control the mean value of the process as desired. At the same time, we do not require knowledge of the system parameters or the initial probability density. Furthermore, controlling the mean value of the process is indeed sufficient to influence the entire probability density function. Since only the drift term in the Fokker-Planck equation is influenced by the control input, the variance of the process is independent of it. We will show that it is the sum of a constant and an exponentially decaying term. Indeed, simulations show that the shape of the probability density does not change after some initial time, and is essentially only shifted according to the movement of the mean value.
The control law to achieve this is based on the funnel control methodology developed in [24]. The funnel controller is an output-error feedback of high-gain type. Its advantages are that it is model-free (i.e., it requires no knowledge of the system parameters or the initial value), it is robust and of striking simplicity -for the Fokker-Planck equation we will show that robustness can be guaranteed w.r.t. additive disturbances "with zero mass". The funnel controller has been successfully applied e.g. in temperature control of chemical reactor models [26], control of industrial servo-systems [20] and underactuated multibody systems [8], voltage and current control of electrical circuits [13], DC-link power flow control [37] and adaptive cruise control [11,12].
Funnel control for infinite-dimensional systems is a hard task in general. A simple class of systems with relative degree one and infinite-dimensional internal dynamics has been considered in the seminal work [24]. Linear infinite-dimensional systems for which an integer-valued relative degree exists have been considered in [25]. In fact, it has been observed in the recent work [10] that the existence of an integer-valued relative degree is essential to apply known funnel control results as formulated e.g. in [7]. It is then shown in [10] that a large class of systems which exhibit infinite-dimensional internal dynamics is susceptible to funnel control. A practically relevant example is a mowing water tank system, which is shown to belong to the aforementioned class in [9]. However, not even every linear infinite-dimensional system has a well-defined relative degree, in which case the results from [7] cannot be applied. For this class of systems -to which the Fokker-Planck equation belongs -the feasibility of funnel control has to be investigated directly for the (nonlinear and time-varying) closed-loop system; see e.g. [34] for a boundary controlled heat equation, [33] for a general class of boundary control systems and [6] for a system of monodomain equations (which represent defibrillation processes of the human heart).
1.1. Nomenclature. The set of natural numbers is denoted by N and N 0 = N ∪ {0}. For a Banach space X, X stands for its dual. For a measurable set Ω ⊆ R, a measurable function w : Ω → R ≥0 and p ∈ [1, ∞], L p (Ω; w) denotes the w-weighted Lebesgue space of (equivalence classes of) measurable and p-integrable functions f : Ω → R with norm f L p (Ω;w) = Ω w(x) |f (x)| p dx 1/p , f ∈ L p (Ω; w), if p < ∞ and f L ∞ (Ω;w) = ess sup x∈Ω w(x) |f (x)| if p = ∞. Additionally, for k ∈ N 0 , W k,p (Ω; w) denotes the w-weighted Sobolev space of (equivalence classes of) k-times weakly differentiable functions f : Ω → R with f, f , . . . , f (k) ∈ L p (Ω; w). If w ≡ 1, then we write L p (Ω; 1) = L p (Ω), W k,p (Ω; 1) = W k,p (Ω) and, when Ω is clear from the context, · L ∞ (Ω;1) = · ∞ . For an interval J ⊆ R, a Banach space X and p ∈ [1, ∞], we denote by L p (J; X) the vector space of equivalence classes of strongly measurable functions f : J → X such that f (·) X ∈ L p (J); the distinction between L p (J; X) and L p (Ω; w) should be clear from the context. Note that if J = (a, b) for a, b ∈ R, the spaces L p ((a, b); X), L p ([a, b]; X), L p ([a, b); X) and L p ((a, b]; X) coincide, since the points at the boundary have measure zero. We will simply write L p (a, b; X), also for the case a = −∞ or b = ∞. We refer to [1] for further details on Sobolev and Lebesgue spaces.
By C(J; X) we denote the space of continuous functions f : J → X. For p ∈ [1, ∞], W 1,p (J; X) stands for the Sobolev space of X-valued equivalance classes of weakly differentiable and p-integrable functions f : J → X with p-integrable weak derivative, i.e., f,ḟ ∈ L p (J; X). Thereby, integration (and thus weak differentiation) has to be understood in the Bochner sense, see [16,Sec. 5.9.2]. The spaces L p loc (J; X) and W 1,p loc (J; X) consist of all f whose restriction to any compact interval K ⊆ J are in L p (K; X) or W 1,p (K; X), respectively.
For an interval J ⊆ R we denote by C ∞ (J) the set of all infinitely times continuously differentiable functions f : J → R, and by C ∞ c (J) the set of all functions f ∈ C ∞ (J) with compact support on J.
1.2. The Fokker-Planck equation for a controlled stochastic process. We consider a controlled stochastic process described by the Itô stochastic differential equation (cf. [30,Sec. 11.]) where X t : Ω → R n , t ≥ 0, are random vectors and Ω is the sample space of a probability space (Ω, F, P ). (W t ) t≥0 denotes a d-dimensional Wiener process with zero mean value and unit variance, b : R ≥0 × R n × R m → R n is a drift function and σ : Using the framework presented in [2] we can formulate the control problem for the probability density function of the stochastic process (X t ) t≥0 as a partial differential equation, the Fokker-Planck equation. This approach is feasible under appropriate assumptions on the functions b and σ as shown in [31,32]. Define x, u) , then the probability density function p : R ≥0 × R n → R associated with the process (X t ) t≥0 evolves according to the Fokker-Planck equation and additionally, since p is a probability density, we require The second condition in (1.3) is the conservation of probability, while the first requires any probability to be non-negative. Some conditions for the existence of nonnegative solutions of the Fokker-Planck equation are given in [3,14,19] for instance.
1.3. The Ornstein-Uhlenbeck process. As a specific stochastic process, in this work we consider the Ornstein-Uhlenbeck process and we assume that it can be controlled via the drift term only. Then it is modelled by an equation of the form (1. This form is often encountered in the literature, see e.g. [2,3,18] and the references therein. However, let us stress that the equation is restricted to a bounded spatial domain in many works such as [2,3], and Dirichlet boundary conditions are used; this is not the natural framework, cf. also Section 2. Let c := 1 2 σ 2 , then the associated Fokker-Planck equation (1.2) is given in the form For later use we define the function Since it is unrealistic to assume that we can measure p(t, x) for all t ≥ 0 and all x ∈ R, we associate an output function y : R ≥0 → R with (1.4). The output should be chosen in such a way that, by manipulating it via the control input, it is possible to influence the collective behavior of the process. As mentioned in [2], the mean value E[X t ] "is omnipresent in almost all stochastic optimal control problems considered in the scientific literature". Therefore, it is a reasonable choice for the output, i.e., We assume that the measurement of the output y(t) is available to the controller at each time t ≥ 0. In practice, the corresponding integral cannot be calculated exactly, thus the mean value will typically be approximated by data-driven methods such as Monte Carlo integration.
Note that controlling the Fokker-Planck equation via the drift term with mean value as output is indeed sufficient to influence the shape of the solution density, since the variance of the process is independent of the control input. In fact, we will show in Proposition 3.3 that the variance of the solution is of the form for some K ∈ R, provided (1.3) holds.

Control objective.
The objective is to design a robust output error feedback such that in the closed-loop system the tracking error e(t) evolves within a prescribed performance funnel which is determined by a function ϕ belonging to The robustness requirement on the control essentially means that it is feasible under bounded additive disturbances "with zero mass", which influence the Fokker-Planck equation. This is made precise in Section 6. The boundary of the funnel F ϕ is given by the reciprocal of ϕ, see Fig. 1.1. We explicitly allow for ϕ(0) = 0, meaning that no restriction on the initial value is imposed since ϕ(0)|e(0)| < 1; the funnel boundary 1/ϕ has a pole at t = 0 in this case.
An important property of the class Φ is that the boundary of each performance funnel F ϕ with ϕ ∈ Φ is bounded away from zero, i.e., because of boundedness of ϕ there exists λ > 0 such that 1/ϕ(t) ≥ λ for all t > 0.
It is of utmost importance to notice that the function ϕ ∈ Φ is a design parameter in the control law (stated in Section 6), thus its choice is completely up to the designer. Typically, the specific application dictates the constraints on the tracking error and thus indicates suitable choices. We stress that the funnel boundary is not necessarily monotonically decreasing, while such a choice may be convenient in most situations. However, widening the funnel over some later time interval might be beneficial, for instance in the presence of strongly varying reference signals or periodic disturbances. A variety of different funnel boundaries are possible, see e.g. [23,Sec. 3.2]. 1.5. Organization of the present paper. In Section 2 we introduce the mathematical framework around the Fokker-Planck operator associated to the equation (1.4). We emphasize that we consider an unbounded spatial domain in (1.4), without any boundary conditions. Using weighted Lebesgue and Sobolev spaces, first an auxiliary operator is defined via a suitable coercive sesquilinear form. This operator, along with the spaces, is then transformed to the desired Fokker-Planck operator. Special emphasis is put on the spaces after the transformation, which are again weighted Lebesgue and Sobolev spaces, but the latter is not equipped with the natural norm. We highlight that this seems to be the natural framework for the equation (1.4). The definition of a weak solution is given in Section 3 and it is shown that any solution satisfies (1.3) and that its variance exponentially converges to c/γ. A simple feedforward control approach is then discussed in Section 4, which may be favourable when the system parameters are known and no disturbances are present. As a basis for the feasibility proof of the robust funnel controller in Section 6, we consider Galerkin approximations of possible solutions of (1.4) under disturbances in Section 5 and derive an energy estimate. We emphasize that the closed-loop system corresponding to the application of the funnel controller is a nonlinear and time-varying PDE, thus proving existence and uniqueness of solutions is a nontrivial task. We illustrate our results by some simulations in Section 7.
2. The Fokker-Planck operator. In this section we introduce an operator which can be associated with the PDE (1.4) in the uncontrolled case, i.e., u = 0. To this end, we invoke form methods for which we frequently refer to [4] and [5]. Consider the system (1.4) with c > 0, γ > 0 and φ as defined in (1.5). To begin with, let and use the short-hand notation v = ∂v ∂x for the weak derivative of v ∈ V . Define the sesquilinear form to which we may associate an operator as follows.
Proposition 2.1. Consider the form (2.1), then there exists exactly one operator A : Moreover, (i) A is self-adjoint, positive and has compact resolvent, (ii) there exists a monotonically increasing sequence (λ j ) j∈N0 ∈ (R ≥0 ) N0 , which satisfies lim j→∞ λ j = ∞ and the spectrum of A reads σ( Proof. We show that the operator A exists as stated. By the Cauchy-Schwarz inequality we have and hence the form a is bounded. Since the injection j : V → H is clearly continuous with dense range, it follows from [4, Prop. 5.5] that A exists and is positive since a is positive. We show (i): As above, there exists an operator B :  In the following we explicitly derive the eigenvalues and eigenfunctions of A. To this end, recall the Hermite polynomials defined by It is well known that these polynomials have, for all x ∈ R and all n, m ∈ N 0 , the properties π2 n n! δ n,m , where δ n,m denotes the Kronecker delta. We obtain the following representation.
Proposition 2.2. Use the notation from Proposition 2.1. Then we have, for all j ∈ N 0 and all x ∈ R, that Proof. Define w j (x) := α j H j (θx) and κ j := 2jθ 2 for j ∈ N 0 and x ∈ R.
Step 2 : We show that w j ∈ D(A) and Aw j = κ j w j for all j ∈ N 0 . First note that w j ∈ V is clear since H j = 2jH j−1 . By definition of A the two assertions hold if, and only if, a(w j , z) = κ j w j , z H for all z ∈ V . For j = 0 this is clear since κ 0 = 0 and w 0 is constant, thus a(w 0 , z) = 0 for all z ∈ V . Now, fix j ∈ N and z ∈ V . Then we have, invoking the properties of the Hermite polynomials, Step 3 : We show that (w j ) j∈N0 is an orthonormal system in H. Invoking the properties of the Hermite polynomials, this follows immediately from Step 4 : From Steps 1-3 it follows that the sequence (κ j ) j∈N0 is a subsequence of (λ k ) k∈N0 and (w j ) j∈N0 is a subsequence of (v k ) k∈N0 . We show equality. Seeking a contradiction, assume that there exist λ ∈ R ≥0 and v ∈ V \{0} such that Av = λv and λ = κ j for all j ∈ N 0 . It is well known that the Hermite polynomials constitute a complete orthogonal system in Step 2) for some k ∈ N 0 in the following and obtain, as in Step 2, which by completeness of (w j ) j∈N0 implies v = 0, a contradiction.
Step 5 : Assertion (ii) is an immediate consequence of the properties of the Hermite polynomials, (iii) is shown in (2.2) and (iv) in Step 1. It remains to show (i), i.e., that If j = 0 or k = 0, then v j or v k is constant and hence v j , v k H = 0. This finishes the proof. Now we turn to transform the operator A so that it becomes a suitable Fokker-Planck operator. To this end, define the spaces together with the inner products We will show that V is equal to W 1,2 (R; e φ ), although not equipped with the natural Sobolev norm. We postpone the proof of this equality and first transform the operator A and collect some results on it. To this end, define the sesquilinear form as well as D(A) := T (D(A)) and the operator Then we have that, for v ∈ D(A) and y ∈ H, From Proposition 2.2 we immediately obtain the following result on the eigenvalues and eigenfunctions of A.
We now turn to the proof of in general. Nevertheless, we will additionally show that the norm · W 1,2 (R;e φ ) can be estimated by the norm · V .
Proposition 2.4. The following statements are true: Proof. We show (i), so let v ∈ V . Use the notation from Propositions 2.1 and 2.2. Since and this implies, again invoking Parseval's identity, Now consider the sequence w n := φ n j=0 β j v j , n ∈ N. We show that (w n ) n∈N has a weakly convergent subsequence in H. To this end, first observe that for all x ∈ R we have where we have used the properties of the Hermite polynomials, 2(j + 1)α j+1 = α j and the convention v −1 (x) := 0. Therefore, As a consequence, (w n ) n∈N is a bounded sequence in H and hence has a weakly convergent subsequence, which we again denote by (w n ) n∈N . This means that there exists w ∈ H such that We show that w = φ v, which proves the claim φ v ∈ H. To this end, let j ∈ N 0 and observe that where β −1 := 0, and, for any n ≥ j + 1, The proof of this statement is analogous to the proof of (i), utilizing the complete orthonormal system (z j ) j∈N0 in H from Proposition 2.3. Since z, z ∈ L 2 (R; e φ ) = H we may compute, for β j := z, z j H , that z 2 H = ∞ j=1 λ j β 2 j−1 . Then, as above, w n := φ n j=0 β j z j is bounded by 4 z 2 H +4θ 2 z 2 H and hence has a weakly convergent subsequence, the limit of which is φ z, which is hence an element of H.
We show (iii). First observe that Then the claim follows from We show (iv). As shown in (ii), for any v ∈ V we have that v ∈ H. Furthermore, we may apply the findings of (i) to e φ v ∈ H, so that with w = φ e φ v ∈ H and the corresponding sequence (w n ) n∈N we have, since lim n→∞ w n , z H = w, z H for all z ∈ H, Finally, (v) is a consequence of (iv) upon observing that v 2 Attention now turns to the operator −cA, which will serve as the Fokker-Planck operator. In view of the right-hand side in (1.4), this is justified by the following property.
Lemma 2.5. Let z ∈ V be such that e φ z ∈ V . Then we have that Proof. We calculate that for any j ∈ N 0 where the last equality follows from the assumption e φ z ∈ V and Proposition 2.2 (iv). Since (2.4) is true for all j ∈ N 0 , we have proved the first equality in the statement. The second is a straightforward calculation.
Recall that φ (x) = 2θ 2 x, and hence cφ (x) = γx. Therefore, with the operator the Fokker-Planck equation (1.4) can be rewritten as with state space H, whereṗ = ∂p ∂t . System (2.6) fits into the framework of bilinear control systems as considered for the Fokker-Planck equation e.g. in [14,22]. However, we stress that the assumptions required in the aforementioned works are not satisfied for (2.6).
We also note that (2.6) admits solutions with potentially less regularity requirements than equation (1.4), because of Lemma 2.5. The specific definition of a weak solution that we use in the present paper is given in the following section.
where a : V × V → R is the sesquilinear form defined in (2.3) and B : V × R → H is the operator defined in (2.5). A function p is called solution of (2.6) on R ≥0 , if p| [0,T ] is a solution of (2.6) on [0, T ] for all T > 0, in particular p ∈ L 2 loc (0, ∞; V) andṗ ∈ L 2 loc (0, ∞; V ). Remark 3.2. We like to note that sinceṗ ∈ L 2 (0, T ; V ), where V is the dual of V with respect to the pivot space H, i.e., V → H → V is a Gelfand triple, the duality pairing between V and V is compatible with the inner product in H, which means using an appropriate identification via the Riesz representation theorem. For brevity, we will always use the latter expression.
We may infer the following properties of a solution of (2.6). (iii) If ∞ −∞ p 0 (x)dx = 1, then for y : [0, T ] → R as in (1.6) there exists K ∈ R, which is independent of T , such that Moreover, since p is a solution of (2.6), Furthermore, we can estimate for almost all x ∈ R, it follows that p − (t) = 0 ∈ H for all t ≥ 0, thus the claim is shown. We show (iii). To this end, consider the system of eigenfunctions (z j ) j∈N0 of A from Proposition 2.3 and define µ i (t) := p(t), z i H for t ∈ [0, T ] and i ∈ {0, 1, 2}. Then it follows from [21, Thm. 1.32] thaṫ for almost all t ∈ [0, T ] and i ∈ {0, 1, 2}, where µ −1 (t) := 0. In particular, since λ 0 = 0, we haveμ 0 (t) = 0, thus µ 0 (t) = p 0 , z 0 H for all t ∈ [0, T ]. Now, the variance is given by where the last equality follows from the assumption. Invoking that the second Hermite polynomial is given by H 2 (x) = 4x 2 − 2 for all x ∈ R, it follows that Inserting this gives Furthermore, invoking H 1 (x) = 2x for all x ∈ R, we find that From the assumption it also follows that and, by definition of θ, we have 2θ 2 = γ/c. Therefore, we obtain for t ∈ [0, T ] and calculatė where we have used that λ 1 = 1 2 λ 2 . Furthermore, invoking α 1 = 2α 2 and α 0 = for almost all t ∈ [0, T ]. Finally, with cλ 2 = 4cθ 2 = 2γ and K := g(0) (which is independent of T ) we obtain that

A simple feedforward controller.
In this section we present a very simple, yet effective feedforward control strategy. We stress that the presented control law does not achieve the control objective -it is not robust and does not guarantee error evolution within the prescribed performance funnel. Nevertheless, we will show that it guarantees fast (exponential) convergence of the tracking error to zero, provided the system parameters are known, no disturbances are present and the derivative of the reference signal is available to the controller. For γ > 0 as in (1.4) and reference signal y ref ∈ W 1,∞ (R ≥0 ) the controller is given by Note that (4.1) is not a feedback controller, it is completely determined by y ref . We show that (2.6) with (4.1) admits a solution. The proof can be found in Appendix A. We like to emphasize that the feasibility result of Proposition 4.1 is independent of the initial value p 0 ∈ H. Moreover, if p 0 satisfies ∞ −∞ p 0 (x)dx = 1, then the control (4.1) achieves exponential convergence of the tracking error e(t) = y(t) − y ref (t) to zero for all initial probability densities. Furthermore, the solution p exhibits the properties derived in Proposition 3.3; thus its mean value and variance exponentially converge to y ref and c γ , resp. Although the controller (4.1) requires knowledge of γ andẏ ref and the absence of disturbances, its simplicity may justify its application in real-world examples. On the other hand, in the presence of uncertainties and disturbances, a feedback control strategy is more suitable, for which we refer to Section 6.

Energy estimate.
In this section we consider the Galerkin approximation of a possible solution of (2.6) under the influence of a disturbance (the definition of a solution in this case will be given Section 6) and show that these approximations satisfy a certain energy estimate. These arguments follow the spirit of [16,Sec. 7.1]. Let n ∈ N, p 0 ∈ H and such that µ i is absolutely continuous on every interval [0, T ], T > 0, the disturbance has "zero mass" in the sense we have, for all i = 0, . . . , n, are the eigenfunctions of A from Proposition 2.3. Observe that, compared to Section 3, here d acts as an additive disturbance on the Fokker-Planck equation (2.6). Although, admittedly, the set of disturbances d ∈ L ∞ (0, ∞; H) satisfying (5.1) is chosen such that "things work out", it is nevertheless very large: If d is chosen such that is only restricted to the orthogonal complement of a one-dimensional subspace of H. We proceed in several steps.
Step 2 : We show that sup t≥0 p n (t) H ≤ C 1 for C 1 > 0 independent of n and T . First observe that We calculate that, writing d ∞ := d L ∞ (0,∞;H) for brevity, for i ∈ N as well as β 0 := 0. Let i 0 ∈ N be the smallest index such that β i0 < 0. Set for almost all t ≥ 0, wherec 3 : where the last inequality follows from Parseval's identity.
Step 3 : We show that p n L 2 (0,T ;H) ≤ C 2 for C 2 > 0 independent of n. This is a direct consequence of Step 2 as  Step 4 : We show that p n L 2 (0,T ;V) ≤ C 3 for C 3 > 0 independent of n. Multiplying with µ i (t) in condition (B) and summing for i = 0, . . . , n gives for almost all t ≥ 0 and hence Finally, this implies which finishes the proof.
6. Funnel control. The controller that we propose in order to achieve the control objective formulated in Subsection 1.4 is the funnel controller. It has the advantage that it is model-free, i.e., we may state the control law without any further information about the equation (1.4). Therefore, it is inherently robust and hence able to handle both uncertainties in the system parameters as well as disturbances in the PDE itself. In particular, we do not need any knowledge of the parameters c > 0 and γ > 0, or of the initial probability density p 0 (·). Utilizing the version from [7], we only require the relative degree in order to state the appropriate control law. For finite dimensional systems we refer to [27] for a definition of the relative degree; this notion can be extended to systems with infinite-dimensional internal dynamics, see e.g. [10]. However, for general infinite-dimensional systems a concept of relative degree is not available. Nevertheless, for the system (2.6), (1.6) we may calculate -formally at this point - where µ = 1 2θα1 , η = µ √ λ 1 p 0 , z 0 H and we have used that p(t), Proposition 3.3. The input appears explicitly in the above equation forẏ, which suggests that (2.6), (1.6) at least exhibits an input-output behavior similar to that of a relative degree one system. This justifies to investigate the application of the funnel controller to (2.6), (1.6), where ϕ ∈ Φ. We stress that the control gain k in (6.1) is not dynamically generated and it is not monotone. It is only large when the error e(t) is close to the funnel boundary 1/ϕ(t) at some time t > 0.
For feasibility we seek to show that for any y ref ∈ W 1,∞ (R ≥0 ), ϕ ∈ Φ and any initial probability density p 0 such that ϕ(0)|e(0)| < 1 we have that the closed-loop system consisting of (2.6), (1.6) and (6.1) has a global and bounded solution p which satisfies the conditions (1.3) and the tracking error e evolves uniformly within the funnel boundaries, i.e., Hence, even if a solution exists on a finite time interval [0, T ), it is not clear that it can be extended to a global solution. Moreover, the closed-loop system (2.6), (1.6) and (6.1) is a time-varying and nonlinear PDE. This renders the solution of the above problem a challenging task.
Furthermore, we study the robustness of the controller (6.1) w.r.t. disturbances d ∈ L ∞ (0, ∞; H) that satisfy the zero-mass condition (5.1) and influence the Fokker-Planck equation (2.6) in an additive way, cf. also Section 5. That is, we consider the system where a : V × V → R is the sesquilinear form defined in (2.3) and B : V × R → H is the operator defined in (2.5). Note that, in the presence of disturbances, it cannot be expected that the solution p(t) is a probability density function for any t ≥ 0 in general, i.e., conditions (1.3) will typically not hold. We introduce solutions of the closed-loop system resulting from the application of the funnel controller (6.1) to system (6.2) with output (1.6) as follows.
Proof. We divide the proof into several steps.
Step 1 : We construct a Galerkin approximation of a solution candidate, which has the properties as in Section 5. To this end, fix n ∈ N, consider the eigenvalues (λ j ) j∈N0 and the system of eigenfunctions (z j ) j∈N0 of A from Propositions 2.2 and 2.3, and define µ 0 (t) := p 0 , z 0 H as well as d i (t) := d(t), z i H for t ≥ 0 and i ∈ N 0 . Furthermore, consider the one-dimensional initial value problem Observe that µ 0 = α 0 ∞ −∞ p 0 (x)dx > 0 by assumption, |d 1 (t)| ≤ d(t) H ≤ d L ∞ (0,∞;H) for almost all t ≥ 0 and Therefore, existence of a solution to (6.3) follows from [7,Thm. 3.1], that is there exists a function µ 1 ∈ C(R ≥0 ; R) which is absolutely continuous on [0, T ] for all T > 0 and satisfies the initial condition and differential equation in (6.3) for almost all t ≥ 0. Moreover, we have that u ∈ C(R ≥0 ; R) and µ 1 , u ∈ L ∞ (R ≥0 ) as well as the estimate Now consider the system of n − 1 ordinary differential equations given by It follows from the theory of ordinary differential equations, see e.g. [40, § 10, Thm. XX], that there exists a solution (µ 2 , . . . , µ n ) ∈ C([0, ω); R n−1 ) with maximal ω ∈ (0, ∞], which is absolutely continuous on [0, T ] for all 0 < T < ω and satisfies the initial condition and differential equation in (6.4) for almost all t ∈ [0, ω). Furthermore, the closure of the graph of (µ 2 , . . . , µ n ) is not a compact subset of R ≥0 × R n−1 .
Step 2 : We show that there exists a solution (p, u, y) of (6.2), (1.6), (6.1) on R ≥0 , where u is the function defined in Step 1. Fix T > 0 and observe that by Step 1 and Proposition 5.1 the sequence (p n ) n∈N satisfies the energy estimate (5.3). Therefore, the sequence (p n ) n∈N is both bounded in L 2 (0, T ; H) and L 2 (0, T ; V), and the sequence (ṗ n ) n∈N is bounded in L 2 (0, T ; V ). Hence there exists p ∈ L 2 (0, T ; V) withṗ ∈ L 2 (0, T ; V ) and subsequences of (p n ) n∈N and (ṗ n ) n∈N , resp., again denoted in the same way, such that p n → p weakly in L 2 (0, T ; H), p n → p weakly in L 2 (0, T ; V), p n →ṗ weakly in L 2 (0, T ; V ).
Step 2d : It remains to show that (1.6), (6.1) are satisfied for all t ∈ [0, T ]. To this end, it suffices to show that y as in (1.6) satisfies y(t) = δµ 1 (t) for δ as defined in (6.3). Let t ∈ [0, T ] and observe that we have p n → p weakly in L 2 (0, t; H). Therefore, and the fundamental theorem of calculus gives that y(t) = δµ 1 (t).

A numerical example.
In this section, we illustrate the applicability of the funnel controller by means of a numerical example. We simulate the evolution of a given initial probability density p 0 under the Fokker-Planck equation (6.2) with the mean value as output (1.6) and under the influence of the controller (6.1). To show the universality of Theorem 6.2 we consider an initial density that is in H, but not in V, namely a uniform distribution on − 1, − 1 2 ∪ 1 4 , 3 4 given by Therefore, feasibility of funnel control, i.e., the application of (6.1) to (6.2), (1.6), is guaranteed by Theorem 6.2.
For the simulation the PDE is solved using a finite difference method with a uniform time grid (in t) with 10.000 points for the interval [0, 10] and a uniform spatial grid (in x) with 2.000 points for the interval [− 5,5]. The simulation has been performed in MATLAB, where in each time step an ODE is solved by using the command pdepe with (artificial) Dirichlet boundary conditions. Relative and absolute tolerance are set to the default values 10 −3 and 10 −6 , resp. Fig. 7.1 (a) shows the error e(t) = y(t) − y ref (t) between mean value and reference signal and the input values u(t) generated by the controller are depicted in Fig. 7.1 (b). Several snapshots of the solution p, are shown in Fig. 7.1 (c) and (d). It can be seen that, in the presence of disturbances, p(t) is not a probability density function for t > 0 in general, since it takes negative values. Nevertheless, the controller guarantees that the error stays within the prescribed funnel boundaries, while the control input shows an acceptable performance.
A simulation of the same configuration, but without disturbance can be seen in Fig. 7.2. Here, the simulations of the undisturbed equation show that p(t) is always a probability density and its variance exponentially converges to c γ = 0.2, as stated in Proposition 4.1. Video clips of the simulations showing the evolution of p(t) for t ∈ [0, 10] can be found in the supplementary material.
Appendix A. Proof of Proposition 4.1.
Proof. The existence and uniqueness of a solution p of (2.6) with (4.1) on R ≥0 that satisfies p ∈ L ∞ (0, ∞; H) follows along the lines of the proof of Theorem 6.2 by observing that the linear ordinary differential equatioṅ  has a unique and bounded solution on R ≥0 . Therefore, it remains to show (ii). To this end, define g(t) := µ 1 (t) − √ λ 1 µ 0 y ref (t) for t ≥ 0 and observe thaṫ for almost all t ≥ 0, since λ 1 = γ c . Thus we find that