Guillaume Cortes
email: guillaume.cortes@inria.fr

Ferran Argelaguet
email: ferran.argelaguet@inria.fr

Eric Marchand
email: marchand@irisa.fr

Anatole Lécuyer
email: anatole.lecuyer@inria.fr

Virtual Shadows for Real Humans in a CAVE: Influence on Virtual Embodiment and 3D Interaction

Keywords: Human-centered computing → User studies, Virtual reality, virtual reality, shadows, embodiment, 3D interaction

In immersive projection systems (IPS), the presence of the user's real body limits the possibility to elicit a virtual body ownership illusion. But, is it still possible to embody someone else in an IPS even though the users are aware of their real body? In order to study this question, we propose to consider using a virtual shadow in the IPS, which can be similar or different from the real user's morphology. We have conducted an experiment (N=27) to study the users' sense of embodiment whenever a virtual shadow was or was not present. Participants had to perform a 3D positioning task in which accuracy was the main requirement. The results showed that users widely accepted their virtual shadow (agency and ownership) and felt more comfortable when interacting with it (compare to no virtual shadow). Yet, due to the awareness of their real body, the users have less acceptance of the virtual shadow whenever the shadow gender differs from their own. Furthermore, the results showed that virtual shadows increase the users' spatial perception of the virtual environment by decreasing the inter-penetrations between the user and the virtual objects. Taken together, our results promote the use of dynamic and realistic virtual shadows in IPS and pave the way for further studies on "virtual shadow ownership" illusion.

mixed realities research have focused in the simulation of virtual shadows in order to reach a more accurate representation of the reality [START_REF] Thompson | Visual perception from a computer graphics perspective[END_REF]. One particular shadow is the one cast by our own body which in combination with a virtual avatar can reinforce the user's virtual experience [START_REF] Slater | The influence of dynamic shadows on presence in immersive virtual environments[END_REF] and theoretically it can also enhance spatial perception. However, the user's avatar representation is way harder to introduce in Immersive Projection Systems (IPS) such as CAVE displays [START_REF] Cruz-Neira | Surround-screen projection-based virtual reality: the design and implementation of the CAVE[END_REF]. Indeed in projective displays the users are equipped with stereoscopic glasses and they are able to see their own body. Such configuration makes it harder for the user to embody someone else since they are always aware of their real body.

In this paper we propose to study the influence of the user's virtual and dynamic shadow in a VR CAVE display. Is it possible to embody someone else in a CAVE environment even though the users are aware of their own body? Does the virtual shadow influence their interaction behavior? In order to answer these questions we carried out an experiment in order to assess the user's virtual embodiment and the user's 3D performance in presence of a virtual shadow. The virtual shadows enabled to provide a virtual representation of the user which differed from their own physical body even though they were still able to see their body. In particular, we studied how the users appropriate different virtual shadows (male and female shadow) and how does the virtual shadow affects the user behavior when performing a 3D positioning task. The results showed that the shadow can have an influence on the user's behavior while interacting, and that participants seemed to prefer virtual shadows which were closer to their own body.

In the remainder of this paper we first make an overview of previous work that has been done on adding user shadows in VR environment. Second we describe the experiment that aims to study the influence of virtual shadows on the user comfort and presence and on the environment understanding. Third we present the results of the experiment. The paper ends with a discussion.

RELATED WORK

Increasing the embodiment of the users in virtual reality applications has been studied in a number of different works. Although the majority of works have been focusing on the user's avatar [START_REF] Kilteni | The Sense of Embodiment in Virtual Reality[END_REF], several works have also addressed the use of virtual shadows to reinforce the effect. For example, [START_REF] Slater | The influence of dynamic shadows on presence in immersive virtual environments[END_REF] carried out a study on the influence of the presence of shadows in HMD virtual environment. They did not find any influence of the shadows on depth perception but they found that adding a static shadow increases the user presence and that adding a dynamic shadow increases it even more. This study focused on the object shadows and no user dynamic shadow was considered. In later work they introduced the dynamic shadow of the user in HMD to confirm that the realism of the scene has an impact on the user behavior [START_REF] Slater | Visual realism enhances realistic response in an immersive virtual environment[END_REF]]. Even though users virtual shadows have not been widely studied, avatars are commonly used in virtual reality HMDs. There even were studies on the influence of the morphology of the virtual avatar on the user behavior and embodiment in HMD [START_REF] Peck | The Effect of Gender Body-Swap Illusions on Working Memory and Stereotype Threat[END_REF]]. Nevertheless they did not carry out any study on immersive projection systems.

Indeed using avatars in IPS or screen displays can be harder since the users are aware of their own body. When using such systems, casting shadows can still be a solution to enhance the user experience. Even though, in 1995, Slater et al. did not find any influence of the shadows on depth cue, later studies [START_REF] Hubona | The relative contributions of stereo, lighting, and background scenes in promoting 3D depth visualization[END_REF] found out that when using screen displays adding objects shadows increases the accuracy during positioning tasks. However the study focused on the object shadows and no user shadow was considered. Moreover the study was carried out on screen displays that do not provide any immersion. Such results were confirmed in later studies carried out on augmented reality displays [START_REF] Diaz | Designing for Depth Perceptions in Augmented Reality[END_REF][START_REF] Sugano | The effects of shadow representation of virtual objects in augmented reality[END_REF]]. Adding shadows to the virtual objects integrated in the real world increases the objects presence and provides depth cues that increase the spatial perception. [START_REF] Hu | Visual cues for imminent object contact in realistic virtual environment[END_REF]] also confirmed the results in VR HMD. Regarding the user shadow, altering the shadow behavior compared to the user's body behavior can modify the user perception of the environment. [START_REF] Ban | Modifying a body perception and an actual motion by visual stimuli of distorted physical exercise using shadow metaphor[END_REF]] carried out a study where the shadow was more or less independent from the user. The shadow was then able to move differently from the user movement. The users were then confused and were not always able to tell if they or their shadow were moving. Moreover their movement were altered by the shadow movements. Such study was carried out by projecting shadows on a wall but no virtual reality environment nor interaction were considered.

[ [START_REF] Steinicke | Virtual reflections and virtual shadows in mixed reality environments[END_REF] were the first ones to introduce user shadows on IPS. They added the presence of users' virtual shadows and reflections on a responsive workbench. The real reflection of the user (captured with a camera) was added on a metallic surface and a virtual shadow of the user's hand was cast on the same surface. The authors claim that it increases the realism of virtual objects but no study was made to evaluate how this approach increases users perception and improves objects interaction. More recent work from [START_REF] Yu | Visual realism enhances realistic response in an immersive virtual environment-part 2[END_REF] proposed to increase the realism of virtual environments in CAVE displays. They confirmed that the user presence was increased when having shadows and reflections that corresponded to the users' body movements. The body movements were forced by the application and the task consisted in naturally walking in the display to avoid a collision with a virtual character. Nevertheless no complex 3D interaction task was proposed to study the influence on depth perception. Later work from [START_REF] Kwon | The Use of Shadows on Real Floor as a Depth Correction of Stereoscopically Visualized Virtual Objects[END_REF]] enhanced wall-sized VR application by adding objects shadows on the real floor in front of the display. They carried out a study were users had to touch the virtual objects with a direct touch metaphor. Their results suggested that the shadow cue is even more important than the stereoscopic cue. Regarding the shadow of the user, no study was considered.

According to the previous work, and to the best of authors knowledge adding user shadows in a CAVE while performing complex 3D interaction task has not been studied yet. Thus we propose to study, for the first time, the influence of the presence and the morphology of the user's virtual shadow on the virtual embodiment and on a complex 3D interaction task in a CAVE display.

EXPERIMENT: INFLUENCE OF THE VIRTUAL SHADOWS ON VIRTUAL EMBODIMENT AND 3D INTERACTION

In this experiment, our goal is to study the influence of a virtual shadow on the presence, the embodiment and the precision of the participants when performing a 3D interaction in an IPS, a CAVE display. We designed a 3D positioning task in which participants had to place a physical ball over virtual targets placed on planar surfaces, such as tables or walls. The goal of the positioning task was to place the ball as close as possible to the target areas without going through them. During the experiment the participants were presented with three virtual shadow conditions: A male shadow, a female shadow and no shadow. We hypothesized that the ability to see their virtual shadow can influence both the user's perception on their shadow but also on the way they perceive and interact in the virtual environment, thus, two main research questions were addressed:

• Q1: Is it possible to embody someone else in an IPS even though the users are aware of their own body? • Q2: Does a user's virtual shadow increases the user's spatial perception of the virtual environment in an IPS?

Apparatus

Participants were immersed in the virtual environment using a 9.6×2.9×3.1 meters CAVE display. The CAVE system is built with 4 screens: one on the floor, one on the front and one on each side. The projection on the screens is made using Barco F90-4K13 laser projectors. Every screen but the floor is back-projected. The tracking data is provided by an infrared optical tracking system from Optitrack1 . The optical tracking is composed of 12 cameras (4 Optitrack Prime 13W and 8 Optitrack Prime 13).

The dynamic virtual shadow is created from a virtual 3D model of a humanoid. For the purpose of the experiment we chose to have a male and a female 3D model (see Figure 2). Such models have been chosen to fit the human proportions and to relieve the experiment from a cumbersome calibration process. Thus the matching between both the virtual and the real body are simply made by scaling the model so that its height corresponds to the participant's height. Regarding the dynamic part of the shadow, an inverse kinematics (IK) algorithm was used to provide movement to the shadow and to make its position correspond to the user's one. The Final IK2 Unity asset was used to optimize the position of the rigged models to fit the actual position of the user. The Final IK algorithms were able to converge to an optimized solution by providing the position of both feet, both hands and the head of the user. Thus, participants were equipped with tracking devices on the feet, hands and head (see Figure 3-left). Finally due to experimental testing we noticed that providing the orientation of the pelvis to the IK algorithm gave better visual performances. Thus the user was also equipped with an additional tracking marker in the waist to track the pelvis. To simplify the experiment and the recorded data, a physical ball was placed on the participants right hand as an extension of their arm and body. A controller was hold on their left hand (see Figure 3-right). The 3D model that matches the user position is introduced in the virtual environment but no rendering of the model's mesh was done, only the cast shadow is displayed.

Since the ball was not perfectly rigidly attached to its tracking constellation and that the participants were able to grab it as they wanted to, a calibration step was introduced during the experiment. The calibration step aimed to compute the distance between the physical ball and the virtual ball (used to cast the ball shadow). The participants were asked to place 3 times the physical ball over a physical table whose height was perfectly known (see Figure 4). Then the offset between the physical and virtual balls was computed and used to correct the recorded data. The calibration step was performed at the beginning and at the end of each experimental condition.

Regarding the lighting of the scene we chose to use 3 directional lights to provide the participants with 3 virtual shadows of themselves. Two lights were oriented of around 45 • according to the wall normal vector and one light was almost collinear with the wall normal. Such lighting configurations provided the scene with two shadows cast half on the floor and half on the wall and one shadow that is almost only cast on the wall (see Figure 5). The VR application was developed using Unity 5.6.1f and ran at an average frame rate of 60 fps.

Participants

27 participants from the university campus took part in the experiment (Age: min = 23, max = 55, and avg = 31 ± 8), recruited both among general students and staff. For the purpose of the experiment we chose to recruit 14 women and 13 men since the participant were confronted to a male and a female shadow. Participants were recruited asking for minimal previous experience in VR: 18 subjects had none to very limited previous experience with virtual reality, 7 had some previous experience, and only 2 were familiar with VR. None of the participants knew about the experiment being tested, or that they would be presented with virtual shadows. All participants were right-handed since the ball was placed in their right hand for the positioning task.

Experimental Task

Participants were asked to position a physical ball (7cm radius, see Figure 3-right) over circular green target areas (2.5cm radius). The target areas were positioned over 2 virtual tables and a virtual Figure 5: Virtual environment of the experiment. Users were asked to place the physical ball over the green target areas that were displayed on both tables and on the wall. Three different point lights generated three different virtual shadows at the same time.

wall (Figure 5). Only one circular target was displayed at a time in order to help the user know which target he/she had to focus on. Every run of the experiment started with a target on the left table, then a target on the wall and then a target on the right table and so on. Once participants were satisfied with the placement of the target, they had to validate the positioning of the ball by pressing the trigger of the controller placed on their left hand (see Figure 3-right). In order to reduce the required time to perform a timer was added. The timer was depicted by changing the color of the current target area. The target areas appeared green, they went orange after 3 seconds and red after 6. Participants were asked to try and validate the positioning of the ball before the target became red. Nevertheless they were told to be as accurate as possible even if they had to spend more time for each target.

In order to decrease learning effects, participants had to face a different number of target configurations. Three target positions were defined for each table and for the wall. Moreover the tables were positioned at a variable height (90cm or 110 cm).

Experimental Protocol

An informed consent form was signed by each participant before starting the experiment. The form stated the participants' right to withdraw and presented the experiment and the main goal of the research. In addition, it also asked their consent regarding image and video copyright. In order to minimize the priming of participants, little details were provided regarding the purpose of the shadow. Mainly, participants were told that the experiment aimed to assess people precision when performing 3D positioning tasks. They were also told that the virtual shadow conditions of the scene could vary but no additional details about the shadows were given.

The experiment was divided into 3 blocks. During each block the participants were presented with one virtual shadow condition which was either No shadow (N), Male shadow (M) or Female shadow (F) (see Figure 6). Each participant performed entirely the positioning task for each one of the 3 conditions. Considering all possible target combinations, three targets (wall, left table, right table), two heights and three positions, participants had to perform the placement task 18 times for each condition. Moreover, as three repetitions were considered, each block resulted in 54 trials. Finally, a training period of half a run (9 targets) was present at the beginning of each block. To counterbalance the influence of the running order of the conditions on the participants behavior, the participants were divided into 6 groups (M/F/N, M/N/F, F/M/N, F/N/M, N/M/F and N/F/M). Each group was composed of at least 2 male and 2 female participants.

After each block participants were asked to fill in a subjective questionnaire (see Table 1) in order to evaluate their subjective appreciation of the experiment and collect their feedback. The questionnaires began with 5 question to evaluate the presence of the user following the suggestions of Usoh et al. [START_REF] Usoh | Using presence questionnaires in reality[END_REF]]. Then there were questions regarding the ownership and some questions about the task and the user comfort. Finally the participants were free to comment their strategy to perform the task and to detail their feelings and comments in presence or absence of a shadow.

While performing the positioning task the participants were immersed in the CAVE. After each condition the tracking constellations were removed and the users had to fill in the questionnaire on an independent laptop. Then they were reequipped with the tracking constellation and reintroduced in the CAVE. The whole experiment including the questionnaires lasted around 45 min in total (15 min per virtual shadow condition).

In addition to the subjective questionnaires assessments the following information was recorded for each positioning task:

• The depth error (Y axis for the tables and Z axis for the wall) between the ball and the target area when the position is validated by the participant. The error is positive when the ball is positioned over the surface and negative when its position inside the surface. • The time the participant took to perform one positioning task. As the trials were performed sequentially, the task completion time matches the time between two validations. There were times during the experience when the living room space was the reality for me P 3

The living room space seems to me to be more like images that I saw or somewhere that I visited P 4 I had a stronger sense of being elsewhere or being in the living room space P 5

During the experience I often thought that I was really standing in the living room space S A 1 When positioning the ball on the tables I felt that the virtual shadow was useful S A 2 When positioning the ball on the wall I felt that the virtual shadow was useful S A 3 I felt that the virtual shadow was a good indicator of the proximity of the ball with the tables S A 4 I felt that the virtual shadow was a good indicator of the proximity of the ball with the wall T A 1 I felt that I was accurate on positioning the ball T A 2 I felt that the positioning of the ball was rather easy O 1 I felt as if the virtual shadow was my own shadow O 2 I felt as if the virtual shadow was from someone else's

A 1 I felt as if the virtual shadow moved just like I wanted A 2 I expected the virtual shadow to react in the same way as my own body A 3 I felt like I controlled the virtual shadow as if it was my own shadow

Results

During the analysis we explored the effect of the participants' gender on the results. If the gender did not significantly influenced the results, data was pooled. Regarding the ANOVA analysis, effect sizes are expressed using the partial eta squared (η 2 p ). The general rules of thumb given by [START_REF] Miles | Applying regression and correlation: A guide for students and researchers[END_REF] state that the qualifiers "small", "medium" and "large" correspond to cases where η 2 p > 0.01, η 2 p > 0.06 and η 2 p > 0.14 respectively. Only significant effects are discussed. We first discuss performance measurements and then the subjective appreciations of participants.

Performance Measurements.

The main indicators of task performance were the final depth position (see Figure 7) and the task completion time. We first analyzed the effect of the Shadow type and the Task on the depth error using a two-way ANOVA analysis considering participants as a random factor. The ANOVA analysis showed a main significant effect for Shadow [F 2, 52 = 8.99, p < 0.001, η 2 p = 0.25] and Task [F 2, 52 = 39.67, p < 0.001, η 2 p = 0.60], no interaction effect was found [F 4, 104 = 0.26, p = 0.9]. Tukey post-hoc tests showed that for the male shadow condition participants were more conservative while performing the task M = 3.2cm; S D = 3cm compared to the female shadow M = 1.4cm; S D = 3.3cm and without shadow M = 0.9cm; S D = 3.7cm. In general, in the condition without shadows participants were more prone to go through the target surface. On the contrary when a virtual shadow was present the user tend to stop their movement before going through the target surface. Regarding the Task, post-hoc tests show that participants were able to place the ball closer to the target for the Left table M = 0.8cm; S D = 2.3cm and the Right table M = 0.0; S D = 2.8cm compared to the Wall M = 3.7cm; S D = 3.6cm. The higher depth error for the wall condition can be explained by the fact that the wall was a flat homogeneous surface which did not provide enough depth cues.

Task

Shadow

Right Table Front Regarding the task completion time (see Figure 8), the ANOVA analysis showed a main effect on the Task [F 2, 52 = 68.24, p < 0.001, η 2 p = 0.72] and an interaction effect [F 4, 104 = 4.13, p < 0.01, η 2 p = 0.13],

Wall Left Table N M F N M F N M
there was no effect on Shadow [F 2, 52 = 0.72, p = 0.49]. Tukey posthoc tests showed that participants required significantly more time to perform the task in the Left table condition M = 2.65s; S D = 0.51s compared to the Right table condition M = 2.28s; S D = 0.49s and the Wall condition M = 2.32s; S D = 0.48s. This result can be explained by the fact that all participants were right handed and required more time to access the left table. Post-hoc tests were not conclusive for the interaction effect. 3.5.2 User Experience Questionnaires. The different questions of the subjective questionnaires have been classed into several categories: Shadow Appreciation (SA), Task Appreciation (T A), Agency (A), Ownership (O) and Presence (P). Table 1 gathers all the questions for the different categories. In the following, the statistical analysis of each questionnaire category is presented: Shadow Appreciation. In general participants considered that the shadow helped them to perform the task (SA 1 : M = 5; I Q R = 2, SA 2 : M = 5; I Q R = 2) and also that it was a good indicator of the proximity of the ball with respect to the targets (SA 3 : M = 5; I Q R = 2, SA 4 : M = 5.5; I Q R = 2) (Figure 9). Wilcoxon signed rank test showed that the male shadow was perceived to provide significantly better assistance when performing the table task (SA 3 , p < 0.05). A similar trend was observed for the wall task, but results were not significant (SA 4 , p = 0.09) Agency. In overall, participants felt that the virtual shadow moved (A 1 : M = 6; I Q R = 2), reacted (A 2 : M = 6; I Q R = 2) and that they could control it (A 3 : M = 6; I Q R = 1) as if it was their own shadow (Figure 9). Wilcoxon signed rank tests did not show any significant differences between the male and female shadows.

Task

Shadow

Right Table Front Wall

Question Shadow A A A SA SA SA SA M F M F M F M F M F M F M F 7 6 5 4 3 2 1 7-Likert Scale 1 2 3 4 1 2 3
Figure 9: Boxplot summarizing ratings for the shadow appreciation (SA) and agency (A) questionnaires. In general, participants appreciated the fact of having a shadow and felt a strong sense of agency.

Task Appreciation. Friedman rank sum test was used to analyze how participants perceived their accuracy while performing the task (T A 1 ) and the perceived difficulty (T A 2 ) considering each level of Shadow (Figure 10). The Friedman analysis of T A 1 showed that the virtual shadow condition had a significant effect [χ 2 (2) = 10.89; p > 0.01]. Pairwise Wilcoxon tests showed that the condition without shadows was perceived as less accurate (both p < 0.05). Similarly, the analysis of T A 2 also showed a significant effect on Shadow [χ 2 (2) = 12.11; p > 0.01]. Again, pairwise Wilcoxon tests showed that the condition without shadows was perceived to be more difficult (both p < 0.05).

Presence. In general, participants experienced a moderate sense of presence. Although participants seemed to rate lower some of the questions for the condition without shadow (see Figure 10), the analysis of the presence questionnaire results did not show any significant on the Virtual shadow condition.

Ownership. In the results related to the feeling of ownership (O 1 and O 2 ), we observed an interaction effect due to the gender of the participant. In order to analyze the results we considered whether the shadow was consistent with the gender of the participant or not (see Figure 11). A Wilcoxon signed rank test showed that participants had a higher feeling of ownership (O 1 ) when the shadow gender was consistent with theirs (p < 0.05). The control question (O 2 ) showed the same result (p < 0.05). Figure 11: Boxplot of the ownership ratings regarding the shadow consistency in terms of the gender. Ownership ratings were significantly higher when the shadow gender was consistent with the participant's gender.

O O

Yes

DISCUSSION

Taken together the results should help answering Q1 and Q2 to give a lead on, respectively, if virtual shadows can provide a sense of virtual embodiment in IPS and if the virtual shadows increase the spatial perception of the users. In the following we discuss the influence of the virtual shadow on both issues.

Virtual Shadows and Virtual Embodiment

One of the major research questions in this paper was: Is it possible to embody someone else in an IPS? The questions from categories A and O are taken into account to discuss the virtual embodiment of the users in presence of the virtual shadows. In terms of agency, subjective ratings showed that participants had a strong feeling of agency towards their virtual shadow. Participants had the feeling that the virtual shadow was moving in a natural way and that it correspond to their shadow position. Some users did not even noticed the shadow at first since it is natural for them to have one:"I did not payed attention to the shadow: it was natural I guess.". This effect was not dependent on the morphology of the shadow. Nevertheless, in an IPS users are always aware of their own body and this limitation is reflected on what kind of body the users are able to appropriate. Indeed the ownership measurements depict that when the morphology of the virtual shadow (or the gender) is not consistent with the user's morphology the user tends to feel that the shadow is from someone else. However, as long as the virtual shadow morphology is close enough to the users' one they feel that the virtual shadow is their own. During the experiment some users reacted to the fact that the shadow was of the opposite gender or that the shadow did not had the same hair style for example: "Is the bun hair style made on purpose ?", "Oh. I'm a male now.", "I clearly have a female outline", "I've got muscles !", "Do I wear a bun ?". Finally, the results in the literature have shown that virtual shadows can increase the sense of presence. Our results show that there is a trend to rate lower the presence question in absence of the shadow which is consistent with the previous work. Some users even commented the enhanced realism of the scene in presence of the virtual shadow: "The room is not realistic because of the absence of the shadows.", "I felt like the experiment was less realistic without the shadow.", "I felt it is more realistic with shadows than no shadow at all.". To sum up, adding dynamic virtual shadows in IPS, such as CAVE displays, can enable the user to embody a virtual shadow. Nevertheless, in order to achieve a higher degree of ownership, the virtual shadow should be close enough to the users body since they are always aware of it.

Virtual Shadows and Spatial Perception

On the other hand, does the presence of a virtual shadow increase the spatial perception of the users? The performance results, the SA and T A assessments are taken into account to answer it. If we consider SA ratings, participants felt that the virtual shadow could be a good indicator of proximity from the targets and that the shadow was a good assistant to position the ball: "I mainly used the shadow to position the ball over the targets.", "The shadow has been useful even if it was not mine.", "The task is more complicated without the presence of a shadow, the distances were harder to estimate.". The participants had an overall feeling of better perceiving and understanding the virtual environment physical limitations. Moreover, although performance measurements did not show any significant results in terms of task completion time, participants found it easier to perform the task with the presence of a virtual shadow. Some of them even commented it in the questionnaires: "The shadow was helpful for the ball "It is helpful to have the shadow to place the ball, particularly on the wall.". Finally, the analysis of the depth error showed that participants had a more conservative behavior when placing the ball on the targets in the presence of virtual shadow. Indeed, the presence of the virtual shadow can warn the users that they are approaching a rigid object and that they may not be able to go through it, as if the object was physically there. An interesting result was the fact that the participants were less accurate when the target was placed in the wall. The most plausible explanation is that it is harder to estimate the actual position of the target when it is placed on the wall, but the actual reasons remain unknown. In summary, these results show that the presence of virtual shadows provides an increased awareness of the spatial relations between the users and the virtual environment (less inter-penetrations) and are positively perceived by the users.

Limitations and Future Work

In this paper we considered arbitrary human 3D models to generate the virtual shadow of the users. Nevertheless some users may have not been morphologically identified to either the female or the male shadow. It might then be of interest to test our approach with the scanned 3D model of the users in the virtual environment. In the other side the 3D models used were not excessively different from what a human can expect from a shadow. Thus the morphology of the shadow was generally not disturbing. A user study with a remarkably different shadow (see Figure 12) could lead to different results in terms of ownership.

As previously mentioned the lighting of the virtual environment was chosen to provide a noticeable virtual shadow whatever the user's position. Nevertheless we did not carried out any study to evaluate the influence of the lighting conditions on the virtual shadow perception. Such study could help create an even more natural and realistic virtual shadow configuration.

Some participants pointed out that their real shadow, cast on the display floor, was visible when no virtual shadow was present and that it could be awkward and disturbing. A user study should be carried out to validate the hypothesis that the presence of a virtual shadow could overcome the problems and limitations brought by the user's real shadow in front-projected IPS.

For the positioning task we chose to add a real ball in the participants right hand as an extension of their arm. Thus when the virtual shadow of the participant was removed (condition N), the shadow of the ball was also removed. Therefore we did not propose a condition with only the virtual shadow of the ball. According to the previous work the presence of the shadow of the objects adds a depth cue and a study with the shadow of the ball only should lead to results that correspond to the previous studies.

CONCLUSION

Compared to HMD, in immersive projection systems the presence of the user's body makes it more difficult to embody a virtual character. We proposed to try and provide the users with a virtual body by adding a dynamic virtual shadow of the user in the environment. We carried out a user study to evaluate the influence of user's dynamic virtual shadows on virtual embodiment and 3D interaction. The experiment showed that, the users had a better spatial perception since they were less prone to go through objects with the virtual shadow. Moreover they appropriate the virtual shadow whenever its morphology was close enough to the user one. The user also felt more comfortable when using the application and they generally felt that the experience was more realistic with their virtual shadows.

In a nutshell, the results of the experiment promote the use of dynamic virtual shadows in IPS and lead the way for further studies on "virtual shadow ownership".

Figure 1 :

 1 Figure 1: Participants interacting in a virtual environment inside a CAVE system. They can embody a male (left) or a female (right) virtual character.

Figure 2 :

 2 Figure2: 3D models -Two models were used to cast a virtual shadow in the virtual environments : a male model (left) and a female model (right).

Figure 3 :

 3 Figure 3: Tracking devices -Six infrared constellations (red) were placed over the participants (left) to provide tracking data to perform the inverse kinematics optimization. A ball was placed in their right hand, as an extension of their arm, and a controller in their left one (right).

Figure 4 :

 4 Figure 4: Calibration step -The calibration step enables to compute the offset between the physical ball and the virtual ball at the beginning and end of each virtual shadow condition (N, F or M).

Figure 6 :

 6 Figure 6: Virtual shadow conditions -The participants performed the positioning task with 3 different virtual shadow conditions: None (N) (left), Male (M) (middle), Female (F) (right). The real shadow of the user is visible on the floor but does not match the natural behavior of a shadow in the virtual environment and is not taken into consideration.

Figure 7 :

 7 Figure 7: Mean interval plot (CI 95%) for the depth positioning error, grouped by the Virtual shadow condition and the Task.

Figure 8 :

 8 Figure 8: Mean interval plot (CI 95%) for the task completion time, grouped by the Virtual shadow condition and the Task.

Figure 10 :

 10 Figure10: Boxplot summarizing the ratings for the presence (P) and task appreciation (TA) questionnaires.

Figure 12 :

 12 Figure12: Virtual embodiment through virtual shadows in a VR entertaining application -The virtual shadow of a cowboy (Inspired from the famous Lucky Luke comic book) is displayed in a far west virtual scene in our CAVE display.

Table 1 :

 1 Summary of the subjective questionnaire. (P: Presence, SA: Shadow Appreciation, T A: Task Appreciation, O: Ownership, A: Agency)

	ID	Question
	P 1	I had a sense of "being there" in the virtual house living room
		space
	P 2

G.Cortes, F. Argelaguet, E. Marchand, and A. Lécuyer

Optitrack tracking hardware http://optitrack.com/hardware/

Final IK asset http://www.root-motion.com/final-ik.html

ACKNOWLEDGMENTS

The authors wish to thank every participant to the experiment. They also would like to thank Florian Nouviale from Inria for his technical support during the experiment.


Jean Bégaint

Franck Galpin

Philippe Guillotel

Christine Guillemot

Technicolor

Region-based models for motion compensation in video compression

Video codecs are primarily designed assuming that rigid, block-based, two-dimensional displacements are suitable models to describe the motion taking place in a scene. However, translational models are not sufficient to handle real world motion types such as camera zoom, shake, pan, shearing or changes in aspect ratio. We present here a region-based interprediction scheme to compensate such motion. The proposed mode is able to estimate multiple homography models in order to predict complex scene motion. We also introduce an affine photometric correction to each geometric model. Experiments on targeted sequences with complex motion demonstrate the efficiency of the proposed approach compared to the state-ofthe-art HEVC video codec.

I. INTRODUCTION

The efficiency of video compression tools heavily relies on their ability to reduce the temporal redundancy between consecutive frames. The classical and current approach consists in predicting the motion between frames by estimating translational motion vectors. A prediction is then performed by translating blocks from the reference frame according to the corresponding motion vectors. A potential residue is added when this prediction is inaccurate in respect to the targeted encoding bit-rate or quality. However, classical translational models cannot compensate accurately some specific motion types such as camera zoom, shake, shearing, pan, changes in aspect ratio. Such complex motion are currently handled by splitting large objects into multiple coding blocks compensated with translational motion models. This requires more side information to code the block splitting tree and produces inaccurate predictions, which consequently result in costly residues.

Using more complex transformation models has long been investigated by the video compression community. Early attempts were proposed for MPEG-4 [START_REF] Richardson | 264 and MPEG-4 video compression: video coding for next-generation multimedia[END_REF] to apply homographic global motion compensation to sprites [START_REF] Dufaux | Efficient, robust, and fast global motion estimation for video coding[END_REF], [START_REF] Watanabe | Sprite coding in object-based video coding standard: Mpeg-4[END_REF] and for an associated global and local compensation [START_REF] Jozawa | Twostage motion compensation using adaptive global mc and local affine mc[END_REF] in H.263 [START_REF] Rijkse | H. 263: video coding for low-bit-rate communication[END_REF]. These approaches were discarded at that time in favor of translational models and dense block partitioning, both for coding performances and complexity reasons. Recent works have demonstrated that coding improvements could still be achieved by using global homographic motion models in current state-of-the-art video codecs.

In the ongoing work to improve the compression efficiency of the High Efficiency Video Coding (HEVC) [START_REF] Sullivan | Overview of the high efficiency video coding (HEVC) standard[END_REF] codec, an affine mode was proposed in [START_REF] Chen | Affine SKIP and MERGE modes for video coding[END_REF] to the Joint Video Exploration Team (JVET) [START_REF]on Future Video Coding[END_REF]. Chen et al. proposed a simplified affine prediction model estimated at the block level which achieves significant gains on the targeted sequences and good results on the common test condition (CTC) sequences [START_REF] Bossen | Common test conditions and software reference configurations[END_REF].

Recently, support for global (frame-based) and local (blockbased) homography models has also been proposed and integrated in the emerging AV1 codec from the Alliance for Open Media [START_REF] Parker | Global and locally adaptive warped motion compensation in video compression[END_REF], [START_REF]Alliance for open media[END_REF]. Parker et al. demonstrated the effectiveness of such motion prediction tools on videos with complex or steep camera motion. However, they noted that a single global motion model may not be sufficient to handle geometric distortions in scenes with strong parallax.

In a previous work [START_REF] Bégaint | Region-based prediction for image compression in the cloud[END_REF], the authors proposed a novel prediction method for compressing highly correlated images as found in photo albums or image cloud databases. In order to compensate for strong distortions such as differences of viewpoint, focal length or cameras, a region-based compensation method was developed. We adapt here the previously proposed scheme to be used for video compression in a classical stateof-the-art video codec (HEVC).

In this paper, we present a region-based prediction mode for motion compensation in video compression. The proposed approach aims at extracting multiple homographic transformation models between video frames. A photometric compensation model is also estimated for each segmented region. We demonstrate the efficiency of the proposed scheme on sequences with non-translational motion, compared both to the classical translational compensation and to a single global homographic motion model. Coding results were obtained with the reference HEVC implementation (HM software1 ).

II. REGION-BASED MODELS ESTIMATION

A. Single homography model estimation

To model large geometric distortion between frames, a homography model is often used as it can handles many types of distortions. A homography model H can be defined by the following 3 × 3 matrix

H =   s x . cos(θ) -s y . sin(θ + σ) t x s x . sin(θ) s y . cos(θ + σ) t y k x k y 1   (1)
where (t x , t y ) represent the translation coefficients, θ the rotation, (s x , s y ) the scale parameters, σ the shear, and (k x , k y ) the keystone distortion coefficients. To estimate the homography parameters, the first step consists in matching common landmarks between the frames. The model parameters are then estimated from the matched locations by minimizing the projection error between the projected keypoints of the reference frame and the target frame keypoints.

To detect and match common landmarks, local feature descriptors are often used as they are more robust to geometric distortions (e.g. translation, rotation, zoom, scale) and illumination variations than pixel values [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. These keypoint descriptors are used to detect and match common landmarks between frames. Feature vectors (or descriptors) are extracted from both images at the detected keypoint locations and then matched exhaustively. The SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] algorithm is used as it has been proven to be robust and efficient. In order to further improve the matching, we use the recent RootSIFT variant proposed by Arandjelovic et al. in [START_REF] Arandjelovic | Three things everyone should know to improve object retrieval[END_REF].

A homography matrix H is then estimated via the RANSAC [START_REF] Fischler | Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography[END_REF] algorithm from the matched keypoints. To robustly compute the homography model parameters with RANSAC, the Symmetric Transfer Error (STE) [START_REF] Harltey | Multiple view geometry in computer vision[END_REF] is used to measure the distances (the l2-norm here) between matched keypoints. Since the STE takes into account both the forward and backward projections of the matched keypoints, this distance is well suited for real-world data where local features detection and their matching will likely contain errors [START_REF] Harltey | Multiple view geometry in computer vision[END_REF].

To increase the robustness of the estimation, the determinant of the homography matrix is also used to discard invalid models. As pointed out by Vincent et al. in [START_REF] Vincent | Detecting planar homographies in an image pair[END_REF], homographies with high or low determinants can be discarded as they correspond to degenerated cases, i.e. when the absolute value of the determinant of the matrix (or its inverse) is close to zero. Following the recommendation of [START_REF] Vincent | Detecting planar homographies in an image pair[END_REF], a threshold of 10 is used.

B. Multiple homography models estimation

A single homography is not always sufficient to describe the possible geometric distortions between frames, especially for non-planar scenes and scenes with strong parallax. To robustly estimate multiple homography models between a reference image and a target image, we propose to use a region-based estimation [START_REF] Bégaint | Region-based prediction for image compression in the cloud[END_REF], consisting of a semi-local approach to better capture correlation between the two images. The main idea is to detect the major regions (planes or objects) found in both images and estimate the geometric transformations between them. We present here the main steps of the multiple models estimation, more details about the method can be found in [START_REF] Bégaint | Region-based prediction for image compression in the cloud[END_REF]. The target image is first segmented into small homogeneous regions by leveraging the efficient SLIC super-pixels algorithm [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF]. SLIC uses a combined colorimetric and spatial distance metric to cluster similar and neighbouring pixels.

For each extracted super-pixel, a projective transformation, i.e. a homography model, is estimated from the SIFT keypoints located inside the super-pixel boundaries. We use the single homography estimation method previously presented in subsection II-A. However, some super-pixels may not have enough keypoints to estimate correctly a homography, may contain outliers, or may share similar or close homographies.

To improve the robustness as well as to reduce the number of homographies, the extracted models are recursively reestimated and fitted to the matched keypoints via the energy minimization method proposed in [START_REF] Delong | Fast approximate energy minimization with label costs[END_REF]. The expansion (assignment) and re-estimation steps are performed iteratively until convergence of the minimization or until a maximum number of iterations is reached. At each iteration, the keypoints are assigned to the homography model that minimizes a combined energy function, the sum of three terms: data cost, smoothness cost and label cost. The data cost is a fidelity term, computed from the STE, which ensures that the model properly describes the projection. The smoothness cost is defined from the Delaunay triangulation of the matched keypoints. It penalizes neighboring points with different assigned homographies in order to preserve spatial coherence. The label cost is used to restrict the number of models. Due to the likely presence of outliers in the matches, an additional model is introduced to fit their distribution (a priori estimated). An example of the resulting labelling is shown in Figure 1d, where one can observe that 3 regions are detected successfully.

C. Photometric compensation

Once the finite set of homographies describing the geometric transformations between frames has been determined, predictions can be constructed. However, disparities due to illumination differences between the constructed blocks image and the current block to encode may exist. During the encoding, these disparities will result in a highly energetic residue, limiting the use of the predictor by the encoder.

To compensate these distortions, we estimate an affine photometric compensation model for each previously estimated region. The two model coefficients, α and β are computed by minimizing the sum of absolute errors (SAD) on the matched pixel regions with a linear solver:

argmin α,β P |Y ′ (x ′ p ) -(α.Y (x p ) + β)| (2)
This scale-offset model is estimated and applied only on the main component channel (Y) of the prediction blocks. We use the sum of absolute differences (SAD) to decide whether to enable or not the photometric model for each region during the prediction. If selected, the compensation is performed for all the blocks in the region. The SAD is used in this context as a fast estimator for the quality of reconstruction, compared to the full RDO process. As the SAD tends to favour more compact residues, it is preferred over the sum of squared differences (SSD).

III. CODING SCHEME

When compressing a video sequence with a classical video codec, most of the sequence frames are coded with the interprediction mode enabled, to make use of the temporal redundancy. All frames coded with inter-prediction leverage a set of reference frames, from which block predictions are performed by estimating and transmitting motion vectors and residues in the bit-stream. The proposed region-based estimator was implemented in this context.

In our setup, the region-based models are estimated between the original current frame and the original first frame of the group of picture (GOP). Although the predictor could be used for all the frames in the reference pictures buffer of the current image, we choose to use only one frame for implementation reasons, focusing on demonstrating the efficiency of the proposed mode. Potential prediction blocks are generated for each extracted model by warping and interpolating the reconstructed (encoded/decoded) blocks of the reference frame from the homography model, then compensating the luminance channel.

Once all the blocks of the current frame have been encoded, we determine which models have been actually used by the encoder through the rate distortion optimization (RDO)

B Cactus 1920x1080 0.02% -0.03% -3.16% -3.17% city 1280x720 0.13% 0.13% 0.00% -0.26% in to tree 1280x720 -0.28% -0.28% -0.54% -0.54% shields 1280x720 -2.06% -2.06% -2.13% -2.13% average -0.55% -0.56% -1.46% -1.52%

Total average -2.16% -2.27% -3.04% -3.37%

process. The default inter-prediction modes of HEVC are often efficient enough to predict the blocks and sending the parameters for multiple models is more expensive than simple motion vector parameters. As such, our mode competes with the other inter-prediction modes in the RDO loop and is only activated when the classical translational estimation fails to predict correctly the current block. A specific syntax is added in the HEVC bit-stream to signal the used models, so that the stream can be decoded. The geometric and photometric models parameters are also encoded and stored in the bitstream for each frame, as halfprecision floating point (16 bits).

IV. EXPERIMENTAL RESULTS

The coding experiments are performed on common test sequences [START_REF] Bossen | Common test conditions and software reference configurations[END_REF], [START_REF]Xiph.org test media[END_REF] and proposed User Generated Content (UGC) sequences [START_REF] Ma | Huawei test sequences of UGC feature for video coding development[END_REF]. The selected sequences display a wide variety of motion caused by camera zooms, camera rotations, camera shakes, and classical 2D translational motion.

The HEVC HM software version 16.16 was used for all the experiments. The rate-distortion performances presented here are computed with the Bjontegaard metric [START_REF] Bjontegaard | Calculation of average psnr differences between rdcurves[END_REF] using the common 22, 27, 32, 37 Quantization Parameter (QP) values. The PSNR is computed on the Y channel only. The default HM random access configuration mode [START_REF] Bossen | Common test conditions and software reference configurations[END_REF] is used as a baseline in all the following tests, with a GOP size of 16. The parameters for the region-based models estimator are fixed for all the experiments, more details about the adjustment of these parameters can be found in [START_REF] Bégaint | Region-based prediction for image compression in the cloud[END_REF].

A. Coding results

Experimental results for the coding experiments are reported in Table I, the first set (1) of sequences corresponds to targeted sequences with known affine content, other sequences are placed in the second set [START_REF] Dufaux | Efficient, robust, and fast global motion estimation for video coding[END_REF]. For comparison, we also introduce a global motion estimator as a second baseline, estimated with a classical SIFT+RANSAC approach (Section II-A). BDrates reductions are presented for the global motion estimator (GLB), the global motion estimator with a global luminance compensation (GLB-L), the proposed region-based approach (RB) and the region-based approach with the luminance compensation (RB-L). First, one can note that the GLB scheme brings an improvement of -2.88% over the default translational motion models of HEVC on the targeted sequences, highlighting the need for more complex motion prediction models. On the whole dataset, improvements go up to -11.75% with a mean BD-rate reduction of -2.16%.

The proposed region-based prediction mode achieves a greater improvement, with an average gain of -3.04%, up to -12.13%. Most of the sequences benefit from the multiple models prediction, with an average improvement of -0.88% over the single model mode. Although the gains are limited for most sequences, videos with affine motion display significant gains such as "station2" (-12.13%), "LakeWalking" (-9.26%) and "bluesky" (-4.32%).

The efficiency the luminance compensation is low in the context of the global estimator, with only a marginal improvement of -0.11%, whereas a higher gain of -0.33% can be obtained with the region-based approach. As the luminance compensation is estimated from the matched regions content and not on the global frame, the estimation is more precise and robust.

Overall, our scheme achieves an average BD-rate gain of -3.37%, with -4.19% on targeted affine sequences and especially -1.52% on the second set.

To illustrate the use of the proposed models by the encoder, we adapted an HEVC bit-stream analyzer to display the use of the mode for each block. Examples are shown Figure 3 on 4 sequences. One may note that our prediction tool is enabled for a large number of the blocks within the reference frame "footprint". For example, in the "station2" and "bluesky" sequences, the borders are not available for the prediction as a zoom and a rotation were respectively performed by the camera.

B. Complexity study

We present here a brief complexity study of the proposed scheme. The prediction tool was implemented in the HM software (version 16.16) without particular optimization.

As it is often the case, the main complexity overhead of the prediction scheme resides on the encoder side. The mean complexity increases are reported in Table II. The mean runtime increase of the RB-L scheme is of ∼300% compared to the default HM encoder. Most of the overhead is spend estimating the region-based models and in the increased RDO loop. Numerous improvements are possible to optimize the region-based estimator, especially the keypoints detection, extraction and matching process.

The complexity on the decoder side has a mean increase of ∼200% for the RB-L method. Again, our implementation is not optimized. For example, we warp a whole frame for each model instead of warping only the selected blocks. Moreover, compared to the encoder, the decoder performs only a few extra operations. The model parameters are first decoded, then the blocks are generated by warping and interpolating the reference frame, and the luminance pixel values are finally corrected. All these operations can be optimized for fast processing. Besides, as the computation is still linear on the input (O(n)), an hardware implementation would have almost no overhead.

V. CONCLUSIONS

In this paper we present a novel video prediction mode to describe complex motion in video sequences. The efficiency of the proposed solution was demonstrated against state-ofthe-art video coding tools on multiple sequences, with an average gain of -3.37% over HEVC. The complexity of the prediction mode is limited, especially on the decoder side, with respect to the gains that can be obtained. Although the regionbased prediction is currently limited to one reference frame, it could be extended to use more frames from the reference pictures buffer. Improving the speed and the robustness of the prediction also constitutes an important future work.

Fig. 1 .

 1 Fig. 1. Region-based estimation process: (a) current frame, (b) reference frame, (c) and (d) matched descriptors.

Fig. 2 .

 2 Fig. 2. Coding scheme: illustration for a GOP of 8 frames with the default random access configuration. The dashed lines represent the default references, the straight lines the extra reference for the proposed mode.

Fig. 3 .

 3 Fig. 3. Region-based prediction mode usage. Red and orange blocks are coded with classical intra-and inter-prediction, respectively. Predictions made by our mode are depicted with the other colors.

TABLE I BD

 I -RATE REDUCTION COMPARED WITH HEVC. AFFINE SEQUENCES IN SET (1), NON-AFFINE IN SET (2).

	Sequence	GLB GLB-L	RB	RB-L
		CStoreGoods 720x1280	-2.29% -2.29% -2.58% -2.71%
		DrivingRecorder1 720x960	0.16%	0.16% -0.90% -1.05%
		DrivingRecorder2 720x960	0.11%	0.07% -1.63% -1.81%
		LakeWalking 720x960	-7.06% -7.06% -9.26% -10.45%
	(1)	ParkSunny 720x1280	-0.13% -0.37% -0.28% -0.74%
		ParkWalking 720x1280	-2.04% -2.04% -2.73% -3.11%
		bluesky 1920x1080	-3.90% -4.21% -3.97% -4.32%
		station2 1920x1080	-11.28% -11.75% -12.13% -13.27%
		tractor 1920x1080	0.53%	0.16% -0.27% -0.27%
		average	-2.88% -3.04% -3.75% -4.19%

TABLE II MEAN

 II COMPLEXITY INCREASES COMPARISON AGAINST HEVC

	Method	Complexity
		Encoding	Decoding
	GLB	189.92%	174.10%
	GLB-L	190.27%	173.18%
	RB	299.93%	196.01%
	RB-L	301.45%	198.39%

https://hevc.hhi.fraunhofer.de/svn/svn HEVCSoftware/


Franz Bethke
email: bethke@math.hu-berlin.de

Axel Kr Öner
email: axel.kroener@math.hu-berlin.de

SUFFICIENT OPTIMALITY CONDITIONS FOR BILINEAR OPTIMAL CONTROL OF THE LINEAR DAMPED WAVE EQUATION

Keywords: 2010 Mathematics Subject Classification, 49K20 and 35L05, damped wave equation and bilinear optimal control and sufficient optimality conditions and Goh transformation and singular arcs

In this paper we discuss sufficient optimality conditions for an optimal control problem for the linear damped wave equation with the damping parameter as the control. We address the case that the control enters quadratic in the cost function as well as the singular case that the control enters affine. For the non-singular case we consider strong and weak local minima, in the singular case we derive sufficient optimality conditions for weak local minima. Thereby, we take advantage of the Goh transformation applying techniques recently established in Aronna, Bonnans, and Kröner [Math. Program. 168(1):717-757, 2018] and [INRIA research report, 2017]. Moreover, a numerical example for the singular case is presented.

Introduction

Let Ω be an open subset of R n , n ≤ 3 with sufficiently smooth boundary ∂Ω and T > 0. We consider optimal control problems for the damped linear wave equation in which the control u enters as a damping parameter, i.e. equations of type (1.1) ÿ(t, x) -∆y(t, x) = u(t)b(x) ẏ(t, x) + f (t, x) in (0, T ) × Ω, y(t, x) = 0 on (0, T ) × ∂Ω, y(0, x) = y 0,1 (x) in Ω, ẏ(0, x) = y 0,2 (x) in Ω with suitable chosen f , b and initial data y 0,i , i = 1, 2. In one space dimension this equation describes the motion of a string with forced damping and additional source term in the space-time domain. In the damping term we have a bilinear coupling of the control and the velocity. We consider optimal control problems for (1.1) with cost function 2 dt subject to (1.1) and u ∈ U ad , with U ad := u ∈ L 2 (0, T ) : u m ≤ u(t) ≤ u M , a.e. in (0, T ) the set of admissible controls, given constant control bounds 0 ≤ u m < u M , and sufficiently smooth desired states y d,i : (0, T ) × Ω → R, y dT,i : Ω → R for i = 1, 2, α 1 ∈ R, α 2 ≥ 0, and β j ≥ 0, j = 1, . . . , 4.

(1.2)                J(u, y) := β 1 2 y -y d,1 2
L 2 (0,T ;L 2 (Ω)) + β 2 2 y(T ) -y 2 dT,1 2 L 2 (Ω) + β 3 2 ẏ -y d,2 2
L 2 (0,T ;H -1 (Ω)) + β 4 2 ẏ(T ) -y 2 dT,2 2
H -1 (Ω) + T 0 α 1 u(t) + α 2 2 u(t)
The main contribution of this paper is the application of techniques developed in Aronna, Bonnans, and Kröner [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF][START_REF] Aronna | Optimal control of PDEs in a complex space setting; application to the Schrödinger equation[END_REF] to the optimal control problem of the damped wave equation given in (1.2). We derive sufficient optimality conditions for weak and strong local minima in the case α 2 > 0 as well as for weak local minima in the singular case α 1 = 0. For the non-sigular case we rely on techniques developed in [START_REF] Aronna | Optimal control of PDEs in a complex space setting; application to the Schrödinger equation[END_REF], where control problems for the Schrödinger equation are considered. In the singular case the classical techniques of the calculus of variations do not lead to the formulation of second-order sufficient optimality conditions. By applying the Goh transformation [START_REF] Goh | Necessary conditions for singular extremals involving multiple control variables[END_REF] (introduced in the context of optimal control of ordinary differential equations) and following ideas in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF], we derive sufficient optimality conditions guaranteeing weak quadratic growth. Thereby, a commutator plays an important role. However, this commutator is a differential operator of qualitative difference to the one considered in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]; while in the latter reference singular optimal control problems for wave equations with bilinear coupling of control and state are considered which leads to a zero-order operator, here the control is coupled bilinearly with the velocity which gives a second-order differential operator for the commutator. Consequently, different regularity properties are necessary to obtain compactness results. We will restrict the presentation to differences to the setting developed in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]. At the end, a numerical example for the case α 1 = α 2 = 0 is presented.

The first extension of the Goh calculus to optimal control problems to PDEs was done in Bonnans [START_REF] Bonnans | Optimal control of a semilinear parabolic equation with singular arcs[END_REF], where a semilinear heat equation with scalar control was considered and then further extended in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF] to problems for strongly continuous semigroups. In [START_REF] Aronna | Optimal control of PDEs in a complex space setting; application to the Schrödinger equation[END_REF] the approach was transferred to a complex setting and applied to optimal control problems for the linear Schrödinger equation.

To complete the list of references we mention some applications of the Goh transformation in the context of ordinary differential equations, see, e.g., [START_REF] Kelley | A second variation test for singular extremals[END_REF][START_REF] Goh | Necessary conditions for singular extremals involving multiple control variables[END_REF][START_REF] Frankowska | The Goh necessary optimality conditions for the Mayer problem with control constraints[END_REF][START_REF] Aronna | Second order analysis of control-affine problems with scalar state constraint[END_REF].

In the context of optimal control of PDEs there exist only a few papers on sufficient optimality conditions for control-affine problems, see Bergounioux and Tiba [START_REF] Bergounioux | General optimality conditions for constrained convex control problems[END_REF], Tröltzsch [START_REF] Tröltzsch | Regular Lagrange multipliers for control problems with mixed pointwise controlstate constraints[END_REF], Bonnans and Tiba [START_REF] Bonnans | Control problems with mixed constraints and application to an optimal investment problem[END_REF], and Casas, Wachsmuth, and Wachsmuth [START_REF] Casas | Second-order analysis and numerical approximation for bang-bang bilinear control problems[END_REF].

For bilinear optimal control of wave equations, see, e.g., Lenhart and Protopopescu [START_REF] Lenhart | Identification problem for a wave equation via optimal control[END_REF] where an identification problem for a coefficient in the wave equation via optimal control is considered, and Sonawane [START_REF] Sonawane | Optimal control for a vibrating string with variable axial load and damping gain[END_REF] where a bilinear optimal control problem for a vibrating string is analyzed. Furthermore, we refer to Lasiecka and Triggiani [START_REF] Lasiecka | Control theory for partial differential equations: Continuous and approximation theories[END_REF] and Bales and Lasiecka [START_REF] Bales | Continuous finite elements in space and time for the nonhomogeneous wave equation[END_REF] for optimal control of wave equations in a semigroup setting.

For results on controllability of the damped wave equation see, e.g., [START_REF] Vancostenoble | Exact controllability of a damped wave equation with distributed controls[END_REF] and for stabilization, e.g., [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF][START_REF] Zuazua | Exponential decay for the semilinear wave equation with locally distributed damping[END_REF][START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF].

The paper is organized as follows: In Section 2 the damped wave equation is introduced. In Section 3 existence of optimal controls is verified. In Section 4 we derive first-and second-order necessary optimality conditions. In Section 5 sufficient optimality conditions for the non-singular case (α 2 > 0), in Section 6 for the singular case (α 2 = 0) are derived, and in Section 7 we present a numerical example for the singular case.

Notation: For given Hilbert space H, with norm • H , we denote by H * its topological dual and by h * , h H the duality product between h ∈ H and h * ∈ H * . For the inner product we write (•, •) H . We omit the index H if there is no ambiguity. By L(H) we denote the space of linear and continuous operators on H. If A is a linear (possibly unbounded) operator from H into itself, its adjoint operator is denoted by A * . We use the standard notation for Lebesgue and Sobolev spaces. The Euclidean norm is denoted by | • | and for diagonal matrices D ∈ R n×n we use the notation diag(d 1 , . . . , d n ).

The damped wave equation

Let Ω ⊂ R n , n ≤ 3 be a bounded domain with smooth boundary T > 0, and

H := L 2 (Ω), V := H 1 0 (Ω). We define the unbounded operator T -1 := -∆ in H with domain dom(T -1 ) := H 2 (Ω) ∩ H 1 0 (Ω) as T -1 : dom(T -1 ) ⊂ H → H. (2.1)
Then T : H → H is a bounded operator. Let

H := H × V * . (2.2)
In the following we will consider for given b ∈ L ∞ (Ω), u ∈ L 1 (0, T ), f ∈ L 1 (0, T ; V * ), and (y 0,1 , y 0,2 ) ∈ H the general second-order hyperbolic equation

(2.3)      ÿ1 (t, x) -∆y 1 (t, x) = f (t, x) + u(t)b(x) ẏ1 (t, x) in (0, T ) × Ω, y 1 (0, x) = y 0,1 (x), ẏ1 (0, x) = y 0,2 (x) in Ω, y 1 (t, x) = 0 on (0, T ) × ∂Ω.
Setting y 2 (t) := ẏ1 (t), we can reformulate the state equation formally as a firstorder system in time given by (2.4) ẏ + Ay = F + uBy t ∈ (0, T ), y(0) = y 0 , with (2.5)

A := 0 -id T -1 0 , B := 0 0 0 b , F := 0 f , y 0 := y 0,1 y 0,2 .
From the Hille Yosida theorem we can derive by classical arguments that A is the generator of a contraction semigroup e -tA with dom(A) = H 1 0 × L 2 (Ω) ⊂ H. We define the mild solution of (2.4) as the function y ∈ C(0, T ; H) such that, for all t ∈ [0, T ]:

(2.6) y(t) = e -tA y 0 + t 0 e -(t-s)A F(s) + u(s)By(s) ds.
The existence of a mild solution follows by a fixed-point argument, cf. Ball, Marsden, and Slemrod [START_REF] Ball | Controllability for distributed bilinear systems[END_REF]. Furthermore, we have the following estimate; cf. [4, Thm. 2].

Theorem 2.1. There exists c > 0 such that the solution y of (2.6) satisfies (2.7)

y C([0,T ];H) ≤ c y 0 H + f L 1 (0,T ;H -1 (Ω)) + B L(H) u L 1 (0,T ) e c u L 1 (0,T ) .
The dual semigroup on H * is well-defined (see [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) and generated by A * = -A with dom(A * ) = dom(A), identifying H with H * (cf. Appendix A).

From Ball [START_REF] Ball | Strongly continuous semigroups, weak solutions, and the variation of constants formula[END_REF] we recall that any mild solution of (2.4) coincides with a weak solution, i.e. y ∈ Y := C(0, T ; H) satisfies y(0) = y 0 and, for any φ ∈ dom(A * ), the function t → φ, y(t) is absolutely continuous over [0, T ] and

d dt φ, y(t) + A * φ, y(t) = φ, F + u(t)By(t) , for a.a. t ∈ [0, T ]. (2.8)
In the following we denote for given u ∈ U the corresponding mild solution by y[u] and its components by y 1 [u] and y 2 [u].

We introduce the linearized state equation at a point (ŷ, û) with ŷ = y[û], û ∈ U for given v ∈ U given by (2.9) ż(t) + Az(t) = û(t)Bz(t) + v(t)Bŷ(t); z(0) = 0, to be understood in the sense of mild solutions. It is easily checked that the equation (2.9) has a unique solution denoted by z[v], and using the implicit function theorem, that the mapping

u → y[u] from U to Y is of class C ∞ with Dy[u]v = z[v].
We recall the notion of restriction property from [4, Def. 2].

Definition 2.2. Let W be a Hilbert space with norm • W and continuous inclusion in H. Assume that the restriction of e -tA to W has image in W , and that it is a continuous semigroup over this space. We let A denote its associated generator, and e -tA the associated semigroup. Then we have that dom(A ) ⊂ dom(A), and A is the restriction of A to dom(A ). We have that (2.10) e -tA L(W ) ≤ c A e λ A t for some constants c A and λ A . Denote by B the restriction of B to W , which is supposed to have image in W and to be continuous in the topology of W , that is,

(2.11) B ∈ L(W ).
In this case we say that W has the restriction property. This allows to prove higher regularity. Let

E 1 := H 1 0 (Ω) × L 2 (Ω), (2.12) H 2 0,1 := H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω). (2.13) Hypothesis 2.1. We assume (2.14) y 0 ∈ E 1 , b ∈ L ∞ (Ω), f ∈ L 2 (0, T ; L 2 (Ω)).
Lemma 2.4. The space E 1 has the restriction property with restricted semigroup A and domain dom(A ) := H 2 0,1 . Proof. We refer to Pazy [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF].

The optimal control problem

Let q and q T be continuous quadratic forms over H, with associated symmetric and continuous operators Q, Q T ∈ L(H); q(y) := (Qy, y); q T (y) := (Q T y, y).

(3.1) Given (3.2) y d ∈ L ∞ (0, T ; H); y dT ∈ H,
we introduce the cost function

(3.3) J(u, y) := 1 2 T 0 q(y(t)-y d (t))dt+ 1 2 q T (y(T )-y dT )+ T 0 (α 1 u(t)+ α 2 2 u(t) 2 )dt
with α 1 ∈ R, and α 2 ≥ 0; this includes in particular the case (1.2). Then, recalling the set of admissible controls For optimal control problems of type (P) we consider weak local minima. We call û ∈ U ad a weak local minimum if there exists an ε > 0 such that

(3.4) U ad = u ∈ U : u m ≤ u(t) ≤ u M ,
(3.5) F (û) ≤ F (u) for all u in U ad with u -û L ∞ (0,T ) ≤ ε.
To obtain existence we need a compactness result.

Lemma 3.1. Let for given control u ∈ U and initial condition y 0 ∈ E 1 the function y = y[u] denote the corresponding solution of (2.4). Then the mapping

(3.6) U → L 2 (0, T ; H), u → by 2 [u],
is compact.

Proof. We have

(3.7) U → L 2 (0, T ; H), u → by 2 [u],
with y[u] being the solution of

(3.8) ẏ1 = y 2 , ẏ2 -∆y 1 = uby 2 + f.
We check the compactness hypothesis. We have

(3.9) y[u] ∈ C(0, T ; E 1 ), ẏ[u] ∈ L 1 (

First and second-order necessary optimality conditions

The costate equation is given in H as

-ṗ + A * p = Q(y -y d ) + uB * p; p(T ) = Q T (y(T ) -y dT ). (4.1)
The corresponding mild solution is well-defined in C(0, T ; H) (cf. the comments on the dual semigroup in Section 2) and is denoted by p[u] = p.

Next, for given u ∈ U we introduce

(4.2) Λ(t) := α 1 + α 2 u(t) + y 2 [u](t)b(x)p 2 [u](t).
Since u → F (u) is of class C ∞ by the implicit function theorem, we have the representation

(4.3) DF (u)v = T 0 Λ(t)v(t)dt for all v ∈ U.
By standard arguments we obtain the necessary optimality: The contact sets are (4.4)

I m (u) := {t ∈ (0, T ) : u(t) = u m }; I M (u) := {t ∈ (0, T ) : u(t) = u M }.
and we have the following proposition.

Proposition 4.1. Let û ∈ U ad be a weak local minimum of (P). Then, up to a set of measure zero, there holds

(4.5) {t; Λ(t) > 0} ⊂ I m (û), {t; Λ(t) < 0} ⊂ I M (û).
To formulate second-order conditions we introduce the second variation of the Lagrangian Q : C(0, T ; H) × U → R by

(4.6) Q(z, v) := T 0 z(t) 2 H + α 2 v(t) 2 + 2v(t)(p 2 (t), z 2 (t)) V * dt + z(T ) 2 H .
Given a feasible control u, the critical cone is defined as

(4.7) C(u) := v ∈ U | Λ(t)v(t) = 0 a.e. on [0, T ],
v(t) ≥ 0 a.e. on I m (u), v(t) ≤ 0 a.e. on I M (u) .

Then we can formulate the second-order necessary optimality conditions.

Theorem 4.2. Let û be a weak local minimum of (P). Then we have,

(4.8) Q(z[v], v) ≥ 0 for all v ∈ C(û).
Proof. See [4, Thm. 6].

5.

Second-order sufficient optimality conditions for the non-singular case α 2 > 0 Next, we formulate sufficient optimality conditions for weak as well as strong local minima. Therefore we introduce a positive definiteness condition for the second variation of the Lagrangian: Let α 0 > 0 be such that (5.1)

Q(z[v], v) ≥ α 0 v(t) 2 L 2 (0,T )
for all v ∈ C(û).

Theorem 5.1. Let û ∈ U satisfy the first order optimality conditions in Proposition 4.1, and let the positive definiteness condition (5.1) hold. Then û is a weak local minimum of problem (P) that satisfies the quadratic growth condition.

Proof. It follows by an adaption of [START_REF] Bonnans | Optimal control of a parabolic equation with time-dependent state constraints[END_REF]Thm. 4.3] or Casas and Tröltzsch [START_REF] Casas | Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems[END_REF].

We introduce the notion of strong local minima.

Definition 5.2. A control û ∈ U ad is a strong local minimum if there exists ε > 0 such that, for all u ∈ U ad and y[u] -y[û] C(0,T ;H) < ε we have F (û) ≤ F (u).

Definition 5.3. A control û ∈ U ad satisfies the quadratic growth condition for strong solutions if there exists σ > 0 and ε > 0 such that for any feasible control u:

(5.2)

F (û) + σ u -û 2 L 2 (0,T ) ≤ F (u), whenever y[u] -y[û] C(0,T ;H) < ε. Theorem 5.4.
Let û ∈ U ad satisfy the first order necessary optimality condition (4.5), and the positive definiteness condition (5.1). Then û is a strong solution that satisfies the above quadratic growth condition.

Proof. The proof follows the same ideas as in the complex setting presented in [2, Thm. 5.10] based on the decomposition principle and using techniques by Bonnans and Osmolovskiȋ [START_REF] Bonnans | Second-order analysis of optimal control problems with control and initial-final state constraints[END_REF].

6. Second-order sufficient optimality conditions for the singular case α 2 = 0

Here, we reformulate the second-order necessary optimality conditions for weak local minima by a transformed quadratic form, apply the Goh transformation, and derive sufficient optimality conditions.

Given û ∈ U, let ŷ = y[û] and p = p[û] be the associated state and costate.

Hypothesis 6.1. In the sequel we assume (additional to Hypothesis 2.1) that the costate p is in C(0, T ; E 1 ).

Remark 6.1. If Q(y -y d ) ∈ L 2 (0, T ; E 1 ) and Q T (y -y dT ) ∈ E 1 this can be guarenteed by the semigroup property. For Q and Q T given by cost functions as in (1.2) and y d ∈ E 1 this is the case.

The space E 1 ⊂ H with continuous inclusion has the restriction property (see Definition 2.2). Using for the restriction of A and B to E 1 the same notation we have by the regularity of b given in Hypothesis 2.1 that (6.1)

B k dom(A) ⊂ dom(A), (B k ) * dom(A * ) ⊂ dom(A * ), k = 1, 2.
This allows to define the following operators with domains dom(A) and dom(A * ):

(6.2) [A, B k ] := AB k -B k 2 A, [(B k ) * , A * ] := (B k ) * A * -A * (B k ) * . Let us define for k = 1, 2 (6.3) M k y := [A, B k ]y
considering the closure of the operator in E 1 . Then we have (6.4)

M k = 0 -b k -b k T -1 0 ; [M k , B] = 0 -b k+1 b k+1 T -1 0 ; k = 1, 2.
We observe that here the commutator is a second-order, self-adjoint differential operator (cf. Appendix A), i. e.

(6.5) M * k = M k . Thus, using the fact that p ∈ C(0, T ; E 1 ) (see Hypothesis 6.1) we have in particular (6.6) M * k p ∈ C(0, T ; H).

To illustrate the dependencies we restrict the presentation to cost functions of type (1.2), nevertheless corresponding statements can be derived for general cost functions as given in (3.3). We define the space

W := L 2 (0, T ; E 1 ) ∩ C([0, T ]; H) × L 2 (0, T ) × R (6.7)
and introduce a quadratic form over W by (6.8)

Q(ξ, w, h) := Q T (ξ, h) + Q a (ξ, w) + Q b (w),
where (6.9)

Q b (w) := T 0 w 2 (t)R(t)dt and (6.10) Q T (ξ, h) := β 2 ξ 1 (T ) 2 H + β 4 ξ 2 (T ) + hbŷ 2 (T ) 2 V * + h(p 2 (T ), bξ 2 (T )) V * + h 2 (p(T ), b 2 ŷ2 (T )) V * , Q a (ξ, w) := T 0 β 1 ξ 1 2 H + β 3 ξ 2 2 V * dt + T 0 2wβ 3 (ξ 2 , bŷ 2 ) V * dt, + T 0 2w β 3 (ŷ 2 -y d,2 , bξ 2 ) V * + (bp 2 , ξ 1 ) H -(bT -1 p2 , ξ 2 ) V * dt, R(t) := β 3 bŷ 2 2 V * + β 3 (ŷ 2 -y d,2 , b 2 ŷ2 ) V * + (p 2 , b 2 (f + T -1 ŷ1 )) V * -(p 1 , b 2 ŷ2 ) H .
We will show that the quadratic form Q is positive semidefinit on a transformed critical cone. Let P C 2 (û) be the closure in the L 2 × R-topology of the set (6.11

) P C(û) := {(w, h) ∈ W 1,∞ (0, T ) × R, ẇ ∈ C(û); w(0) = 0, w(T ) = h}.
Theorem 6.2. Let Hypothesis 2.1 and 6.1 hold and let the cost function be given by (1.2). Let û ∈ U ad be a weak local minimum of (P) and z solution of the linearized state equation at (û, y[û]). Furthermore, let for given v ∈ U the functions (w, ξ) be defined by the Goh transformation (6.12) w(t) := t 0 v(s)ds, ξ := z -wBŷ.

(i) Then ξ is the mild solution of

(6.13) ξ1 -ξ 2 = wbŷ 2 , ξ2 -∆ξ 1 = ûbξ 2 -wb∆ŷ 1 -wbf with ξ(0) = 0. (ii) We have (6.14) Q(ξ, w) = Q(z, v).
(iii) In particular, the transformed second-order variation is positive semidefinit on the transformed critical cone, i.e.

(6.15) Q(ξ, w) ≥ 0 for all (w, h) ∈ P C 2 (û).

Proof. (i) By a direct calculation (cf. [4, (3.28)]) we obtain formally

(6.16) ξ + Aξ = ûBξ -wBF -w[A, B]ŷ, ξ(0) = 0.
We check well-posedness. Since by the regularity assumption on the data in Hypothesis 2.1 we have ŷ ∈ C(0, T ; E 1 ) and since the commutator is a second-order differential operator we derive ξ ∈ C(0, T ; H). Furthermore, (6.16) is, in the case under consideration, equivalent to (6.13).

(ii) We apply [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]Thm. 7] to obtain (6.14). We have to verify the conditions [4, (3.23)], namely: (6.1), which is satisfied, and (6.17)

   (a) for k = 1, 2 : M k has a continuous extension to E 1 , (b) for k = 1, 2 : M * k p ∈ L ∞ (0, T ; H), (c) ŷ ∈ L 2 (0, T ; E 1 ); [M 1 , B]ŷ ∈ L ∞ (0, T ; H).
In (6.4) we have seen that the commutator M k is a second-order differential operator having a continuous extension to E 1 which gives (a). Condition (b) results from the regularity of the costate p ∈ C(0, T ; E 1 ), cf. Hypothesis 6.1. Finally, we have ŷ ∈ C(0, T ; E 1 ) by Hypothesis 2.1 so [M 1 , B]ŷ ∈ C(0, T ; H) which gives (c).

(iii) This follows from (4.8) and (ii), cf. [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]Lem. 6].

Remark 6.3. In comparison to [START_REF] Aronna | Optimal control of PDEs in a complex space setting; application to the Schrödinger equation[END_REF], where the control is coupled bilinearly with the state and not the velocity component, the right hand side of the equation in ξ given in (6.13) is of qualitativ difference, since it involves the Laplacian of y 1 .

Definition 6.4 (Singular arc). A control u ∈ U ad is said to have a singular arc over (t 1 , t 2 ), with 0 ≤ t 1 < t 2 ≤ T , if, for all θ ∈ (0, 1 2 (t 2 -t 1 )), there exists ε > 0 such that (6.18) u(t) ∈ [u m + ε, u M -ε], for a.a. t ∈ (t 1 + θ, t 2 -θ).

We may also say that (t 1 , t 2 ) is a singular arc itself. We call (t

y 0 ∈ E 2 ; f ∈ L ∞ (0, T ; H 1 0 (Ω)), b ∈ W 1,∞ (
Ω). Corollary 6.5. Let Hypothesis 6.2 hold and û ∈ U ad be a weak local minimum for problem (P). Then, (6.21) the mapping w → ξ[w] is compact from L 2 (0, T ) to L 2 (0, T ; H).

Let (t 1 , t 2 ) be a singular arc. Then R ∈ L ∞ (0, T ; H) defined in (6.10) satisfies (6.22) R(t) ≥ 0 for a.a. t ∈ (t 1 , t 2 ).

Proof. We know ξ ∈ C(0, T ; H). Since E 2 has the restriction property (by the Hille-Yosida theorem) [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Cor. 3.8] we have by Hypothesis 6.2 and Lemma 2.3 that the first component y 1 of the solution of (2.4) is in C(0, T ; H 2 0 (Ω)). Consequently, the right hand side of equation (6.13) is in C(0, T ; E 1 ). Similarly as in the proof of Lemma 3.1 we further obtain ξ[w] ∈ L 1 (0, T ; H) and thus the compactness property in (6.21). Using (6.21) we obtain that the terms of Q where ξ is involved are weakly continuous. We conclude as in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]Cor. 5].

To formulate second-order sufficient optimality conditions we make the following hypotheses. Hypothesis 6.3. We assume (1) Regular data: Let Hypotheses 2.1, 6.1, and 6.2 hold and additionally, (6.23) f ∈ C(0, T ; H -1 (Ω)), y d ∈ C(0, T ; H).

(2) finite structure:

(6.24) there are finitely many boundary and singular maximal arcs and the closure of their union is [0, T ],

(3) strict complementarity for the control constraint (note that Λ is a continuous function of time) We say that û satisfies a weak quadratic growth condition if there exists β > 0 such that for any u ∈ U ad , setting v := u -û and w(t) := T 0 v(s)ds, we have (6.29) F (u) ≥ F (û) + β( w 2 L 2 (0,T ) + w(T ) 2 ), if v L 1 (0,T ) is small enough. The word 'weak' refers to the fact that the growth is obtained for the L 2 -norm of w, and not of v.

Similar as in [START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF]Thm. 8] (taking into account [START_REF]Optimal Control of Infinite Dimensional Bilinear Systems: Application to the Heat and Wave Equations[END_REF]) we obtain the following statement. Theorem 6.7. Let û ∈ U ad , the cost function be given by (1.2), and let Hypothesis 6.3 hold. Remark 6.8. When û has no bang-bang switch, the cones P C 2 (û) and P C 2 (û) coincide and, therefore, the necessary and sufficient conditions have no gap.

Numerical example

We present a numerical example for the singular case α 1 = α 2 = 0. Let Ω = (0, 1), T = 1, and for (t, x) ∈ (0, T ) × Ω we set For the approximation in space we use a spectral basis { √ 2 sin(kπx) | 1 ≤ k ≤ N } with N = 10 and for the approximation in time an implicit Euler scheme with 1000 time steps. The numerical simulation was performed with Bocop [START_REF] Bonnans | Bocop, the optimal control solver, open source toolbox for optimal control problems 2011[END_REF] which uses IPOPT [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]. In Fig. 1 we see the control is first on the lower bound, then a singular arc appears, and then it is on the upper bound. The computed control is stable with respect to time and space discretization.

Lemma 2 . 3 .

 23 Let W have the restriction property, y 0 ∈ W , and f ∈ L 1 (0, T ; W ) hold. Then the solution of (2.4) satisfies y ∈ C(0, T ; W ) and the mapping u → y[u] is of class C ∞ from L 1 (0, T ) to C(0, T ; W ). Proof. See [4, Lem. 1].

  a.e. on (0, T ) ; with U := L 2 (0, T ) and defining the reduced cost by F (u) := J(u, y[u]) the optimization problem reads as (P) Min u∈U ad F (u).

( 6 . 6 .

 66 [START_REF] Sonawane | Optimal control for a vibrating string with variable axial load and damping gain[END_REF] Λ has nonzero values over the interior of each boundary arc, and at time 0 (resp. T ) if an initial (resp. final) boundary arc exists, (4) letting T BB denote the set of bang-bang junctions, we assume (6.26) R(t) > 0, t ∈ T BB . h) ∈ L 2 (0, T ) × R; w is constant over boundary arcs, w = 0 over an initial boundary arc, and w = h over a terminal boundary arc. Let û ∈ U ad satisfy (6.24) and (6.25). Then P C 2 (û) defined before(6.11) can be characterized as (6.28) P C 2 (û) = {w ∈ P C 2 (û) : w is continuous on bang-bang junctions} Proof. See [1, Proposition 4].

  (a) If û satisfies the necessary optimality conditions (4.1) and if there exists α > 0 such that(6.30) Q(ξ[w], w, h) ≥ α( w 2 L 2 (0,T ) + h 2 ), for all (w, h) ∈ P C 2 (û), then the quadratic growth condition (6.29) is satisfied.(b) If û is a weak minimum and the quadratic growth condition (6.29) is satisfied, then(6.31) Q(ξ[w], w, h) ≥ α( w 2 L 2 (0,T ) + h 2 ), for all (w, h) ∈ P C 2 (û), holds.Proof. We have to check: (6.32)(i) B 2 f ∈ C(0, T ; H); y d ∈ C(0, T ; H); (ii) M * k p ∈ C(0, T ; H), k = 1, 2.Then we conclude by[START_REF]Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations[END_REF] Thm. 8]. (i) follows immediately from (6.23); (ii) results from the fact that p ∈ C(0, T ; E 1 ), cf. Hypothesis 6.1.

(7. 1 )

 1 y 0,1 = sin 2 (πx); y 0,2 = 0; f = 0; b = 1; y d,1 = 0; y d,2 = 1 10 sin(2πt) sin(πx); Q = id; Q T = 0; u m = 0; u M = 10.

Figure 1 .

 1 Figure 1. Optimal control with singular arc.

  0, T ; H); the second inclusion in (3.9) follows from (2.8) and estimate (2.7) with E 1 instead of H. Since E 1 is compactly embedded in H, we conclude by Aubin's Lemma in the variant given in [26, Remark 2.1, p. 189]; cf. also[24, p. 37].

Corollary 3.2. Problem (P) has a solution. Proof. The existence follows by classical arguments using the compactness result from Lemma 3.1; cf. [2, Thm. 2.15].

  1 , t 2 ) a lower boundary arc if u(t) = u m for a.a. t ∈ (t 1 , t 2 ), and an upper boundary arc if u(t) = u M for a.a. t ∈ (t 1 , t 2 ). We sometimes simply call them boundary arcs. We say that a boundary arc (c, d) is initial if c = 0, and final if d = T .

	Let
	(6.19)	E 2 := H 2 0 (Ω) × H 1 0 (Ω).
	Hypothesis 6.2. We assume
	(6.20)

Acknowledgments

The authors thank Pierre Martinon (INRIA Saclay) for fruitful discussions on Bocop [9] and Philipp Trunschke (HU Berlin) on the implementation. The second author thanks the INRIA team Commands and the Centre Mathématiques Appliquées, Ecole polytechnique (Palaiseau, France) for their hospitality in the years 2017-2018. The second author was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences.

Appendix A. On the adjoint equation

We consider the derivation of the costate equation for problem (P) with cost function (1.2).

Lemma A.1. Given constants c i ∈ R, i = 1, . . . , 4 and the operator

Identifying H with its dual H * the adjoint operator N * is given by

This proves the claim.

Corollary A.2. (i) The adjoint operators of A and M k defined in (2.5) and (6.4) are given as where p takes the role of p 2 .