Florent Sevely

Tao Wu

Felipe Sodre Ferreira De Sousa

Lionel Séguier

Vincent Brossa

Samuel Charlot

Alain Estève

Carole Rossi
email: rossi@laas.fr

Lionel Seguier

Alain Esteve

Developing a highly responsive miniaturized security device based on a printed copper ammine energetic composite

Keywords: transient electronics, anti-reverse engineering, anti-hacking, nanoenergetic materials, PyroMEMS, 3D printing

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Security has become a vital part of electronic products that are employed in critical data storage (personal, bank, etc.) or critical equipment such as military systems. These electronics not only handle sensitive data in uncontrolled environments but also face intellectual protection issues or counterfeiting issues. Several electronic anti-tamper and anti-reverse engineering technologies exist [START_REF] Mcloughlin | Secure embedded systems: the threat of reverse engineering[END_REF][START_REF] Burd | Using evolution to evaluate reverse engineering technologies: mapping the process of software change[END_REF][START_REF] Guo | Investigation of Obfuscation-based Anti-Reverse Engineering for Printed Circuit Boards[END_REF]. In brief, the simplest anti-tamping and anti-reverse engineering solutions are based on cryptographic keys or software [START_REF] Park | Symmetric-Key Cryptographic Routine Detection in Anti-Reverse Engineered Binaries Using Hardware Tracing[END_REF], security fuses or circuit breakers [START_REF] Desai | Anti-counterfeit Integrated Circuits Using Fuse and Tamper-Resistant Time-stamp Circuitry[END_REF] to prevent nonauthorized access, or reinforcement of the component packaging [START_REF] Desai | Anti-counterfeit Integrated Circuits Using Fuse and Tamper-Resistant Time-stamp Circuitry[END_REF][START_REF] Li | Design of anti-key leakage camouflage gate circuit for reverse engineering based on dummy vias[END_REF] to limit access to sensitive components. More sophisticated solutions, including anti-tamper sensors, are able to detect the type of intrusion, such as Hall-effect switches [START_REF] Patnaik | Advancing Hardware Security Using Polymorphic and Stochastic Spin-Hall Effect Devices[END_REF]. Often, several different anti-tamper protections are integrated to ensure efficient protection [START_REF] Fleck | Controlled Substrate Destruction Using Nanothermite[END_REF].

Unfortunately, all current anti-tamping technologies, while being effective and in constant evolution to adapt to new complex topologies and progressing technologies, are susceptible to being cracked or bypassed, leaving the system vulnerable. That is why an ultimate action must be considered upon the detection of intrusion based on irreversible destruction of the component being attacked. Indeed, irreversibly destroying the component and its stored data is the most secure and cost-effective way to dispose of sensitive information such as topology, access code or private data. From this viewpoint, the use of transient electronic devices [START_REF] Cheng | Self-erasable and rewritable photonic platform for antitamper hardware[END_REF][START_REF] Pandey | Towards Transient Electronics through Heat Triggered Shattering of Off-the-Shelf Electronic Chips[END_REF] was explored to destroy the sensitive component based on various mechanisms, such as light, heat, or mechanical impact. However, the transience technology is mostly too slow, taking several minutes to a few days to act. It also imposes several a priori restrictions in terms of chip design and materials, making it unsuitable for the semiconductor industry at large. Alternative works have explored the release of acids or corrosives [START_REF] Park | Thermally Triggered Degradation of Transient Electronic Devices[END_REF] to chemically degrade an electronic chip. Another study has explored a transient mechanism based on the distribution of thermally actuating material [START_REF] Banerjee | From Chips to Dust: The Mems Shatter Secure Chip[END_REF] within the silicon chip to fracture it under the stress generated. While each of these methods succeeded in either mechanically or chemically destroying the silicon chip, they present severe limitations on the time scale needed to achieve destruction (minutes), on the input energy required to render the device totally inoperable and on required restrictions in term of chip design. Other alternative works have considered the heat generated by the exothermic reaction of an energetic material to mechanically destroy an electronic device [START_REF] Fleck | Controlled Substrate Destruction Using Nanothermite[END_REF][START_REF] Pandey | An Exothermal Energy Release Layer for Microchip Transience[END_REF][START_REF] Pandey | Self-Destructing Secured Microchips by On-Chip Triggered Energetic and Corrosive Attacks for Transient Electronics[END_REF][START_REF] Ma | Additive-Free Energetic Film Based on Graphene Oxide and Nanoscale Energetic Coordination Polymer for Transient Microchip[END_REF]. A few milligrams of energetic layers, such as Al/Bi 2 O 3 nanothermites, were found to be effective in fracturing silicon wafers, whereas the same amount of Al/CuO failed [START_REF] Fleck | Controlled Substrate Destruction Using Nanothermite[END_REF]. In [START_REF] Pandey | An Exothermal Energy Release Layer for Microchip Transience[END_REF], a thick nanothermite-based film (Al/CuO powder mixed with gasoline and benzene) was spin coated on top of a microchip during the microfabrication steps. Authors observed the effect of thermite-enabled destruction prior to and after ignition through microscopic imaging and electrical measurements. This work provides conclusive evidence that when ignited, the thermite layer releases enough heat to melt the surface and damage the surface bound components within 1 s, much shorter than the time required for thermally actuated mechanisms, but fails to confirm the irreversible destruction of data contained in the bulk. In a very recent work [START_REF] Ma | Additive-Free Energetic Film Based on Graphene Oxide and Nanoscale Energetic Coordination Polymer for Transient Microchip[END_REF] graphene oxide-energetic coordination polymer energetic composite was used to demonstrate a transient microchip device. Authors demonstrate the destruction of the chip within one second.

The key objective of this research is to develop and test a modular, highly responsive, safe (no sensitive material such as Al/Bi 2 O 3 ) device that acts as an add-on module that can be placed on any IC (integrated circuit) circuit, off-the-shell component, or memory card without requiring any specialized chip design, substrate type or encapsulation solution. This so-called ultimate security device (USD) compatible with an embedded system environment can be actuated in less than 100 microseconds to mechanically disintegrate any types of components containing sensible data upon the detection of intrusion into the system. The actuation mechanism is provided by using a printed copper ammine complex-based energetic composite [START_REF] Wu | New coordination complexes-based gas-generating energetic composites[END_REF][START_REF] Wu | Enhanced Reactivity of Copper Complex-Based Reactive Materials via Mechanical Milling[END_REF]; this material was chosen as it is a source of reliable, safe and "dormant" energy, exhibiting a long shelf life (decades) and able to very quickly deliver gas and heat through a self-sustained redox reaction. Whereas the Al/CuO thermite generates a very limited amount of gas, the addition of copper ammine complex greatly enhances the pressurization rate and gas generation so that to better achieve the component destruction.

Previous studies have demonstrated the capabilities of nanothermite-based energetic films to provide high-energy density actuations to fracture silicon substrates [START_REF] Glavier | Nanothermite/RDX-Based Miniature Device for Impact Ignition of High Explosives[END_REF] or disconnect circuitry [START_REF] Nicollet | Fast circuit breaker based on integration of Al/CuO nanothermites[END_REF] while being safe and stable until 250 °C [START_REF] Esteve | How Thermal Aging Affects Ignition and Combustion Properties of Reactive Al/CuO Nanolaminates: A Joint Theoretical/Experimental Study[END_REF][START_REF] Wu | Unexpected Enhanced Reactivity of Aluminized Nanothermites by Accelerated Aging[END_REF][START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF], but none have demonstrated the irreversible destruction of data contained in the bulk silicon memory. Herein, we specifically design and elaborate a printable copper ammine complex-nanothermite composite featuring a higher pressurization capability than nanothermite powder to produce fast and highly efficient mechanical actuation. The paper is organized as follows: Section 2 presents the USD concept, operation principle and design. Section 3 describes the thermitebased energetic composite synthesis and printing and the fabrication of each layer of the USD.

Finally, Section 4 presents an ensemble of miniaturized USD test demonstrators (~9.5 cm 3 ) with varying energetic mass loading mounted onto a SD memory card. Tests demonstrate the irreversible physical destruction of the silicon chip, and, thus, all readable data in the memory for an energetic composite mass of greater than 300 mg.

USD operation principle and design

The USD operates autonomously in response to an order-related signal received from a control unit, as illustrated in [START_REF] Pandey | An Exothermal Energy Release Layer for Microchip Transience[END_REF]. Upon the detection of an intrusion, the control unit sends a key code. If the code does correspond to the trigger logic for the activation of the USD operation, the control unit activates the USD initiation through an electrical command signal sent to the initiation module, which then triggers the charging of a capacitor connected to a power supply (power unit) followed by its discharge to the USD initiator that triggers the energetic composite reaction, leading to the destruction process.

The methods used to implement hardware protection using key-based obfuscation have already been developed and well documented in the literature [START_REF] Chakraborty | HARPOON: An Obfuscation-Based SoC Design Methodology for Hardware Protection[END_REF][START_REF] Chakraborty | Security Against Hardware Trojan Attacks Using Key-Based Design Obfuscation[END_REF]. That is why in this paper, we will focus on the novel aspects, USD initiation unit and destruction mechanisms.

The main active part of the USD, i.e., to realize the initiation and chip destruction consists of multiple successive layers, as represented in the schematic diagram shown in Figure 1a: -The nanothermite/copper ammine-based energetic composite is ink-printed onto the device for protection. The mass of the deposited composite varies from 100 to 600 mg as a function of the energy required for the irreversible destruction of the component. Details for the elaboration and deposition of the energetic nanocomposite are given in Section 3.

-The initiation circuitry integrates a pyroMEMS [START_REF] Salvagnac | Layered Al/CuO Thin Films for Tunable Ignition and Actuations[END_REF][START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF][START_REF] Pouchairet | PyroMEMS as Future Technological Building Blocks for Advanced Microenergetic Systems[END_REF][START_REF] Zapata | Self-propagating combustion of sputter-deposited Al/CuO nanolaminates[END_REF] to trigger the reactive composite combustion upon electronic order. The pyroMEMS consists of a stack of 15 Al/CuO bilayers deposited [START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF] onto a thin-film resistive filament. When a current is applied to the resistive filament, the Al/CuO reactive film is ignited by the Joule effect and reacts to produce a flame. Because pyroMEMS is manufactured using microelectronic manufacturing techniques (cf. Section 3), it can be easily interconnected with electronics, which enables control of the entire ignition process, as detailed in Section 2.1.

The operational principle of the device is depicted in Figure 1. The USD is positioned onto the sensitive component to be secured (Figure 1 (2)) and can remain in sleeping mode for years until a threat is detected; once detected and confirmed (Figure 1 (3)), the pyroMEMS is ignited in less than 100 µs and further ignites the energetic composite (Figure 1 (4)). Its combustion generates a heat and pressure impulse that bursts the target component (Figure 1 (5)).

Initiation unit design

Requirements for the initiation circuit are as follows: the pyroMEMS must be ignited with the lowest possible energy and in less than 100 µs. However, pyroMEMS must be totally safe. This latter requirement means that during the possibly long lifetime of the component operation under normal conditions, the risk of undesired ignition must be null. We have chosen to ignite the pyroMEMS through a capacitive discharge using a 10 µF capacitor (maximum voltage of 15 VDC) for its ease of integration into the electronic circuitry. Thus, the circuitry includes charge and discharge switches (Figure 2a). We use 3 MOS transistors: one NMOSFET for the discharge and one NMOSFET driven by a PMOSFET for the charging of the capacitor (Figure 2b). The trigger signal sent by the control unit opens the charge MOSFET, and another signal is sent to close the discharge MOSFET, enabling the capacitor to discharge in the pyroMEMS resistive filament (5 Ω) to trigger the energetic composite reaction. Depending on the application requirements, it is also possible to maintain the capacitor in a continuously charge state, which means that the charging switch must be kept closed to keep the capacitor always connected to the power supply. This solution reduces the USD response time by 10 µs (the time needed for charging the capacitor), but at the cost of increased energy consumption to compensate for the current leakage through both transistors and capacitors. The actual pyroMEMS used in this prototype has a 100% rate ignition success at 15 V. To be compatible with embedded systems, it was implemented with a capacitive discharge circuit, and a tension boost circuit using a LT3580 Linear technology DC/DC boost. The boost component was polarized to raise the input voltage from 5 V to 15 V. Despite this light implementation, we were still able to charge our capacitor to 15 V and maintain a 100% pyroMEMS ignition success rate. For our preliminary tests, a simplified version of the initiation circuitry was considered using only the capacitive discharge circuit powered by a 15 V lab. power supply. The circuitry was mounted onto the main PCB, as shown in Figure 6b. Inside the box, only a pyroMEMS with long connection pads was present, and the connection between the main PCB and the pyroMEMS was made via two holes in the box, leaving the pyroMEMS pads apparent.

Fabrication and assembly

The successive layers, e.g., the PyroMEMS, the energetic composite and the reservoir, were designed, fabricated as detailed in this section before being assembled and mounted onto the memory or component to be destroyed.

PyroMEMS fabrication

A 500-µm thick 4-inch glass substrate was cleaned with oxygen plasma at 800 W for 5 min to remove surface contamination. Next, negative nLof photoresist (AZ nLof 2035, MicroChem Corporation, USA) was spin-coated and patterned using photolithography. Subsequently, 350-nm-thick titanium and 300-nm-thick gold layers were evaporated onto the surface, and then a second photolithography step with positive photoresist was performed to define the Ti resistive filament and Au electrical pads by chemical etching of the gold. Finally, an ~13 mm² stack of 15 Al/CuO bilayers (~300 nm thick) was sputter deposited as described in [START_REF] Salvagnac | Layered Al/CuO Thin Films for Tunable Ignition and Actuations[END_REF][START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF][START_REF] Pouchairet | PyroMEMS as Future Technological Building Blocks for Advanced Microenergetic Systems[END_REF][START_REF] Zapata | Self-propagating combustion of sputter-deposited Al/CuO nanolaminates[END_REF] directly in contact with the Ti electrical resistance. The main process steps are summarized in Figure 3.

Energetic composite preparation and deposition

The aluminum nanopowders (Al, average particle size: 80 nm, purity: 69%) were purchased from Novacentrix (USA) and stored in a glove box before use. Copper oxide (CuO, average particle size: 100 nm), copper nitrate trihydrate, 25% aqueous ammonia, ethanol (CH 3 CH 2 OH, anhydrous, 99.9%), and polyvinylpyrrolidone (PVP, molar weight: 40k) purchased from Merck (Germany) were directly used as received. To prepare Al/CuO nanothermites, 456 mg of Al powder and 1 017 mg of CuO were dispersed in ethanol and then stirred for 45 minutes in a sonication bath cooled by ice. The suspension was then dried at 50 °C. The dry powder (referred to as Al/CuO) was collected and reserved for future use. Note that the masses of Al and CuO were calculated based on an equivalence ratio of 1.2.

The copper ammine complex (CuC), Cu(NH 3 ) 4 (NO 3 ) 2 , was synthesized and cryo-milled to reduce its size, as detailed in [START_REF] Wu | Enhanced Reactivity of Copper Complex-Based Reactive Materials via Mechanical Milling[END_REF]. In brief, 4.83 g (0.02 mol) of copper nitrate trihydrate was dissolved in 10 mL of distilled water followed by the addition of 15 mL of 25% aqueous ammonia solution (0.24 mol). The final product (micron-sized powder) was separated via vacuum filtration and dried in an oven (Carbolite Gero, England) at 70 °C. The assynthesized CuC powders were milled using a Retsch CryoMill machine using ZrO 3 grinding balls (diameter: 2.5 mm). The milling parameters were as follows: frequency of 30 Hz (30/s) for 10 min while subject to liquid-N 2 cooling. The equivalence ratio for the CuC over Al/CuO nanopowder was 25%, which was chosen to obtain the highest pressure burst and burn rate (see ref [START_REF] Wu | New coordination complexes-based gas-generating energetic composites[END_REF]). Notably, the combustion speed under unconfined conditions was characterized by a high-speed camera at ~ 141 m.s -1 .

The next step consists in preparing the energetic ink to be deposited by printing into the USD reservoir. For that purpose, 2 g of Al/CuO/CuC was dispersed with 100 mg (5% wt) of the binding polymer polyvinylpyrrolidone (PVP, average molecular weight 40 000 g mol -1 purchased from Sigma-Aldrich) in ethanol (4 ml) and then stirred for 2 hours in a sonication bath cooled by ice. With 5% wt of PVP, the energetic ink is homogeneous and stable for at least 24 h, enabling reproducible and semiautomatic deposition. The deposition time for 400 mg of Al/CuO/CuC ink is ~1 h using the semiautonomous deposition machine (Figure 4b). This polymer-coated Al/CuO/CuC ink is relatively compact with a final thin film characterized by a density of 65% of its theoretical maximum density, TMD (see Equation 1). It should be noted that without PVP binding, the maximum achievable compaction is close to 45% TMD. However, as a counterpart, being a nonenergetic binder, adding 5% wt PVP with Al/CuO/CuC alters the energetic performance: the combustion speed is reduced by ~80%, i.e., 17 m.s -1 for Al/CuO/CuC +5% wt PVP against 141 m/s without PVP, as detailed in [START_REF] Sevely | Effect of Process Parameters on the Properties of Direct Written Gas-Generating Reactive Layers[END_REF].

Figure 4a summarizes the preparation steps for the energetic composite, and Figure 4b shows the deposition bench specifically developed to deposit the ink into the USD reservoir.

It is composed of a homemade 3D printer with a 3-axis controller, a syringe driver (composed of a stepper motor, a 3D printed structure in PolyLactic Acid & a 5 ml syringe) controlled by an Arduino card and supervised by a LabVIEW program. The ink deposition is first calibrated and then controlled using the motor step. The precision of the deposition is ± 5%, i.e., ± 5 mg for the deposition of 100 mg. Figure 4c gives snapshot images of the energetic ink in the vial with a zoom (electron microscopy images) of the dried ink after deposition.

After drying, the density of the printed material reaches 2.3 g/cm 3 i.e., 65 %TMD.

Reservoir and external packaging fabrication

The USD packaging consists of two parts:

1. The external packaging (brown piece in Figure 5a) is a rectangular cap. 5a) designed to fit the dimension of the chip to be protected (SD memory for demonstrator) with two functions: block the pyroMEMS in its trench and store the energetic composite. The initiation circuitry is inserted between the external packaging and the rectangular reservoir, as illustrated in Figures 5a-b.

A rectangular reservoir (violet piece in Figure

On both sides of the packaging, holes permit electrical connections of the initiation circuitry with the printed board circuitry containing the target component, as described in the next section. Both parts were 3D printed in Nylon (Nylon, Ultimaker, Netherlands) using a 3D printer (Ultimaker S5), as shown in Figure 5b. 3D printing technology was chosen because it offers a good resolution (± 6.9 µm) while being low cost and fast for prototyping a custom-made device. In addition, nylon offers a good compromise between cost and thermal and mechanical resistance (Table 1).

Test device assembly and tests

Ten USD demonstrators were designed and fabricated to validate the concept. An off-theshelf SD memory card (Transcend SDHC memory Card 8GB, Taiwan) was chosen as the component to be destroyed for the demonstration tests. It offers the advantage of being easily readable and programmable with a computer. Its packaging is composed of 500 µm thick epoxy resin reinforced with microsilica balls and fireproof composite, making it resistant to heat and pressure. Therefore, we first chemically etch the packaging to open an ~5 × 10 mm² squared window into the packaging to access the silicon microchip (Figure 6a), taking care to not damage the internal wire bonding and electrical connections or the passive components and the silicon chip that constitute the memory. The prepared SD card is then checked, programmed, mounted and connected to the printed board circuitry (PCB) serving for the test, as shown in Figure 6b. A simplified initiation unit and PCB designed with a USB drive shape are used to demonstrate the destruction of the SD card. One side of the PCB contains the charging and discharging switches with the initiation trigger switch simulating the electrical signal sent by the order unit upon intrusion detection. The SD card is glued onto the PCB, and its contact pads are attached onto the back side of the PCB, enabling the SD to be connected to a computer.

In parallel, a mass of energetic composite ranging from 100 mg to 600 mg is printed into the nylon reservoir (volume of 268 mm 3 ) containing one pyroMEMS and sealed with a cellophane thin film (Figure 7c). Then, the external packaging (in brown in Figure 5a) is glued on the main PCB using a bi-component epoxy glue (3M Scotch-Weld Epoxy Adhesive

Destruction tests

Height demonstrators similar to those shown in Figure 6d were assembled with different masses of energetic composites varying from 13 mg to 600 mg, corresponding to different compaction rates expressed in percentage of the TMD (noted %TMD, Equation 1): 13 mg (%TMD 45%) and 100 mg (%TMD 11%) to 600 mg (%TMD 64%).

M mass of reactive composite

M vol me of the reservoir [START_REF] Mcloughlin | Secure embedded systems: the threat of reverse engineering[END_REF] where TMD is the theoretical maximum density of the reactive composite, equal to 3.51 g/cm 3 for our Al/CuO/CuC composition. The volume of the reservoir is 268 mm 3 .

For each mass, 3 tests were performed. After the ignition and combustion of the copper ammine complex-based energetic composite, the demonstrators are opened, and SD cards are observed visually using a Microscope Hirox RX-100 with ×20 and ×140 magnification.

Pictures were taken using a multifocus technique enabling us to observe rifts/trenches in a fully reconstituted 3D picture. The results are summarized in Figure 8 for configurations with energetic composite masses of 200 mg (a), 300 mg (b), 400 mg (c) and 600 mg (d). Each demonstrator was ignited following the procedure detailed in subsection 4.1. Finally, the USD was disassembled to characterize the state of the chip. The results are given as snapshot images of the SD cards after the tests: (i) non-magnified, (ii) ×20 magnification, (iii) and (iv) ×140 magnification. We observe that with a 600 mg load of energetic composite, the SD card is totally destroyed: a part of the packaging is torn out, and most of the silicon chip is disintegrated. When not disintegrated, the silicon features rifts and holes (Figure 8d (iii) and (iv)). With 400 mg loading, some parts of the silicon chip are disintegrated, and rifts run across the chip.

Although the damage is less intense compared to the case of a 600 mg reactive composite, the SD is sufficiently damaged to guarantee total destruction of the data. With 300 mg, some cracks appear on the silicon chip, and some shards are torn off (Figure 8b (iii) and (iv)). The silicon surface is covered by a superficial scratch, but the silicon chip remains unbroken as well as the data. With a less reactive composite, i.e., < 200 mg in mass, the silicon surface remains completely intact. The reaction damages only the wire bonding.

These results validate the good functioning of our USD module, as we can destroy the silicon component with all the data in it for an energetic composite mass of greater than 400 mg in less than 1 ms.

This work reports the successful implementation of a triggerable, modular and fast ultimate security device capable of destroying a silicon chip containing sensible data in tens of milliseconds. making the data completely unreadable. This destruction is possible thanks to the highly exothermic reaction of a solid-state energetic composite composed of a mixture of Al/CuO nanothermite with an ammine copper complex densely printed onto the chip to be protected. This energetic layer is ignited by capacitance discharge in less than 100 µs. We demonstrate that 400 mg of energetic composite enables irreversible destruction of one silicon chip (~118 mm 3 ) in less than 10 ms. The outcome of this work also outlines the fact that the reaction time and efficiency are highly dependent on the energetic composite loading, i.e., the mass of CuC/nanothermite printed into the device in contact with the memory card.

Importantly, this ultimate security device is a versatile solution that can be implemented for any tunable mechanical actions for release, severance/fracture, jettison, switching, time delay, and actuation. Therefore, it can be successfully implemented for multiple relevant emergency safety responses, which clearly responds to the crucial need to fight against data theft and reverse engineering.

Figure 1 .

 1 Figure 1. Conceptual illustration of the USD operation principle: (a) exploded view of an USD device composed of 2 stacked parts: the initiation circuitry with a pyroMEMS and the copper ammine/nanothermitebased energetic composite printed in a reservoir (inner packaging). (b) Illustration of the different steps of operation, from the detection to the destruction mechanisms.

Figure 2 .

 2 Figure 2. Design of the initiation unit: (a) schematic of the capacitive discharge circuit and (b) electronic circuitry with the discharge transistor (NMOSFET) shown in green and the charge transistors (NMOSFET + PMOSFET) shown in purple.

Figure 3 .

 3 Figure 3. 3D schematic views of the main pyroMEMS fabrication steps: (a) Ti and Au deposition on a glass wafer, (b) etching of Au to define the electrical tracks and Ti resistance, (c) sputter-deposition of the Al/CuO multilayered reactive film, and (d) real view snapshot.

  Then, 1473 mg of Al/CuO was mixed with 427 mg of milled CuC powder to produce energetic nanocomposites, referred to as Al/CuO/CuC. The exothermic self-sustained reaction of Al/CuO/CuC occurs via several steps upon heating:

Figure 4 .

 4 Figure 4. Energetic ink preparation and deposition: (a) schematic of the copper ammine complex and nanothermite ink preparation. (b) Photograph of the homemade 3D printer with the syringe driver pushing the syringe filled with energetic ink and the 3-axis controller.

Figure 5 .

 5 Figure 5. The USD packaging: (a) schematic of the 3 main parts of the USD with the external packaging in brown, the pyroMEMS in blue, the reservoir in which the energetic composite is printed in purple, and (b) a photo of the reservoir with the initiation circuit inside.

  2216) which permits to keep the mechanical integrity of the USD assembly during the operation. The final step involves the electrical connection of the pyroMEMS electrical pads with the initiation circuitry electrical connections.

Figure 6 .

 6 Figure 6. Photographs of (a) the SD card after being locally etched to reveal the silicon microchip that stores the sensible data, (b) the prepared SD card glued onto the demo PCB featuring a simple capacitive discharge circuit, (c) reservoir with printed energetic composite, (d) an assembled test device for demonstration tests, and (e) schematic of the demonstration test configuration.

Figure 7 .

 7 Figure 7. Ignition characterization: (a) electrical command sequence and capacitor discharge. (b) Photograph of the reaction of one pyroMEMS.

Figure 8 .

 8 Figure 8. Photographs of SD cards after tests with energetic composite masses of (a) 200 mg or 90 mm 3 , (b) 300 mg or 130 mm 3 (c) 400 mg or 170 mm 3 (d) 600 mg or 268 mm 3 and with their insert (i) a global photograph of the SD card, (ii) microscope imaging with ×20 magnification and (iii/iv) microscope imaging with ×140 magnification.

Table 1 .

 1 Main physical and thermal characteristics of the nylon used to fabricate the packaging and reservoir.

	Tensile	Tensile stress	Tensile stress	Flexural	Flexural	Izod impact	Melting
	modulus	at yield	at break	strength	modulus	strength	temperature

https://ultimaker.com/materials/nylon

Acknowledgements

The authors acknowledge support from the European Research Council (H2020 Excellent Science) Researcher Award (grant 832889 -PyroSafe). This work was also supported by LAAS-CNRS technology platform, a member of Renatech network. We acknowledge the French Defense Agency which partially funds F.S. scholarship. The authors also acknowledge the help from Pierre Burgaud and Julien Nicolod in opening commercial SD memories packaging.


Nicolas Vanspranghe

Francesco Ferrante
email: francesco.ferrante@unipg.it.

Christophe Prieur

Stabilization of the wave equation through nonlinear Dirichlet actuation *

Keywords: Wave equation, boundary stabilization, saturating feedback, asymptotic stability, polynomial decay rates. AMS subject classifications. 35L05, 93C20, 93D15, 93D20

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L 2 (Ω) × H -1 (Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H 1 (Ω) × L 2 (Ω) of the wave equation with Neumann boundary conditions.

Introduction

Let Ω ⊂ R d (d 2) be a bounded domain with smooth boundary Γ. Given a relatively open nonempty subset Γ0 of Γ, we consider the wave equation subject to non-homogeneous Dirichlet boundary conditions: ∂ttu(x, t) -∆u(x, t) = 0 in Ω × (0, +∞), (

u |Γ (σ, t) = -g(U (σ, t)) on Γ0 × (0, +∞), (1.1b) u |Γ (σ, t) = 0 on Γ1 × (0, +∞), (1.1c)

where Γ1 Γ \ Γ0, U represents a control input, and g is a real function fulfilling the following assumption.

Assumption 1. The scalar mapping g satisfies the following properties:

(i) g is globally Lipschitz continuous and nondecreasing;

(ii) g(s) = 0 if and only if s = 0.

Background. The general problem of this paper is the feedback stabilization of the control system (1.1) in presence of a static pointwise nonlinearity g. Consider the velocity feedback

U (σ, t) = -∂ν[A -1 u ′ ](σ, t), (1.2)
where u ′ is the time derivative of u, ∂ν denotes the outward normal derivative, and A -1 is the inverse of the positive "minus Laplacian with homogeneous Dirichlet boundary conditions" operator. The corresponding linear feedback system (i.e., when g is the identity) was first introduced by Lasiecka and Triggiani in [START_REF] Lasiecka | Uniform exponential energy decay of wave equations in a bounded region with L2(0, +∞; L2(Γ))-feedback control in the Dirichlet boundary conditions[END_REF], with initial data lying in the energy space

H L 2 (Ω) × H -1 (Ω).
(1.

3)

The choice of state space is motivated by optimal regularity results for second-order hyperbolic equations with Dirichlet boundary data in L 2 (0, T ; L 2 (Γ)) -see [START_REF] Lasiecka | Non homogeneous boundary value problems for second order hyperbolic operators[END_REF]. It was proved in [START_REF] Lasiecka | Uniform exponential energy decay of wave equations in a bounded region with L2(0, +∞; L2(Γ))-feedback control in the Dirichlet boundary conditions[END_REF] that the linear version of (1.1)-(1.2) gives rise to an exponentially stable semigroup of operators on H under the assumption that the whole boundary is actuated (i.e., Γ = Γ0) and that Ω satisfies suitable geometrical conditions. The proof relies on the analysis of a new variable p defined as p A -1 u ′ (1.4) which is smoother and solves a wave-type equation as well. The result was later refined by the same authors in [START_REF] Lasiecka | Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions[END_REF] where feedback acting only on a subset of the boundary is allowed and, most importantly, specific geometrical conditions related to the analysis of the p-variable by multipliers are relaxed. This was achieved by the mean of another change of variable operating at the level of pseudodifferential calculus. In short, after transposing problem (1.1)-(1.2) to the half-space via partition of unity and truncating the solution with respect to the time variable, one defines a new variable w by F[w](ξ, ω; x) = λ(ξ, ω)F[u](ξ, ω; x), ξ ∈ R d-1 , ω ∈ R, x 0, (1.5)

where F denotes the Fourier transform in both tangential and time variables and λ is a carefully constructed symbol. While transformations (1.4) and (1.5) are quite different in nature, both enable computations on variables with H 1 (Ω) × L 2 (Ω)-regularity.

As far as we know, there has been no attempt to extend the stability analysis of the closed-loop system (1.1)-(1.2) to the nonlinear case. Yet, one can see problem (1.1)-(1.2) as a natural Dirichlet counterpart to the wave equation with nonlinear Neumann boundary dissipation ∂ttu(x, t) -∆u(x, t) = 0 in Ω × (0, +∞) (1.6a) ∂νu(σ, t) = -g(∂tu(σ, t)) on Γ0 × (0, +∞), (1.6b)

u |Γ (σ, t) = 0 on Γ1 × (0, +∞), (1.6c) which, in contrast, have been extensively studied in the literature. To cite only a few, when the nonlinearity g has linear growth at infinity, uniform decay of the H 1 (Ω) × L 2 (Ω)-energy of solutions to (1.6) can be achieved, as in [START_REF] Zuazua | Uniform stabilization of the wave equation by nonlinear boundary feedback[END_REF] or [START_REF] Lasiecka | Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping[END_REF] -see also [START_REF] Komornik | Exact controllability and stabilization: the multiplier method[END_REF] and the references therein, or more recently [START_REF] Daoulatli | Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions[END_REF]. In the onedimensional settings, arguments based on Riemann invariants are available, and the decay of the energy can be analyzed via appropriate iterated sequences. See for instance [START_REF] Chitour | One-dimensional wave equation with setvalued boundary damping: well-posedness, asymptotic stability, and decay rates[END_REF], where g is allowed to be a multivalued monotone mapping, or [START_REF] Vancostenoble | Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks[END_REF], where it is proved, in particular, that exponential or polynomial uniform decay cannot be achieved when g represents a pointwise saturation mapping -see also [START_REF] Xu | Saturated boundary feedback stabilization of a linear wave equation[END_REF] or [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] for a stability analysis in the saturated case.

Outline of the paper and contributions. This paper aims at bridging the gap between Neumann and Dirichlet boundary conditions as far as nonlinear boundary stabilization is concerned. First, we prove that the nonlinear dynamics (1.1)-(1.2) generate a strongly continuous semigroup of contractions on the energy space H (Theorem 2.1) that is globally asymptotically stable around the zero equilibrium (Theorem 2.2). The proof relies on LaSalle's invariance principle and unique continuation for the wave equation.

Next, having in mind the more specific problem of saturated boundary stabilization, in Section 3, we work under the assumption that g has linear growth around zero (see Assumption 2 below). Then, by analogy with the Neumann case, we focus on non-uniform decay rates for solutions with "smooth" initial data. We establish a polynomial decay rate for smooth solutions (Theorem 3.1) that holds under standard geometrical conditionssee Assumption 3 below, which is however always satisfied when Γ0 = Γ, i.e., the whole boundary is actuated.

To do so, we consider the change of variable (1.4) and we derive appropriate integral inequalities using suitable multipliers. Note that the question of uniform stability when g has linear growth at infinity is out of the scope of the paper -this is discussed in Section 4 below. Throughout the paper, one can think of the "hard" saturation mapping satS with threshold S > 0, defined by satS(s)

   s if |s| S, S s |s| otherwise, (1.7)
as a prototype nonlinearity satisfying Assumptions 1 and 2, highlighting the fact that no differentiability condition on g is prescribed.

Notation and elements from elliptic theory. We end this section by introducing some notation and recalling useful results from elliptic theory.

First, if H is a given Hilbert space, we denote by • H its norm, and its scalar product is written (•, •)H . For T > 0, we denote by W 1,p (0, T ; H) the subspace of L p (0, T ; H) composed of (classes of) H-valued functions φ such that, for some ξ in H and ψ in L p (0, T ; H), φ(t) = ξ + t 0 ψ(s) ds almost everywhere in (0, T ). Such a class φ is identified with its continuous representative and we say that φ ′ = ψ in the sense of H-valued distributions. Note that vector-valued integrals are intended in the sense of Bochner. Also, the space of bounded linear operators between two normed spaces E and F is denoted by L(E, F ).

In this paper, all scalar functions are real-valued. The notation dx indicates the standard Lebesgue measure on R d while dσ denotes the induced surface measure on Γ. By H s (Ω) (resp. H s (Γ)) we denote the L 2 (Ω)-based (resp. L 2 (Γ)-based) Sobolev space of order s. The space of compactly supported and infinitely differentiable functions on Ω is written C ∞ c (Ω). We also recall that H 1 0 (Ω) is defined as the closure of C ∞ c (Ω) in H 1 (Ω). Furthermore, H -1 (Ω) is the topological dual of H 1 0 (Ω). The unbounded operator A can be defined 1 as follows: having set

D(A) H 2 (Ω) ∩ H 1 0 (Ω), (1.8)
we let Au -∆u ∈ L 2 (Ω) for all u ∈ D(A). Then, A is a closed strictly positive self-adjoint operator on L 2 (Ω). Its dense domain D(A) is equipped with the norm A • L 2 (Ω) , which is equivalent to the norm induced by H 2 (Ω). The operator A possesses fractional powers A s , s ∈ R -see for instance [Tem97, Chapter II, Section 2.1]. Those are also strictly positive self-adjoint operators. For s 0, D(A s ) are dense subsets of L 2 (Ω), which we equip with the norm A s • L 2 (Ω) . In particular, we have

D(A 1/2 ) = H 1 0 (Ω), with ∇w 2 L 2 (Ω) d = A 1/2 w 2 L 2 (Ω) = w 2 H 1 0 (Ω)
for all w ∈ H 1 0 (Ω).

(1.9)

Then, we let D(A -s ) D(A s ) ′ and we can extend A s as an isomorphism between L 2 (Ω) and D(A -s ). Here, H -1 (Ω) is equipped with the scalar product

(v1, v2) H -1 (Ω) (A -1/2 v1, A -1/2 v2) L 2 (Ω) ,
(1.10) which induces a norm equivalent to the dual one; we also recover D(A -1/2 ) = H -1 (Ω). Throughout the paper, we use the following chain of continuous embeddings:

D(A) ֒→ H 1 0 (Ω) ֒→ L 2 (Ω) ֒→ H -1 (Ω) ֒→ D(A -1 ). (1.11)
Finally, we define the Dirichlet map D, which is a continuous right inverse for the trace. For any f in H 1/2 (Γ), there exists a unique u in H 1 (Ω) solving -∆u = 0 and u |Γ = f ; and we let Df u. The mapping D defined in this way is continuous from H 1/2 (Γ) to L 2 (Ω) and can be extended as a continuous operator from L 2 (Ω) into L 2 (Γ). We define its adjoint D * by (D * u, f ) L 2 (Γ) = (u, Df ) L 2 (Ω) for all u in L 2 (Ω) and f in L 2 (Γ). Extensions on fractional Sobolev spaces are denoted with the same symbols:

D ∈ L(H s (Γ), H s+1/2 (Ω)), D * ∈ L(H s (Ω), H s+1/2 (Γ)) for all s ∈ R.
(1.12)

2 Well-posedness and asymptotic stability

In this section, we give the operator-theoretic formulation of the evolution problem (1.1a) with initial data in L 2 (Ω) × H -1 (Ω) and feedback control (1.2). After that, we state and prove the well-posedness and asymptotic stability properties of the feedback system (1.1)-(1.2).

Operator model and well-posedness

We shall recast the closed-loop evolution equations (1.1)-(1.2) into a first-order abstract Cauchy problem on the energy space H and state well-posedness results by relying on nonlinear semigroup theory. With a little abuse of notation, we denote by g the nonlinear Lipschitz mapping on L 2 (Γ) defined by g(f )(σ) g(f (σ)) for any f in L 2 (Γ). We also define a projection operator P on L 2 (Γ) by [P f ](σ) = 1Γ 0 (σ)f (σ) for any f ∈ L 2 (Γ). From the Green formula, it follows that -D * Ap = ∂νp for all p ∈ D(A).

(2.1) Therefore, the boundary conditions associated with the feedback law (1.2) can be rewritten as follows:

u |Γ = -P g(D * u ′ ). (2.2)
Next, we introduce the nonlinear operator A associated with the closed-loop system (1.1)-(1.2). Recalling the chain of embeddings (1.11) and that A maps L 2 (Ω) onto D(A -1 ), we define A by

D(A) [u, v] ∈ H : v ∈ L 2 (Ω), A[u + DP g(D * v)] ∈ H -1 (Ω) (2.3a) A[u, v] [-v, Au + ADP g(D * v)]. (2.3b) Equivalently, D(A) is the set of all [u, v] in L 2 (Ω) × L 2 (Ω) such that u + DP g(D * v) belongs to H 1 0 (Ω).
Note that the feedback law (1.2) appears as a natural choice when (formally) differentiating the energy functional

E (u, v) 1 2 { u 2 L 2 (Ω) + v 2 H -1 (Ω) }, [u, v] ∈ H, (2.4)
along "trajectories" of the open-loop system (1.1). Indeed, this leads to the energy identity

d dt E (u, u ′ ) = Γ 0 g(U (t))∂ν[A -1 u ′ ] dσ (2.5)
and since g satisfies Assumption 1, we see that (1.2) renders the energy E nonincreasing along the trajectories.

In the sequel, we employ the standard nonlinear semigroup terminology: by a strong solution to (1.

d dt E (u, u ′ ) = - Γ 0 g(D * u ′ )D * u ′ dσ = Γ 0 g(-∂ν[A -1 u ′ ])∂ν[A -1 u ′ ] dσ (2.8)
in the scalar distribution sense on (0, +∞).

Remark 2.1. If we also assume that, say, |g(s)| α|s| for all s ∈ R and some α > 0, the energy identity (2.8) provides a uniform estimate of the L 2 (0, +∞; L 2 (Γ0))-norm of ∂ν [A -1 u ′ ] for strong solutions. From there, one can prove that (2.8) holds for generalized solution as well by passing to the limit and recovering the traces u |Γ and ∂ν[A -1 u ′ ] in L 2 (0, +∞; L 2 (Γ0)) -see for instance [START_REF] Chueshov | On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation[END_REF] for similar arguments in the Neumann case.

Proof of Theorem 2.1. Once proven that A is maximal monotone, existence and uniqueness of strong and generalized solutions to (1.1)-(1.2), together with the appropriate semigroup properties, follow from Kato's theorem and standard nonlinear semigroup theory -see, e.g., [START_REF] Edwin | Monotone operators in Banach space and nonlinear partial differential equations[END_REF]Chapter IV].

Step 1:

Monotonicity. Let [u1, v1] and [u2, v2] in D(A). Then, (A[u1, v1] -A[u2, v2], [u1, v1] -[u2, v2])H = -(v1 -v2, u1 -u2) L 2 (Ω) +(A 1/2 [u1 -u2 + DP g(D * v1) -DP g(D * v2)], A -1/2 [v1 -v2]) L 2 (Ω) .
(2.9)

Now we use that

A -1/2 [v1 -v2] belongs to D(A 1/2 ) and that A 1/2 is self-adjoint to obtain (A[u1, v1] -A[u2, v2], [u1, v1] -[u2, v2])H = (DP g(D * v1) -DP g(D * v2), v1 -v2) L 2 (Ω) = (P g(D * v1) -P g(D * v2), D * v1 -D * v2) L 2 (Γ) = (g(D * v1) -g(D * v2), D * v1 -D * v2) L 2 (Γ 0 ) 0,
(2.10) the right-hand side being nonnegative by nondecreasingness of g, which proves that A is monotone.

Step 2: Range condition. Let λ > 0 and [f1, f2] ∈ H. To solve the equation

A[u, v] + λ[u, v] = [f1, f2], it suffices to find v ∈ L 2 (Ω) such that λ -1 v + DP g(D * v) + λA -1 v = A -1 f2 -λ -1 f1.
(2.11) This is seen by substituting -v + λu = f1 into the second coordinate of the equation and applying A -1 to the result. If such an element v ∈ L 2 (Ω) is found, then [u, v] belongs to D(A) (and solves the desired equation). Indeed, we then have u + DP g(D * v) + λA -1 v = A -1 f2, which implies that u + DP g(D * v) ∈ H 1 0 (Ω) since A -1 f2 and λA -1 v both belong to H 1 0 (Ω). We define a nonlinear operator Θ on L 2 (Ω) by Θ(v) λ -1 v + DP g(D * v) + λA -1 v for all v ∈ L 2 (Ω). Then, Θ enjoys the following properties:

(i) Θ maps bounded sets into bounded sets;

(ii) (Θ(v1) -Θ(v2), v1 -v2) L 2 (Ω) 0 for all v1 and v2 in L 2 (Ω); (iii) The scalar function t → (Θ(v1 + tv2), v2) L 2 (Ω) is continuous for all v1 and v2 in L 2 (Ω). Also, we have (Θ(v), v) L 2 (Ω) λ -1 v 2 L 2 (Ω)
for all v ∈ L 2 (Ω).

(2.12) Thus, it follows from [Sho13, Lemma 2.1 and Theorem 2.1] that Θ is onto. Consequently, the equation

A[u, v] + λ[u, v] = [f1, f2] has a solution in D(A).
Step 3: Denseness of the domain.

Let [u, v] ∈ H and ǫ > 0. Since A -1 v ∈ H 1 0 (Ω) and C ∞ c (Ω) is dense in H 1 0 (Ω), we can pick φ ∈ C ∞ c (Ω) such that A -1 v -φ 2 H 1 0 (Ω)
ǫ, and thus v -Aφ 2

H -1 (Ω) Cǫ (2.13) where C > 0 comes from A ∈ L(H 1 0 (Ω), H -1 (Ω)). Besides, there exists ψ ∈ C ∞ c (Ω) such that u -ψ 2 L 2 (Ω) ǫ. Since φ ∈ C ∞ c (Ω) ⊂ D(A), we have Aφ ∈ L 2 (Ω) and also, using (2.1), g(D * Aφ) = g(-∂νφ) = 0. Thus, [ψ, Aφ] ∈ D(A); also, we have [u, v] -[ψ, Aφ] 2 H (1 + C)ǫ. It is now proved that D(A) is dense in H. Step 4: Energy identity. Let [u, v] be a strong solution to (1.1)-(1.2). We recall that E (u, u ′ ) = 1 2 [u, u ′ ] 2 H .
Consequently, by the chain rule, E (u, u ′ ) is an absolutely continuous scalar function and

d dt E (u, u ′ ) = (-A[u, u ′ ], [u, u ′ ])H a.e.
(2.14) Thus, the desired identity (2.8) follows from (2.10).

Additional properties of the semigroup

Now, we establish some compactness and regularity properties that are useful in the proof of the stability results presented in Subsections 2.3 and 3.1. We start by introducing the following proposition, which enables us to prove asymptotic stability of the feedback system (1.1)-(1.2) using LaSalle's invariance principle.

Proposition 2.1 (Compactness). For any λ > 0, the (nonlinear) resolvent operator (A + λid) -1 is well-defined on H and compact. In particular, for all initial data [u0, v0] ∈ H, the (semi)trajectory {St[u0, v0]} t 0 is relatively compact in H.

Proof. Assume for a moment that (A + λid) -1 is well-defined and compact for some λ > 0. Then, since A(0) = 0, relative compactness of the trajectories follows from [DS73, Theorem 3]. Let λ > 0. We already know from the proof of Theorem 2.1 that the equation

A[u, v] = [f1, f2] (2.15)
has a solution in D(A) for all [f1, f2] ∈ H.

Step 1: Uniqueness. Consider two solutions [u1, v1] and [u2, v2] to (2.15). Then, we recall from (2.11) in the proof of Theorem 2.1 that

λ -1 [v1 -v2] + DP [g(D * v1) -g(D * v2)] + λA -1 [v1 -v2] = 0.
(2.16)

Taking the scalar product in L 2 (Ω) of (2.16) with v1 -v2 yields

λ -1 v1 -v2 2 L 2 (Ω) + (g(D * v1) -g(D * v2), D * v1 -D * v2) L 2 (Γ 0 ) + λ v1 -v2 2 H -1 (Ω) = 0.
(2.17)

In particular, since g is nondecreasing, we infer from (2.17) that v1 = v2; thus, [u1, v1] = [u2, v2] and (A + λid) -1 is well-defined.

Step 2: Compactness of the resolvent operator. In what follows, we let [u, v] (A + λid) -1 [f1, f2] and we look for estimates of [u, v] ∈ D(A) in stronger norms. First, as in the previous step, we obtain

λ -1 v 2 L 2 (Ω) + (g(D * v), D * v) L 2 (Γ 0 ) + λ v 2 H -1 (Ω) = (A -1/2 f2, A -1/2 v) L 2 (Ω) -λ -1 (f1, v) L 2 (Ω) , (2.18)
where it is used that A -1/2 is self-adjoint. From (2.18), using Cauchy-Schwarz and Young inequalities with appropriate constants, we obtain the estimate

v 2 L 2 (Ω) 1 2 f2 2 H -1 (Ω) + f1 2 L 2 (Ω) . (2.19)
The remainder of the proof relies on elliptic regularity theory and in particular [LM61, Théorème 10.1]. Since [u, v] ∈ D(A), we know that u + DP g(D * v) belongs to H 1 0 (Ω) and

u + DP g(D * v) = A -1 f2 -λA -1 v.
(2.20)

Picking an arbitrary test function φ in C ∞ c (Ω) ⊂ D(A), taking the scalar product in L 2 (Ω) of (2.20) with Aφ and using again that D * Aφ = -∂νφ = 0 leads to -∆u = f2λv in the sense of distributions on Ω. Besides, since -∆u ∈ H -1 (Ω), u |Γ is well-defined in H -1/2 (Γ); then, we infer from u + DP g(D * v) ∈ H 1 0 (Ω) that u |Γ = -P g(D * v) ∈ L 2 (Γ). Applying the aforementioned theorem, we obtain u ∈ H 1/2 (Ω) along with the estimate

u 2 H 1/2 (Ω) C1 P g(D * v) 2 L 2 (Γ) + f2 -λv 2 H -1 (Ω) (2.21)
where C1 > 0 is solution independent. Since P and g are Lipschitz continuous on L 2 (Γ), g(0) = 0 and

D * is linear continuous from L 2 (Ω) into L 2 (Γ), plugging (2.19) into (2.21) yields u 2 H 1/2 (Ω) C1C2 v 2 L 2 (Ω) + 2C1λ 2 v 2 H -1 (Ω) + 2C1 f2 2 H -1 (Ω) , (2.22)
where C2 > 0 is some other constant. Combining (2.19) and (2.22), we see that (A + λid) -1 maps bounded sets of

H = L 2 (Ω) × H -1 (Ω) into bounded sets of H 1/2 (Ω) × L 2 (Ω)
, the latter being compactly embedded into H. Thus, the result is proved.

The next proposition is meant for use in Section 3, where we work under additional assumptions on Ω; however, since it is a direct continuation of the proof of Proposition 2.1, we introduce it here.

Proposition 2.2 (Regularity). Suppose that Γ0 ∩ Γ1 = ∅. Then, the following explicit characterization of D(A) holds:

D(A) = [u, v] ∈ H : v ∈ L 2 (Ω), u ∈ H 1 (Ω), u |Γ = -1Γ 0 g(D * v) .
(2.23)

Thus, strong solutions [u, u ′ ] take values in H 1 (Ω) × L 2 (Ω).
Furthermore, there exists a constant K > 0 such that any strong solution to

(1.1)-(1.2) satisfies [u(t), u ′ (t)] H 1 (Ω)×L 2 (Ω) K A[u(0), u ′ (0)] H for all t 0. (2.24)
Remark 2.2. Since H 1 (Ω) × L 2 (Ω) is continuously embedded into H, it follows from (2.24) evaluated at t = 0 that for some constant K ′ > 0,

E (u0, v0) K ′ A[u0, v0] 2 H for all [u0, v0] ∈ D(A). (2.25) Proof of Proposition 2.2. Let u ∈ H 1 (Ω) and v ∈ L 2 (Ω) such that u |Γ = -1Γ 0 g(D * v) = -P g(D * v).
By trace regularity, P g(D * v) ∈ H 1/2 (Γ), and by (1.12), DP g(

D * v) ∈ H 1 (Ω). It follows that u + DP g(D * v) ∈ H 1 0 (Ω), i.e., [u, v] ∈ D(A).
Conversely, let [u, v] ∈ D(A). Recalling calculations made in Proposition 2.1, we already know that [u, v] must satisfy -∆u ∈ H -1 (Ω) and u |Γ = -P g(D * v). Therefore, in comparison with the proof of Proposition 2.1, it suffices to show that u |Γ belongs to H 1/2 (Γ) instead of L 2 (Γ) and apply the elliptic regularity theorem to gain the desired extra half-unit of regularity. By virtue of (1.12), we have

D * v ∈ H 1/2 (Γ).
First, recall that pointwise Lipschitz nonlinearities such as g map bounded sets of H 1/2 (Γ) into bounded sets of H 1/2 (Γ). Indeed, using the definition of Sobolev spaces on manifold by local charts and the Sobolev-Slobodeckij characterization of the fractional spaces H s (R d-1 ) (see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]), we know that for a given f in

H 1/2 (Γ), g(f ) ∈ H 1/2 (Γ) if and only if R d-1 ×R d-1 |φi(x1)g([f • ψi](x1)) -φi(x2)g([f • ψi](x2))| 2 x1 -x2 d dx1 dx2 < +∞, (2.26)
for all suitable (φi, ψi), where the functions φi ∈ C ∞ c (R d-1 ) are chosen from a partition of unity subordinate to some (finite) covering of Γ and the functions ψi are corresponding local representations of the surface.

The integral term in (2.26) is finite because g and φi are globally Lipschitz continuous; hence, g(f ) ∈ H 1/2 (Ω). Furthermore, taking the integral term in (2.26) plus some appropriate lower-order L 2 -term defines a norm on H 1/2 (R d-1 ) equivalent to the one given by interpolation. Thus, after coming back to functions on Γ, it follows from (2.26) that g(f

) H 1/2 (Γ) K f H 1/2 (Γ) for all f ∈ H 1/2 (Γ), (2.27)
where K is some positive constant coming from the Lipschitz continuity of g and norm equivalence.

Next we have to check that P ∈ L(H 1/2 (Γ)). Again, this is a consequence of (2.26): we observe that since Γ0 ∩ Γ1 = ∅, there exists m > 0 such that x1 -x2 > m whenever (ψi(x1

), ψi(x2)) ∈ [Γ0 × Γ1] ∪ [Γ1 × Γ0].
Finally, combining (1.12) for s = 1/2, the estimate (2.27), the fact that P ∈ L(H 1/2 (Γ)) together with the elliptic regularity theorem, we obtain u ∈ H 1 (Ω) and the stronger estimate

u H 1 (Ω) C ∆u H -1 (Ω) + v L 2 (Ω) = C A[u + DP g(D * v)] H -1 (Ω) + v L 2 (Ω) C ′ A[u, v] H (2.28)
where C and C ′ are some positive constants that do not depend on [u, v]. The set equality in (2.23) is now proved and the property (2.24) readily follows from (2.28) and (2.7).

Asymptotic stability

Next, we state the second main result of the section, which asserts that the zero equilibrium of the closed-loop system (1.1)-(1.2) is globally asymptotically stable.

Theorem 2.2 (Asymptotic stability of the closed-loop system). Let [u0, v0] ∈ H. Then,

St[u0, v0] H → 0 as t → +∞.
(2.29)

Together with the contraction property of {St}, (2.29) implies that 0 is a globally asymptotically stable equilibrium point for the feedback system (1.1)-(1.2).

Proof. By the contraction property of the semigroup {St} and denseness of D(A) in H, it suffices to prove (2.29) for initial data [u0, v0] in D(A).

To do so, we use a Lasalle-type invariance approach. Let us recall the classical line of arguments. We consider the ω-limit set ω([u0, v0]) of [u0, v0], which can be characterized as follows: [w0, z0] ∈ H belongs to ω([u0, v0]) if there exists an increasing sequence {tn} ∈ R N such that tn → +∞ and

St n [u0, v0] → [w0, z0] in H as n → +∞. ( 2
τ 0 Γ 0 g(-∂ν[A -1 w ′ ])∂ν[A -1 w ′ ] dσ dt = 0 for all τ 0.
(2.31)

It follows from Assumption 1 that g(s)s > 0 for every nonzero s. Thus, letting p A -1 w ′ , p ∈ C(R + , H 1 0 (Ω)) ∩ L ∞ (0, +∞; D(A)), (2.31) leads to ∂ν p(σ, t) = 0 for a.e. (σ, t) ∈ Γ0 × (0, +∞).

(2.32)

Next, we recall that w ′ ∈ W 1,∞ (0, +∞; H -1 (Ω)) and, using (2.32) together with the operator-theoretic formulation of (1.1)-(1.2), we obtain w ′′ + Aw = 0. Hence, p ∈ W 1,∞ (0, +∞; H 1 0 (Ω)) satisfies p ′ + w = 0, which in turn implies that p ∈ W 2,∞ (0, +∞; H -1 (Ω)) and solves p ′′ + Ap = 0 in H -1 (Ω), i.e., the standard variational formulation of the wave equation with homogeneous Dirichlet boundary conditions. In particular, p ∈ C 1 (R + , L 2 (Ω)) and solves the following boundary value problem:

∂ttp -∆p = 0
in Ω × (0, +∞), (2.33a)

p |Γ = 0 on Γ × (0, +∞), (2.33b) ∂νp = 0 on Γ0 × (0, +∞).

(2.33c)

The subset Γ0 being relatively open in Γ, a unique continuation argument for waves yields p = 0 -for instance, one can directly apply [Rob91, Théorème 2]. Therefore, w ′ = 0, Aw = 0 and finally w = 0, which concludes the proof.

Polynomial decay rates for strong solutions

This section is dedicated to the analysis of the decay rate of strong solutions under additional assumptions on the feedback nonlinearity and the geometry of the problem.

Statement of the result and outline of the proof

In what follows, we work under stronger assumptions that are given next.

Assumption 2. There exist positive constants S, α1 and α2 such that where Cu is a positive constant depending only on E (u0, v0) and A[u0, v0] H.

Theorem 3.1 is a Dirichlet counterpart to non-uniform polynomial decay results that are well-known in the case of Neumann boundary conditions -see, e.g., [Kom94, Theorem 9.10]. Generally speaking, by non-uniform we mean that the right-hand side of the inequality (3.4) depends not only on the natural energy of the initial data [u0, v0] (i.e., E (u0, v0)) but also on higher-order terms (here, A[u0, v0] H). As mentioned in the introduction, in the context of the wave equation with nonlinear boundary control, this type of decay is expected when the feedback nonlinearity g has no linear lower bound at infinity (for instance, when g is bounded and (3.1) cannot hold for all real numbers). Note that Theorem 3.1 is different from uniform polynomial decay results such as [Rao93, Theorem 5.1] or [Zua90, Theorem 2.1], where the corresponding feedback nonlinearities may have power growth around zero (typically leading to polynomial instead of exponential decay rate) but are still required to grow linearly at infinity (allowing uniform decay). In our problem, the non-uniform polynomial decay rate of Theorem 3.1 is related to the lack of linear dissipation at infinity -this will be further discussed in Section 4.

Let Then,

E(t) E(0) T + γt T + γT -1/γ
for all t T.

(3.7)

We already know from Section 2 that Eu(t) converges to 0 as t goes to +∞. Our subsequent efforts focus on estimating

τ 2 τ 1 E (r+1)/2 u (t) dt for arbitrary 0 τ1 τ2, (3.8)
where we recall that

Eu(t) = 1 2 { u 2 L 2 (Ω) + u ′ 2 H -1 (Ω) }. (3.9)
As mentioned in the introduction, the proof is based on an analysis of the variable p defined by

p = A -1 u ′ (3.10)
which solves, at least formally, the following boundary-value problem:

∂ttp -∆p = -∂t[DP g(-∂νp)]
in Ω × (0, +∞), (3.11a)

p |Γ = 0 on Γ × (0, +∞). (3.11b)
If u ′ takes values in L 2 (Ω), recalling the formula (2.1), we have

-∂νp = -∂ν[A -1 u ′ ] = D * Ap = D * u ′ . (3.12)
To alleviate notation, in the sequel we denote by Φ the term

Φ(t) DP g(D * u ′ (t)). (3.13)
In regards to (2.2), we see that -Φ is the harmonic extension of the trace u |Γ . As mentioned earlier, the p-variable is smoother, which permits, in regards to the wave-type equation (3.11a) satisfied by p, the use of a differential multiplier technique to obtain estimates of the integral over time of

1 2 Ω ∇p 2 + |p ′ | 2 dx (3.14)
premultiplied by an appropriate power of Eu. The quantity (3.14) is the natural energy of [p, p ′ ] at the H 1 (Ω) × L 2 (Ω)-level (i.e., the standard variational framework); from there, we will be able to deduce a suitable integral estimate of the energy Eu associated with the less regular u-variable.

Remark 3.1. In fact, since we want to avoid differentiating terms involving g so that our results remain valid when the nonlinearity is only continuous, we will rather multiply an integrated version of (3.11a), namely the formula u = -[p ′ + Φ], by the time derivative of the multiplier -see Lemma 3.2 below. In particular, p ′ need not be continuous.

The multiplier identity

In this subsection, we give more precise properties of the p-variable and derive an expression of (3.8) in the form of a identity obtained by applying an appropriate multiplier to (3.11a). We recall that D(A) is H 2 (Ω) ∩ H 1 0 (Ω) equipped with the norm A • L 2 (Ω) , which is equivalent to the one induced by H 2 (Ω). Lemma 3.2. Let [u, u ′ ] be a strong solution. The corresponding functions p and Φ enjoy the regularity

p ∈ L ∞ (0, +∞; D(A)) ∩ W 1,∞ (0, +∞; H 1 0 (Ω)), Φ ∈ L ∞ (0, +∞; H 1 (Ω)), (3.15)
Also, the following identity holds:

u = -[p ′ + Φ] ∈ C(R, L 2 (Ω)). (3.16) Proof. We infer from Proposition 2.2 that [u, u ′ ] ∈ L ∞ (0, +∞; H 1 (Ω) × L 2 (Ω)). As a consequence, since A -1 is continuous from L 2 (Ω) into D(A), we get p = A -1 u ′ ∈ L ∞ (0, +∞; D(A)).
(3.17)

Besides, u ′ : R + → H -1 (Ω) is absolutely continuous and

u ′′ + Au = -ADP g(D * u ′ ) a.e.
(3.18) Thus, applying A -1 to (3.18) yields

p ′ = A -1 u ′′ ∈ L ∞ (0, +∞; H 1 0 (Ω)), (3.19)
and also u = -[p ′ + Φ], and Φ ∈ L ∞ (0, +∞; H 1 (Ω)), (3.20) which concludes the proof.

Define the usual wave multiplier as follows:

Mp 2h • ∇p + (d -1)p, (3.21)
where h(x) = x -x0 as defined in Assumption 3 and d is the space dimension. Since p satisfies a wave equation, we know that τ 2 τ 1 Ω ∇p 2 + |p ′ | dx dt can be estimated by multiplying (3.11a) by Mp and integrating over Ω × (τ1, τ2). Since we are looking for estimates of Eu at the power (r + 1)/2, we premultiply Mp by Eu at the power (r -1)/2. Thus, we shall multiply (3.11a) by

E (r-1)/2 u (t)Mp(x, t). (3.22)
The resulting identity is given in the next lemma.

Lemma 3.3 (Multiplier identity). The following equality holds for any 0 τ1 τ2:

2 τ 2 τ 1 E (r+1)/2 u dt = E (r-1)/2 u Ω uMp dx τ 2 τ 1 + τ 2 τ 1 E (r-1)/2 u Γ (h • ν)|∂νp| 2 dσ - Γ 0 (h • ν)|g(D * u ′ )| 2 dσ dt - τ 2 τ 1 E (r-1)/2 u Ω (d + 1)Φu + Φ[2h • ∇u] dx dt - (r -1) 2 τ 2 τ 1 E ′ u E (r-3)/2
u Ω uMp dx dt.

(3.23)

Proof. The proof is split into four steps.

Step 1: Integration by parts with respect to time. First, by linearity and continuity of M, Mp belongs to W 1,∞ (0, +∞; L 2 (Ω)). On the other hand, recall that Eu is bounded and absolutely continuous with

E ′ u = - Γ 0 g(D * u ′ )D * u ′ dσ a.e., E ′ u ∈ L ∞ (0, +∞), (3.24)
because strong solutions are Lipschitz continuous with respect to time. Thus, E (r-1)/2 u belongs to W 1,∞ (0, +∞) and

E (r-1)/2 u Mp belongs to W 1,∞ (0, +∞; L 2 (Ω)) with [E (r-1)/2 u Mp] ′ = E (r-1)/2 u Mp ′ + (r -1) 2 E ′ u E (r-3)/2 u Mp a.e. (3.25)
Now, it follows from (3.16) that p ′ + Φ belongs to W 1,∞ (0, +∞; L 2 (Ω)) and

-[p ′ + Φ] ′ = u ′ = Ap. (3.26)
Let 0 τ1 τ2. Recall that, since p ∈ D(A), Ap = -∆p ∈ L 2 (Ω). Thus, taking the scalar product of (3.26) with E (r-1)/2 u Mp in L 2 (τ1, τ2; L 2 (Ω)) and using the integration by parts formula in W 1,2 (τ1, τ2; L 2 (Ω)) leads to

- τ 2 τ 1 E (r-1)/2 u Ω ∆pMp dx dt = E (r-1)/2 u Ω uMp dx τ 2 τ 1 - τ 2 τ 1 E (r-1)/2 u Ω uMp ′ dx dt - (r -1) 2 τ 2 τ 1 E ′ u E (r-3)/2
u Ω uMp dx dt.

(3.27)

Step 2: Multiplier technique for the wave equation. In what follows, we apply classical vector calculus identities to recover the

H 1 0 (Ω) × L 2 (Ω)-energy of [p, p ′ ]. Since p takes values in H 2 (Ω), Rellich's identity yields Ω ∆p[2h • ∇p] dx = (d -2) Ω ∇p 2 dx + Γ ∂νp[2h • ∇p] dσ - Γ (h • ν) ∇p 2 dσ. ( 3
Ω ∆p[2h • ∇p] = (d -2) Ω ∇p 2 dx + Γ (h • ν)|∂νp| 2 dσ. (3.30)
On the other hand,

Ω ∆p(d -1)p dx = -(d -1) Ω ∇p 2 dx, (3.31)
where we use again that p vanishes on the boundary. Summing (3.30) and (3.31) yields

Ω ∆pMp dx = - Ω ∇p 2 dx + Γ (h • ν)|∂νp| 2 dx. (3.32)
Coming back to (3.27) and recalling (3.16), let us write

Ω uMp ′ = - Ω p ′ Mp ′ dx - Ω ΦMp ′ dx = - Ω p ′ [2h • ∇p ′ ] + (d -1)|p ′ | 2 dx - Ω ΦMp ′ dx = Ω |p ′ | 2 dx - Ω ΦMp ′ dx, (3.33)
where we use the identity

Ω φ[2h • ∇φ] dx = Γ (h • ν)(φ |Γ ) 2 dσ - Ω (φ) 2 div h dx for all φ ∈ H 1 (Ω) (3.34)
together with div h = d and p |Γ = 0. Therefore, combining (3.27) with (3.32) and (3.33) leads to

τ 2 τ 1 E (r-1)/2 u Ω ∇p 2 + |p ′ | 2 dx dt = E (r-1)/2 u Ω uMp dx τ 2 τ 1 + τ 2 τ 1 E (r-1)/2 u Ω ΦMp ′ dx dt + τ 2 τ 1 E (r-1)/2 u Γ (h • ν)|∂νp| 2 dσ dt - (r -1) 2 τ 2 τ 1 E ′ u E (r-3)/2
u Ω uMp dx dt.

(3.35)

Step 3: Additional terms. Here, we put the terms involving Φ into a form suitable for further estimation. It follows from (3.16) that

Ω ΦMp ′ dx = - Ω ΦMu dx - Ω ΦMΦ dx.
(3.36) Applying (3.34) to Φ, similarly to (3.33), we obtain

- Ω ΦMΦ dx = Ω |Φ| 2 dx - Γ (h • ν)(Φ |Γ ) 2 = Ω |Φ| 2 dx - Γ 0 (h • ν)|g(D * u ′ )| 2 dσ, (3.37)
where we use that, by definition,

Φ |Γ = g(D * Ap) = g(D * u ′ ) on Γ0, Φ |Γ = 0 on Γ1. (3.38)
On the other hand, we recall that

- Ω ΦMu dx = - Ω Φ[2h • ∇u] + (d -1)Φu dx. (3.39)
Plugging (3.37) and (3.39) into (3.35) leads to

τ 2 τ 1 E (r-1)/2 u Ω ∇p 2 + |p ′ | 2 dx dt = E (r-1)/2 u Ω uMp dx τ 2 τ 1 - (r -1) 2 τ 2 τ 1 E ′ u E (r-3)/2 u Ω uMp dx dt + τ 2 τ 1 E (r-1)/2 u Γ (h • ν)|∂νp| 2 dσ - Γ 0 (h • ν)|g(D * Ap)| 2 dσ - Ω Φ[2h • ∇u] + (d -1)Φu -|Φ| 2 dx dt.
(3.40)

Step 4: Conclusion. We finish the proof by rewriting (3.40) as an estimate of

τ 2 τ 1 E (r+1)/2 u
dt. This is done as follows. First, by definition of the p-variable,

Ω ∇p 2 dx = A 1/2 p 2 L 2 (Ω) = A -1/2 u ′ 2 L 2 (Ω) = u ′ 2 H -1 (Ω) . ( 3
Ω ∆φ[2h • ∇φ] dx (d -2) Ω ∇φ 2 dx + Γ ∂νφ[2h • ∇φ] dσ - Γ (h • ν) ∇φ 2 dσ, (3.44)
holding for any φ satisfying φ ∈ H 1 (Ω), ∆φ ∈ L 2 (Ω), φ |Γ = 0 on Γ1 and (h • ν) -1 ∂νφ ∈ L 2 (Γ0). The reader is referred to [START_REF] Grisvard | Contrôlabilité exacte avec conditions mêlées[END_REF][START_REF] Komornik | A direct method for the boundary stabilization of the wave equation[END_REF] for the proof of (3.44). The main benefit in that case is that the multiplier analysis can be carried out even when Γ0 ∩ Γ1 = ∅, in which case strong solutions may have weaker regularity. 0

Γ 0 |g(D * u ′ )| 2 dσ K Γ 0 |g(D * u ′ )D * u ′ | dσ = K Γ 0 g(D * u ′ )D * u ′ dσ (3.49)
by Lipschitz continuity and nonincreasingness of g, together with g(0) = 0. Moreover, recalling (3.3) in Assumption 3 and looking at the sign of each term, we observe that |∂νp| in (3.23) need not be estimated on the uncontrolled boundary Γ1.

That being said, let us start by estimating the term involving 2h • ∇u in (3.23). This is done in the following lemma.

Lemma 3.4. Suppose that r 2. Then, there exists a positive constant K such that

τ 2 τ 1 E (r-1)/2 u Ω Φ[2h • ∇u] dx dt K A[u0, v0] 1/2 H τ 2 τ 1 1 µ Γ 0 |g(D * u ′ )| 2 dσ + µE (r-2)/2 u (0)E (r+1)/2 u dt (3.50)
for all τ2 τ1 0 and µ > 0.

Proof. We start by writing

(Φ, 2h • ∇u) L 2 (Ω) = (DP g(D * u ′ ), 2h • ∇u) L 2 (Ω) = (g(D * u ′ ), D * [2h • ∇u]) L 2 (Γ 0 ) . (3.51)
Thus, applying the Cauchy-Schwarz inequality yields

Ω Φ[2h • ∇u] dx g(D * u ′ ) L 2 (Γ 0 ) D * [2h • ∇u] L 2 (Γ 0 ) g(D * u ′ ) L 2 (Γ 0 ) D * [2h • ∇u] L 2 (Γ) . (3.52)
Next, recall from (1.12) that D * ∈ L(H -1/2 (Ω), L 2 (Γ)).

(3.53) Therefore, it follows from (3.52) that

Ω Φ[2h • ∇u] dx K g(D * Ap) L 2 (Γ 0 ) 2h • ∇u H -1/2 (Ω) . (3.54)
Linear interpolation between the Sobolev spaces L 2 (Ω) and H -1 (Ω) leads to

2h • ∇u H -1/2 (Ω) K 2h • ∇u 1/2 L 2 (Ω) 2h • ∇u 1/2 H -1 (Ω) (3.55) First, by Proposition 2.2, 2h • ∇u 1/2 L 2 (Ω) K h 1/2 L ∞ (Ω) d ∇u 1/2 L 2 (Ω) d K ′ A[u0, v0] 1/2 H . (3.56)
Besides, since 2h • ∇u belongs to L 2 (Ω), we have

2h • ∇u, w H -1 (Ω),H 1 0 (Ω) = (2h • ∇u, w) L 2 (Ω) for all w ∈ H 1 0 (Ω). (3.57) Let us write Ω [2h • ∇u]w dx = 2 d i=1 Ω hi∂iuw dx = -2 d i=1 Ω u[w∂ihi + hi∂iw] dx + d i=1 Γ νihiuw dσ = -2 Ω u[w div h + h • ∇w] dx, (3.58)
where hi (resp. νi) denotes the i-th coordinate of the vector field h (resp. the outward normal vector ν). Recall that h ∈ C 1 (Ω). Then, using the Poincaré inequality on w, we obtain that for some K > 0,

(2h • ∇u, w) L 2 (Ω) K u L 2 (Ω) w H 1 0 (Ω) . ( 3
Dg(D * u ′ )[2h • ∇u] dx K A[u0, v0] 1/2 H g(D * u ′ ) L 2 (Γ 0 ) u 1/2 L 2 (Ω) K ′ A[u0, v0] 1/2 H g(D * u ′ ) L 2 (Γ 0 ) E 1/4 u .
(3.61) Therefore, since Eu 0, we have

E (r-1)/2 u Ω Dg(D * u ′ )2h • ∇u dx K A[u0, v0] 1/2 H g(D * u ′ ) L 2 (Γ 0 ) E (r-1)/2+1/4 u . (3.62)
Applying Young's inequality with a parameter µ > 0, we obtain

g(D * u ′ ) L 2 (Γ 0 ) E (r-1)/2+1/4 u 1 2µ g(D * u ′ ) 2 L 2 (Γ 0 ) + µ 2 E r-1/2 u . (3.63)
It is assumed that r 2. Thus, letting η (r -1/2) -(r + 1)/2 0, by nonincreasingness of Eu, we have Lemma 3.5. Suppose that r d -1. Then, there exists K > 0 and η ∈ (0, 1) such that

E r-1/2 u = E η u E (r+1)/2 E η u (0)E (r+1)/2 u . ( 3
τ 2 τ 1 E (r-1)/2 u Γ 0 |∂ν p| 2 dσ dt K A[u0, v0] 2-η H µ 1/η τ 2 τ 1 Γ 0 g(D * u ′ )D * u ′ dσ dt +K A[u0, v0] 2-η H µ -1/(1-η) τ 2 τ 1 E (r+1)/2 u dt + K τ 2 τ 1 Γ 0 |g(D * u ′ )| 2 dσ dt (3.65)
for all 0 τ1 τ2 and µ > 0.

Proof. For each t 0, we set

Γ 0 t {σ ∈ Γ0 : |∂νp(σ, t)| S} and Γ 1 t Γ0 \ Γ 0 t . (3.66)
where we recall that the constant S is introduced in Assumption 2. Then,

Γ 0 (h • ν)|∂νp| 2 dσ K Γ 0 t |∂ν p| 2 dσ + K Γ 1 t |∂νp| 2 dσ. (3.67)
Using (3.1) in Assumption 2, we estimate the first term in (3.67) as follows:

Γ 0 t |∂ν p| 2 dσ α -2 1 Γ 0 t |g(-∂νp)| 2 dσ α -2 1 Γ 0 |g(-∂νp)| 2 dσ = α -2 1 Γ 0 |g(D * u ′ )| 2 dσ. (3.68)
Let us examine the second term in (3.67). Setting a parameter η ∈ (0, 1) to be tuned later on, we have

Γ 1 t |∂ν p| 2 dσ = Γ 1 t |D * u ′ | 2 dσ = Γ 1 t |D * u ′ | 2-η |g(D * u ′ )D * u ′ | η |g(D * u ′ )| η dσ.
(3.69) Equation (3.69) makes sense since |g(D * u ′ )| min{g(S), -g(-S)} > 0 on Γ 1 t . In fact, we have

Γ 1 t |D * u ′ | 2 dσ min{g(S), -g(-S)} -η Γ 1 t |D * u ′ | 2-η |g(D * u ′ )D * u ′ | η dσ K Γ 0 |D * u ′ | 2-η |g(D * u ′ )D * u ′ | η dσ.
(3.70) Using Hölder's inequality with conjugates 1/η and 1/(1η), we infer from (3.70) that In what follows, we rely on (fractional) Sobolev inequalities -see [DNPV12, Theorem 6.5 and Theorem 6.9]. First, we consider the case d 3, where we recall that d denotes the space dimension. We have the continuous embedding for all 0 τ1 τ2 and µ > 0.

Γ 1 t |D * u ′ | 2 dσ K Γ 0 |D * u ′ | 2-η 1-η 1-η Γ 0 g(D * u ′ )D * u ′ dσ η . ( 3
H 1/2 (Γ0) ֒→ L q (Γ0) for all q ∈ 2, 2(d -1) d -2 I d . ( 3
Proof. First, using Cauchy-Schwarz and Young inequalities, we obtain and we conclude the proof by plugging (3.84) into (3.83) and using that Eu is nonincreasing.

Concluding remarks

In this section, we discuss the results of our paper and give some comments and perspectives.

• Theorem 3.1 deals with the decay rate of strong solutions to (1.1)-(1.2), which remain bounded in a stronger norm (here, in H 1 (Ω)×L 2 (Ω)). In particular, this enables the use of Sobolev embeddings to obtain estimates of the boundary term ∂ν[A -1 u ′ ] in L ∞ (0, +∞; L q (Γ)) for some appropriate q. In view of the energy identity (2.8), and as done in Lemma 3.5, we can then derive an estimate involving only the "dissipation term" g(-∂ν[A -1 u ′ ]) and lower-order energy terms, even though no lower bound on the nonlinearity g is prescribed at infinity. Here, using the terminology of [VM00], the feedback is allowed to be weak, i.e., g(s)/s can go to 0 as |s| goes to infinity, as it is the case when g represents a saturation mapping; then, loss of uniformity is to be expected. More precisely, coming back to the Neumann problem, the one-dimensional version of (1.6) with g given by (1.7) is known to possess weak solutions that decay to zero (in the natural energy space H 1 (Ω) × L 2 (Ω)) slower than any exponential or polynomial, whereas strong solutions decay exponentially to zero but in a non-uniform way -see [VM00, Theorem 4.1] or also [START_REF] Chitour | One-dimensional wave equation with setvalued boundary damping: well-posedness, asymptotic stability, and decay rates[END_REF]Theorem 4.33].

Proving a similar result in our Dirichlet case would be interesting.

• Putting aside the matter of saturated feedback and assuming if needed that g has linear growth at infinity, we see that, unfortunately, the strategy followed here is not sufficient to prove uniform decay of solutions to (1.1)-(1.2). Indeed, while estimating the term (Φ, 2h • ∇u) L 2 (Ω) as in Lemma 3.4 is good enough for the purpose of proving Theorem 3.1, it requires, again, that solutions remain bounded in a norm stronger than that of the energy space H. If, instead of (3.50), one manages to prove something in the likes of for some τ > 0, where K(τ ) is allowed to depend on τ and ǫ can be chosen sufficiently small, then, by remarking that the multiplier identity (3.23) is still valid with the time-varying weight E (u, u ′ ) (r-1)/2 replaced by the constant 1, one could easily adapt the rest of our proof to obtain exponential uniform stability. By following the proof of [LT87, Lemma 3.3], we can prove such an estimate when g is the identity, at least under some specific geometrical conditions; however, the argument breaks down in the nonlinear case. Therefore, as mentioned in the introduction, the problem of uniform stability is still open.

  α1|s| |g(s)| α2|s| for all |s| S. (3.1) Assumption 3. The domain Ω ⊂ R d with smooth boundary Γ = Γ0 ∪ Γ1 satisfies the following conditions: 1. The boundary is such that Γ0 ∩ Γ1 = ∅; (3.2) 2. There exists a point x0 ∈ R d such that, setting h(x) x -x0, h • ν 0 on Γ1. (3.3) Then, we can estimate the decay rate of each strong solution. Theorem 3.1 (Non-uniform polynomial decay rate). Let r max{d -1, 2}. Under Assumptions 2 and 3, strong solutions [u, u ′ ] to (1.1)-(1.2) satisfy E (u(t), u ′ (t)) Cut -2

E

  γ+1 (t) dt T E(0) γ E(τ ) for all τ 0. (3.6)

  .28) Furthermore, p |Γ = 0; thus ∇p = (∂νp)ν on Γ. (3.29) Combining (3.28) and (3.29), we obtain

  .41) On the other hand, it immediately follows from u = -[p ′ + Φ] that Ω |p ′ | 2 dx = Ω |u| 2 dx + Ω 2Φu + |Φ| 2 dx a.e. (3.42) Summing (3.41) and (3.42) yields Ω ∇p 2 + |p ′ | 2 dx = 2Eu + Ω 2Φu + |Φ| 2 dx a.e. (3.43) Plugging (3.43) into (3.40), we get the desired identity. Remark 3.2 (Disconnected boundary). The assumption Γ0 ∩ Γ1 = ∅ allows us to work with H 1 (Ω) × L 2 (Ω)-valued strong solution to the Dirichlet problem (1.1)-(1.2). The corresponding regularity for the Neumann problem (1.6) would be H 2 (Ω) × H 1 (Ω). Assume for a moment that h • ν > 0 on Γ0. In [XX19, Zua90], a slightly modified version of the Neumann problem (1.6) is considered, namely ∂ttu -∆u = 0 in Ω, ∂νu = -(h • ν)g(∂tu) on Γ0 and u |Γ = 0 on Γ1. The authors are able to relax the hypothesis Γ0 ∩ Γ1 = ∅ by relying on a Rellich-like inequality that is valid in space dimension d 3 and reads as follows:

  .64) Plugging (3.64) into (3.63) and integrating over (τ1, τ2) yields the desired result. Next, we deal with the term involving |∂νp| = |D * u ′ | on the controlled boundary Γ0. Here, the arguments are very similar to those employed in the case of saturated Neumann feedback -see [Kom94, Theorem 9.10] or [XX19, Theorem 3].

  .73) Furthermore, since r + 1 d, if we choose η = 2/(r + 1), some computations yield (2η)/(1η) ∈ I d ; henceH 1/2 (Γ0) ֒→ L 2-η 1-η (Γ0).(3.74)If d = 2,then the embedding (3.73) holds in fact for any q ∈ [2, +∞); therefore, (3.74) is valid as well. Coming back to (3.71), combining (3.74) with (3.72) yieldsE (r-1)/2 u Γ 1 t |D * u ′ | 2 dσ K A[u0, v0] 2-η H E (r-1)/2 u Γ 0 g(D * u ′ )D * u ′ dσ η . (3.75) Lemma 3.8. There exists a positive contant K such that * u ′ )| 2 dσ dt (3.82)

  that Φ = DP g(D * u ′ ) and that D continuously maps L 2 (Γ) into L 2 (Ω). Therefore,Φ 2 L 2 (Ω) K P g(D * u ′ ) 2 L 2 (Γ) = K g(D * u ′ ) 2 L 2 (Γ 0 )(3.84)

  • ∇u] dx dt K(τ ) τ 0 |g(D * u ′ )| 2 dσ dt + K ′ {Eu(0) + Eu(τ )} + ǫ

Alternatively, A can be defined as a duality mapping between H 1 0 (Ω) and H -1 (Ω), in which case (1.8) is recovered a posteriori by applying elliptic regularity theory.

We can write estimates in terms ofτ 2 τ 1 Γ 0 |g(D * u ′ )| 2 dσ dt instead of τ 2 τ 1 Γ 0 g(D * u ′ )D * u ′ dσ dt since

* This work has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003).

Estimates of the right-hand side and conclusion

Our goal in this subsection is to establish an integral inequality in the form of (1µ) where Ku is a constant that may depend on the initial data and µ is a sufficiently small constant that may depend on u as well. Bearing in mind the full statement of Theorem 3. • Or the boundary dissipation term

which is nonnegative and can be integrated, since

• And also

dt premultiplied by small µ so that it can be absorbed in the left-hand side.

Remark 3.3. In what follows, we shall denote by K, K ′ , etc. generic constants that do not depend on the initial data.

Applying the Young inequality with conjugates 1/η and 1/(1η), we get

Since (r -1)/2(1η) = (r + 1)/2, we conclude the proof by combining (3.76) and (3.68) together with (3.67).

At this point, the proof of Theorem 3.1 is almost complete. Estimates of the remaining terms in (3.23) are given in the next lemmas. Following our remarks at the beginning of the subsection, we claim that Theorem 3.1 is proved once those are established.

Lemma 3.6. There exists a positive constant K such that

Proof. Let τ 0. Then,

where it used that Eu is nonincreasing. Equation (3.77) readily follows from the triangular inequality.

Lemma 3.7. There exists a positive constant K such that

u Ω uMp dx dt KE (r-1)/2 u (0){Eu(τ1) + Eu(τ2)} for all 0 τ1 τ2.

(3.79)

Proof. Again, we write Ω uMp dx KEu.

(3.80) Therefore,

u Ω uMp dx dt K (3.81)

The desired inequality follows from the nonincreasingness of Eu and (3.81).