Optimal Control of Hughes’ Model for Pedestrian Flow via Local Attraction

We discuss the control of a human crowd whose dynamics is governed by a regularized version of Hughes’ model, cf. Hughes (Transp Res Part B: Methodol 36(6):507–535, 2002. https://doi.org/10.1016/s0191-2615(01)00015-7). We assume that a finite number of agents act on the crowd and try to optimize their paths in a given time interval. The objective functional can be general and it can correspond, for instance, to the desire for fast evacuation or to maintain a single group of individuals. We provide an existence and regularity result for the coupled PDE-ODE forward model via an approximation argument, study differentiability properties of the control-to-state map, establish the existence of a globally optimal control and formulate optimality conditions.

models for pedestrian dynamics are microscopic in the sense that they provide constitutive laws for the motion of each pedestrian (e. g., systems of ODEs or cellular automata or the social force model, cf.Burstedde et al., ; Helbing, Molnár, ), Hughes' model starts from a macroscopic approach.It is based on the following three assumptions: () The velocity  of the pedestrians is determined by the density  of the surrounding pedestrian ow and the behavioral characteristics of the pedestrians only.Denoting the movement direction by  ∈ R there holds  =  () , | | = .
() Pedestrians have a common sense of the task (called potential ), i. e., they aim to reach their common destination by () Pedestrians seek to minimize their (accurately) estimated travel time, but modify their velocity to avoid high densities.The potential is thus a solution of the Eikonal equation .
( . ) Due to the previous explanations it is clear that − (, )| (,) is the direction of the individuals in the point (, ) ∈   .
The analysis of this system is quite involved since the derivative of , being the viscosity solution to ( .b), has jump discontinuities on a set that depends on  and is not known a priori.Since we are going to consider optimal control problems, we shall focus on a regularized version of the forward model ( .), given by where we assume that the boundary consists of parts which act as "doors" Ω D , at which pedestrians can exit with a given out ow velocity  > , and "walls" Ω W .We assume Ω D ∪ Ω W = Ω and Ω D ∩ Ω W = ∅ and set Σ W ( , ) × Ω W and Σ D ( , ) × Ω D . .

. O C P
Our optimal control problem is based on the following scenario.We assume that there is a small, given number  > of agents (guides), who are able to locally in uence the motion of pedestrians in their vicinity.Think, for instance, of tourist guides or marked security personnel at large sports events.The -th agent's position is described by a function   () ∈ R ,  = , . . ., , and the interaction is modeled by a radially symmetric and decreasing kernel  Here and throughout,  =  () = ( (), . . .,   ()) ᵀ is the function collecting all agent positions.Furthermore, we need to modify the model to insure that the maximal velocity of the crowd is still normalized to the velocity model  () despite the agents' presence.Indeed, for the unregularized model, holds.When  is replaced by  +   , this is no longer true.Thus we explicitly normalize the transport direction, i. e., instead of ( .), we de ne the transport velocity by  (, , )  () ℎ ∇( +   (; •)) .
( . ) Here, ℎ is a smoothed projection onto the unit ball.Throughout this article we will use with min  a xed smooth approximation of the minimum function.
We assume that the agents move with maximal possible velocity towards a prescribed directions   () ∈ R ,  = , . . ., , which act as control in the system.Since the agents are also part of the crowd, their e ective velocity will depend on the surrounding density in the same way as it does for all other individuals in the crowd.We therefore assume |  ()| ≤ and the law of motion for the agents will be   () =   (,   ())   (),  = , . . ., , ( . where  is a suitable extension of  from Ω to R which will be detailed later.While this extension is necessary to ensure the existence of solutions since the agents may leave the domain Ω on which  is de ned, the precise choice of the extension is clearly irrelevant in terms of modeling.Notice also that the ODE ( . ) does not prevent agents from walking through walls or to attract people from behind a wall.The rst issue can be avoided by imposing additional state constraints in an optimal control problem.
The complete forward system which we are going to consider nally reads as well as boundary conditions ( . ).
The aim of the present paper is to investigate several optimal control problems for this coupled system.We seek an optimal control function  such that the solution triple  = (, , ) is optimal in a certain sense.Depending on the application in mind it remains to de ne a suitable objective functional.We particularize the following two examples: • Minimal evacuation time: In this case one seeks to minimize the time required for the evacuation of a room.The exits where individuals can leave the room are located at the boundary parts Ω D modeling doors.As time-optimal control problems with PDEs are rather challenging, see, e. g., Bonifacius, Pieper, Vexler, ; Knowles, ; Schittkowski, ; Zheng, Yin, , we consider a simpler but closely related model.We x a reasonably large nal time  > and minimize with weighting parameters  ,  > and a regularization parameter  > .The rst term in  penalizes individuals remaining in the room at time  .The second term encourages individuals to leave the room as early as possible.The last terms provides the required regularity for the control variables, so that the forward system ( . ) is well-de ned.From the modeling point of view, these terms also avoid unrealistic trajectories of the agents.
• Optimal binding of a crowd: In some applications it might be desired to keep the group of individuals together, i. e., trying to maintain a single group during an evacuation.This is also motivated by a similar approach which has been used to model the repulsive interaction of dogs in a ock of sheeps, see Burger, Pinnau, et al., .To this end, we de ne the center of mass and variance of  as with total mass  () = ∫ Ω  (, ) d.A crowd is optimally kept together when the functional Clearly, it is also possible to use a combination of the objective functionals ( . ) and ( .).

. . R W
A rst contribution on existence results for the Hughes model for the one-dimensional case is Di Francesco, Markowich, et al., .There it was shown, starting from the regularized version ( .), that in the limit  → a suitable unique entropy solution  exists.The proof is based on a vanishing viscosity argument and Kruzkov's doubling of variables technique to show uniqueness.The results were complemented by more detailed studies on the unregularized problem, also numerically.For instance generalizations to higher spatial dimensions can be found in Colombo, Garavello, Lécureux-Mercier, , even for a slightly more general class of models.Further articles examine Riemann-type solutions to the unregularized problem; see Amadori, Di Francesco, ; El-Khatib, Goatin, Rosini, ; ; Di Francesco, Fagioli, et al., in one spatial dimension.As far as modeling is concerned, slightly di erent models were derived in Burger, Di Francesco, et al., based on a mean eld games approach.In Carrillo, Martin, Wolfram, , a modi ed approach using multiple local potentials   instead of one global potential  is introduced, removing the possibly unrealistic assumption that every pedestrian has complete information of the entire crowd.Moreover, in Carlini et al., , a discrete pedestrian model in a graph network is studied.
From a broader perspective, the optimal control of ( . ) falls into the class of the optimal control of coupled ODE-PDE systems.Such problems, with models from a range of di erent applications have been analyzed, for instance, in Chudej et al., ; Wendl, Pesch, Rund, ; Kimmerle, Gerdts, ; Kimmerle, Gerdts, Herzog, ; Holtmannspötter, Rösch, Vexler, .Also in the context of pedestrian dynamics a couple of contributions exist.We want to mention Albi et al., where a closely related microscopic model with crowds consisting of a ( xed) number of leaders and followers is studied.The interaction between individuals is a short range retraction and a long range attraction.While leaders are not visible to the remaining crowd, they still in uence it by taking part in these interactions.Then an optimal control problem arises as an external force acts on the leaders.The authors also consider, in the limit of many individuals (grazing interaction limit), macroscopic Boltzmann type equations for this interaction, while the number of leaders remains xed and nite.Similar studies can be found in Burger, Pinnau, et al., where external agents act as control.Again, they start from a microscopic ODE model and subsequently obtain a continuous model for the uncontrolled population by means of a mean eld limit.For a general overview on interacting particle systems and control, we refer the reader to Pinnau, Totzeck, .
Closer to our approach is the work of Borsche, Colombo, et al., .There, a system of hyperbolic conservation laws for the density of di erent pedestrian groups is coupled to ODEs accounting for agents.Due to the low regularity of solutions to the hyperbolic equations, a regularization in the ODEs, similar to Lemma . in our case, is used.See also Borsche, Klar, et al., ; Borsche, Meurer, for a similar approach in di erent settings.This paper is organized as follows.In Section we collect the required notation, introduce some assumptions and state a precise existence and uniqueness result for the regularized system ( .).The full forward system involving also the ordinary di erential equation . is investigated in Section and the linearized forward system in Section .The latter is required to establish the di erentiability of the control-to-state map, which in turn is the basis of optimality conditions.Based on this we discuss the optimal control problem in Section and derive rst-order necessary optimality conditions.The presentation of numerical results will be postponed to a forthcoming publication.

. M P
Let us rst state the assumptions on the domain and data.(A ) The initial density satis es  ∈  / , (Ω) and ≤  ≤ a.e. in Ω.
Moreover, we require some assumptions on the potential functions ( . ) of the agents, which depend on the kernel .
Finally, we consider an assumption on the velocity controls of the agents.
Throughout this article we frequently exploit the boundedness and Lipschitz continuity of the functions ,  , ℎ and ()   () and its derivatives.When doing so, we denote the bounds and Lipschitz constants by   ,   ,  ℎ and   respectively.  ,   ,  ℎ and   .
Remark .(Assumptions).() Our results extend to the case  = upon adopting the Sobolev embeddings used in several places.
() Assumption (A ) essentially means that the door and wall parts of the boundary "do not meet", i. e., we do not consider a truly mixed boundary value problem in order to avoid technical conditions ensuring su cient regularity of solutions.(A ) can be replaced, e. g., in two spatial dimensions, by suitable angle conditions on the points where the two parts of the boundary meet.The interested reader is referred to results in Grisvard, .
() While in Assumption (A )  is de ned on all of R, as far as the modeling is concerned, only  | [ , ] is relevant.Indeed, we will later see that the solution to ( .a) satis es ≤  ≤ .
() A reasonable choice in (K ) and (K ) is the kernel function ) where  > is an intensity factor and  > is related to an attraction radius.
- .Spaces with non-integral  and  are de ned, as usual, as (real) interpolation spaces.Of particular interest in our application is the space  ,  (  ) with  >  = , which ful lls the embedding This is needed in order to allow point evaluations of the density , required in the ordinary di erential equation ( .c).Finally, for functions from the Hölder space  , (Ω) we introduce the norm In the following we collect some important properties of the function spaces used in this article.
-  Recall the di erential equation ( . ), where an extension to R of the density function  is used.For theoretical purposes we will use an extension operator ful lling the following result from Lemma . in Gilbarg, Trudinger, : Lemma . .Let  ∈ ( , ) be a xed number.There exists a linear, continuous extension operator holds for all  ∈  , (Ω).For brevity we will write  E . .

A F S
This section is devoted to showing the existence of strong solutions to the forward system ( . ) with boundary and initial conditions ( .), ( .), ( .).We proceed in two steps.First we provide auxiliary results on equation ( .b) as well as on linear parabolic equations.Then we prove existence of solutions to the complete forward system.
--cbna page of
Lemma . .For given  :   → R consider the equation with boundary conditions ( .).We have: with a constant   depending on ,  ,  , Ω and  only.
From now on we shall use the de nition   max   ,   ,   .
Proof.We rst show assertion (i).Note that due to the continuity of  in time it makes sense to de ne, for xed  ∈ [ , ], the function By assumption (A ) on  we then have   ∈  ∞ (Ω).
To obtain the desired  ,  (  )-regularity of the density function  we will need the following lemma taken from Denk, Hieber, Prüss, but adopted to our notation.
admits a unique strong solution  ∈  ,  (  ) depending continuously on the input data ,  and  .
Finally, we need the following regularity result for  = with ux boundary conditions.
The proof mainly uses standard methods but since, to the best of the authors' knowledge, a proof matching our boundary conditions is not available in the literature, we included it into Appendix A.
Next we de ne our notion of solution for the ODE ( .c).
We have the following result about the existence of a solution of ( .).
Lemma . .For given <  < ,  ∈  ∞ ( , ; R ) satisfying assumption (C ) and  ∈  ,  (  ), there exists a unique, absolutely continuous solution  : (  ), the corresponding solutions  and  satisfy where the constant   depends on  ,  E ,  E,∞ and the Lipschitz constants of  and .
Proof.First note that  ∈   ( , ; , (Ω)) ↩→   ( , ;  , (Ω)), for some  > , so that the applica- where   ,E, ()      ∞   () and with  ∞ the embedding constant for  , (Ω) ↩→  , (Ω).The estimate ( . ) implies uniqueness by Ch.I, Thm. . in Hale, . The additional regularity  ∈  ,∞ ( , ; R ) is a consequence of the boundedness in  ∞ ( , ; R ) of the right-hand side of ( .).To establish the stability estimate we show where we used that the extension E is also continuous with respect to the  ∞ -norm.An application of Gronwall's inequality in integral form then yields, for  ∈ ( , ), --cbna page of Next, we state an existence and stability result for a regularized version of ( .).Note that the following result requires less regularity for the density function .
Lemma . .Fix <  < ∞ and  ∈  ( [ , ];  (Ω)).Then for given  ∈  ∞ ( , ),  satisfying assumption (A ) and (C ), and every  > , there exists, a unique, absolutely continuous solution for a.a. ∈ [ , ] and all ,  ∈ Ω.This implies uniqueness by Ch.I, Thm. . in Hale, .To show the stability estimate we apply Lemma .with  and  replaced by   *  and   *  , respectively, which yields with   de ned as in the proof of Lemma . .Applying Young's inequality for convolutions to the norm on the right-hand side allows us to estimate which completes the proof.
We continue with the following Lipschitz estimate for the transport term in ( .a), which is needed in the theorem right after the next.
where   depends on the Lipschitz and boundedness constants of  , ℎ and . .

. E F F S
We are now in a position to show the following existence and uniqueness result.
The structure of ( .a)-( .b) is very similar to chemotaxis models with volume lling, see for instance Painter, Hillen, , except for the additional nonlinearity of the squared gradient term in ( .b), which can, however, be handled using Lemma . .Therefore, the existence and uniqueness of solutions can be proved using Banach's xed point theorem, similar to, e. g., Thm. . in Egger, Pietschmann, Schlottbom, . The main issue in our situation is the additional coupling to the system of ODEs ( .c), which requires  to be regular enough to allow point evaluations.Our strategy is to introduce an additional regularization in ( .c) in order to be able to perform the xed point argument in the relatively "large" space  ( [ , ];  (Ω)).We then show additional regularity and pass to the limit to recover the original system.
Proof.The proof consists of two parts.First we show existence with ( .c) replaced by the regularized version ( .).Then, we perform the limit  → to recover the original problem.
Step : Fixed point argument: We consider the xed-point operator where, for xed  > ,   is the unique weak solution to the system ) The boundary conditions for  and the initial conditions for all variables are as in ( . ) and ( .), ( .), respectively.The equation for   is understood in the weak sense, i. e., shows that there exists a unique weak solution Choosing  itself as a test function in the weak formulation ( . ) yields, after an application of the weighted Young's inequality with parameter / , the trace inequality for  -functions, and an integration in time, the a priori estimate , where now, due to the Gronwall argument in Lemma .,  depends on  but is monotonically decreasing.Thus, we can again nd  small enough so that  is a contraction and Banach's xed point theorem asserts the existence of a unique solution.The box constraints ≤   ≤ a.e. in   follow by applying Lem. in Egger, Pietschmann, Schlottbom, , i. e., by testing with smoothed versions of the positive part of  − and −, respectively.In view of these uniform estimates a standard continuation argument yields existence for arbitrary  > .

So far we have shown
since the xed-point satis es the weak formulation ( .).Note in particular that   is bounded in  ( , ;  (Ω)) ∩  ( , ;  (Ω) * ) ∩  ∞ (  ) by a constant independent of .Furthermore, we denote the corresponding potential and agent trajectories by --cbna page of Step : Additional regularity: To shorten the notation we write the nonlinear term in the form with (  )    (  ) and Φ  ∇(  +   (  ; •)).From the product rule we obtain the following representation for the divergence, Freezing the nonlinear terms allows us to understand ( . ) as a linear equation of the form and with  = − /.It remains to show the regularity claimed for ,  and  .Again, this follows from the Hölder inequality and the regularity already shown for   ,   ,   .Together with the product and the chain rule this leads to ) Note that   and   are independent of  as, in particular,   can be bounded independently of   and thus of .To show the required regularity for  we proceed as follows.First, we show the estimates for  = , , as well as This follows from the regularity already shown for   ,   (see Lemma .) and   (see Lemma . ).
With these considerations we conclude --cbna page of This allows us to apply the trace Lemma ., which provides Collecting the properties ( .)-( .) an application of Thm. . in Denk, Hieber, Prüss, and Lemma .implies We can even further improve the regularity of   .Analogous to ( . ) and ( . ) we show and together with the trace Lemma .we deduce This, ( . ) and ( . ) allow a further application of Thm. . in Denk, Hieber, Prüss, with  ≤ , from which we infer the desired regularity The functions   ,   and   are thus a strong solution of the system that satisfy the boundary and initial conditions pointwise almost everywhere.
Step : Limit  → : In order to recover a solution to ( .), it remains to pass to the limit  → in ( .)-( .).As a rst step in this direction, note that  (, ),  (, ) and  (, ) de ned in ( . ) and ( . ) are bounded independently of .Thus, understanding ( . ) as the linear equation ( . ) we obtain the following estimate, uniformly in , This implies the existence of a sequence   with   → that where the second convergence is a consequence of the Aubin-Lions lemma, Thm.II. . in Boyer, Fabrie, .As, moreover,   is also uniformly bounded in   ( , ; , (Ω)) by   , we also have where the second convergence follows from Lemma A. .We omit the index  in the following to shorten the notation.Passing to the limit  → , in the sense of distributions, in equation ( .b) yields the validity of this equation also for the limit values  and . .
( . ) Next, we pass to the limit in ( .).The convergence of the linear terms is a direct consequence of ( .).For the convection term we note that it can be written as As both   and ℎ are uniformly bounded, the convergences ( .), ( . ) and ( . ) imply It remains to pass to the limit in the boundary conditions.The trace Lemma .implies that  ∇  •  is bounded in   (Σ  ) and thus converges weakly.The compact embedding so that using the uniform boundedness of both   and ℎ as well as ( . ) we conclude Thus, the weak limit of   , namely , is the strong solution of ( . ) with boundary conditions ( .).This completes the proof.
The previous theorem allows us to introduce the solution operator of ( . ) with boundary conditions ( . ) and initial conditions  ( ) =  and  (•, ) =  .The control and state spaces are with / = / + /.Later, the operator  is referred to as control-to-state operator.
We conclude this section with an auxiliary result required later to show the existence of global solutions to an optimal control problem.
Proof.Given a weakly convergent sequence (  )  ∈N ⊂ U with   ⇀  one has to show that the corresponding states (  ,   ,   ) =  (  ) converge weakly in Y to (, , ) =  ().This follows from the same arguments as those in step of the proof of Theorem ., together with the uniqueness of solutions. .

T L S
In order to prove necessary optimality conditions for the optimal control problems introduced later, we investigate the di erentiability of the control-to-state operator .The desired results follow from the implicit function theorem applied to the equation  (, ) = with  =  () = (, , ).Here,  corresponds to the strong formulation of our forward system ( .a)-( .c), more precisely, there holds Notice that from here on we write  () in place of  (, , ).Recall that  ∈ ( , ) is a xed number and the integrability index  of the space of agent trajectories is chosen such that  = +  .
To shorten the notation we also introduce the following constants, whose boundedness is guaranteed due to assumptions (A ), (K ), (K ) and (C ).
We con rm the assumptions of the implicit function theorem in the following lemmas.
In the following we derive bounds for the remainder terms (the terms depending nonlinearly on ,  and  in the above equations), which we denote by  , ( ),  , ( ) and  ,  ( ),  = , . . ., .
First, we apply the product rule, the Hölder inequality and the embedding  ,  (  ) ↩→  ∞ (  ), where we denote by  ∞ the maximum of the embedding constant and .These arguments yield ( . ) Second, to show di erentiability with respect to  we con rm With the Hölder inequality, the Lipschitz properties of ℎ and  ℎ, in particular ( .), and the usual embeddings we can show Third, we derive an estimate for  ,  ( ) with  = ( , . . ., ,   , , . . ., ).We use the notation   =   (; •) and   =   ( +  ; •) as well as   =  (• −   ) and   =  (• −   −    ), reformulate the remainder term by applying the product and chain rule and obtain Here, |•| is an arbitrary vector, matrix or tensor norm, depending on the argument.With the Hölder inequality, the regularity of (, , ), in particular  ∈  ,  (  ) ↩→  ∞ ( , ; , (Ω)) ∩   ( , ; ,∞ (Ω)), and the Lipschitz properties of ℎ and  we deduce The di erentiability of  can be shown with similar arguments.The Taylor formula yields and it remains to estimate the remainder terms, i. e., the last terms on the right-hand sides of the previous equations.We abbreviate these terms by  , ( ) • ,  , ( ) •  and  , ( ) • , respectively.
First, we show  , ( ) . We wish to apply the trace Lemma ., for which we have to show the required time and space regularity of the extension onto   .First, note that there holds which can be concluded from the same arguments as in ( .).Moreover, for the time derivative we show ( . ) With the estimates ( . ) and ( .), the embedding and the trace Lemma .we deduce which implies the di erentiability of  with respect to .
To show an estimate for  , ( ) •  we proceed in a similar fashion.With analogous arguments we deduce the estimates --cbna page of and exploit again ( . ) to arrive at which con rms the di erentiability of  w.r.t..
An analogous procedure is used to deduce an estimate for the remainder term  , ( ) • .For each direction  = ( , . . .,   , , . . . ) ᵀ ,  = , . . ., , a direct calculation taking into account Lemma .and the Lipschitz continuity of the derivatives of  and ℎ yields Using again ( . ) and the trace Lemma .leads to which proves the di erentiability of  w.r.t..
The component  is trivially di erentiable.For  we show Again, we denote the remainder terms (the terms which are nonlinear in  and ) by  , ( ) and  , ( ).We show and both estimates together imply With similar arguments we can show the di erentiability of  with respect to .
Using the Lipschitz continuity of  , assumption (C ) and With similar arguments and the Hölder inequality with / > / we deduce for the temporal derivative of  , For the remainder term  , ( ) with  = ( , . . .,   , , . . . ) ᵀ we show Using Lipschitz estimates for  and the mean value theorem we conclude In the above estimates we used   , (R ) ≤   , (Ω) , the continuity of E :  , (Ω) →  , (R ), see Lemma ., and the embedding  , (Ω) ↩→  , (Ω), which is valid for  ∈ ( , / ] due to  > . The insertion of ( . ) and ( . ) into ( . ) yields The above also uses an application of the Hölder inequality in time and / < .
Finally, the Hölder inequality in time with / = / + / yields the estimate where the last step follows again from ( . ) and ( .).This con rms the partial di erentiability of  .
Collecting the previous estimates proves the partial Fréchet di erentiability of the operator .Based on the properties of  , ℎ and   using again Lipschitz continuity and boundedness, the operators   ,    and     depend continuously on (, , ), which implies the continuous Fréchet di erentiability.
Next, we show that the operator    (, ) is invertible.
Proof.The strategy of the proof is to apply Banach's xed point theorem to the linear system ( . ) to avoid technicalities that arise from the fact that  () depends on  non-locally in time.To this end, we introduce three solution operators.
First, there is   : Y →  ,  (  ) which maps  = ( , , ) to  ∈  ,  (  ), which is de ned as the solution to ( . ) Second, we de ne the operator together with the boundary conditions ( . ) (these are incorporated in the function space).
The idea of the proof is to apply Banach's xed point theorem in the space  ,  (  ) to the operator As the operator  is a ne, it su ces to show the boundedness of the linear part of  by a constant smaller than , which will imply that  ( ) is a contraction.
( . ) In the second estimate  may depend on  , via norms of   ,   , etc.. Since this implies that  is decreasing when  is decreasing, this does not a ect the contraction argument.
As  is an a ne linear operator we directly conclude the contraction property provided that  is su ciently small.Thus, Banach's xed-point theorem provides the existence of a unique solution  ∈  ,  (  ) which can be extended to an arbitrary time-scale with a concatenation argument.
The previous two lemmas, together with the implicit function theorem, imply the di erentiability of the control-to-state operator.The derivative in a direction  ∈  ∞ ( , ; R )  can be computed by means of   (, )  = −  (, ) .
We summarize the nal result in the following theorem.

. O C P
In this section we come back to the optimal control problem outlined in Section . .We consider objective functionals of the form where  > is a regularization parameter and the functional Ψ : Y → R ful lls the following assumption: (J ) The functional Ψ : Y → R is weakly lower semi-continuous and bounded from below on {(, , ) ∈ Y :  ≥ a. e. in   }.
The assumed weak lower semi-continuity is ful lled, e. g., when Ψ is convex and continuous.As the density part  of each solution of the forward model ( . ) is non-negative it su ces to assume boundedness of Ψ on a subset only.
We consider the following optimal control problem:  Using (A. ), we can estimate the second term on the right-hand side which yields the desired regularity for   .We also infer the existence of a weakly converging subsequence in  ( , ;  (Ω)) ∩  ∞ ( , ;  (Ω)) which, by the weak lower semi-continuity of the norms, yields the bound also for the limit .Reinserting this into (A. ) yields the assertion.
We also state the de nition of Carathéodory conditions.
De nition A. .We say that a function  :   → R satis es the Carathéodory conditions whenever • for each xed  ∈ R  , the function  ↦ →  (, ) is measurable, • for a.a. ∈ [ , ], the function  ↦ →  (, ) is continuous, • there exists an integrable function  s. t.Since ≤  holds, this is the desired estimate.Thus, Lemma on p. in Simon, ensures the existence of a subsequence that strongly converges in   ( , ; , (Ω)).By inverting the transformation ( .), the result for   is obtained.

(
Σ  ) de ned by   =  | Σ  and   = ∇ •  Σ  are bounded and have a continuous right inverse.