Oscar Sánchez

Christophe Jégo
email: christophe.jego@enseirb-matmeca.fr

Michel Jézéquel
email: michel.jezequel@telecom-bretagne.eu

Yannick Saouter
email: yannick.saouter@telecom-bretagne.eu

High Speed Low Complexity Radix-16 Max-Log-MAP SISO Decoder

Keywords:

At present, the main challenge for hardware implementation of turbo decoders is to achieve the high data rates required by current and future communication system standards. In order to address this challenge, a low complexity radix-16 SISO decoder for the Max-Log-MAP algorithm is proposed in this paper. Based on the elimination of parallel paths in the radix-16 trellis diagram, architectural solutions to reduce the hardware complexity of the different blocks of a SISO decoder are detailed. Moreover, two complementary techniques are introduced in order to overcome BER/FER performance degradation when turbo decoders based on the proposed SISO decoder are considered. Thus, a penalty lower than 0.05dB is observed for a 8 state binary turbo code with respect to a traditional radix-2 turbo decoder for 6 decoding iterations.

I. INTRODUCTION

The adoption of turbo codes [START_REF] Berrou | Near Shannon limit error-correcting coding and decoding: turbo-codes[END_REF] in last generation wireless communication systems, such as 3GPP-LTE and WiMAX, has prompted research activities towards the design of high throughput Turbo Decoder (TD) architectures. In next generation standards, throughput requirements are significantly higher. For instance, LTE-advance standard targets data rates around 1Gbit/s. Thus, the design of low complexity high throughput TD architectures becomes a major challenge. The decoding of turbo codes is carried out through an iterative process where two Soft-Input Soft-Output (SISO) decoders, operating in natural and interleaved domain, exchange extrinsic information. The inherent recursive nature of MAP-based algorithms (Log-MAP or Max-Log-MAP [START_REF] Robertson | A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain[END_REF]), implemented by each SISO decoder in the TD, is an important issue that limits the TD throughput. To overcome this constraint, radix-2 N T architectures have been proposed [START_REF] Bickerstaff | A 24mb/s radix-4 logMAP turbo decoder for 3GPP-HSDPA mobile wireless[END_REF]- [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF]. In these architectures, NT transitions in the trellis diagram are performed per clock cycle.

Most of the works in the literature that propose radix-2 N T architectures are concerned to high throughput, and only a few explore the hardware complexity reduction. A radix-16 ACS unit to decompose the compare operation among 16 branches into two levels is presented in [START_REF] Tang | A 952ms/s Max-Log MAP decoder chip using radix-4 x 4 ACS architecture[END_REF]. The hardware overhead is minimized thanks to a two-dimensional ACS unit architecture. In [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF], a radix-16 modified log-MAP algorithm and a two-stage compare and select architecture to further decrease the latency and processing power is detailed. The main novelty of our work is the design of a radix-16 SISO architecture based on a radix-8 ACS unit for the Max-Log-MAP algorithm. Since the ACS unit is responsible of the bottleneck in the decoder, a reduction of the SISO critical path with respect to the architectures in [START_REF] Tang | A 952ms/s Max-Log MAP decoder chip using radix-4 x 4 ACS architecture[END_REF] and [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF] is possible. Thus, the SISO decoder throughput can be improved. Furthermore, a hardware complexity reduction of all the block units that form the SISO decoder is proposed. A clever idea is also introduced to avoid TD BER/FER performance degradation.

k k + 4 s (j) s (i) {d (0) i,j , r (0)
i,j } → γ (0) i,j {d (1) i,j , r (1)
i,j } → γ The rest of the paper is organized as follows. In section II the proposed radix-16 SISO decoder architecture is presented. Each block unit of the designed SISO decoder is detailed. Then, a comparison in terms of BER/FER performance of different versions of the turbo decoder is given in section III. Synthesis results are summarized in section IV. Finally, section V concludes the paper. k and takes hard decisions dk . To ensure that the critical path is in the ACS unit, pipeline stages can be included in the BMUs and SOU.

II. PROPOSED RADIX-16 SISO ARCHITECTURE

The 8-state binary turbo code compliant with the LTE standard has been selected to illustrate our propositions. Let S = {s (0) , . . . , s (7) } be the set of convolutional encoder states, and

2 m=0 a (i) m • 2 m
the decimal representation of the state s (i) , with a (i) m ∈ {0, 1} the value of the flip-flops that form the convolutional encoder. Consider Fig. 2 that depicts the state transition s (i) → s (j) in the trellis diagram of a radix-16 ACS unit (NT = 4). For this radix value, two paths (parallel paths) are possible from any state at time k to any state at time k + 4. Let these two paths be due to the systematic bit sequence d path b. As pointed out in [START_REF] Fettweis | Parallel Viterbi algorithm implementation: breaking the ACS-bottleneck[END_REF], parallel paths should be eliminated prior to the ACS unit. Thus, it is possible to replace a radix-16 ACS unit by a less complex radix-8 ACS unit where each pair of states (s (i) , s (j) ), at time k and k + 4 respectively, are connected only by one path 1 . The branch metric value for this path is then

γ max i,j = max(γ (0) i,j , γ (1)
i,j ). This simplification does not affect the result of the ACS unit operation. Furthermore, since a radix-8 ACS unit has a lower critical path compared to a radix-16 ACS unit, it enables to increase the clock frequency of the corresponding decoder. The systematic and redundant bits that define the parallel paths in the transition s (i) → s (j) are given by:

d (b) i,j =(d (i,j,b) k , d (i,j,b) k+1 , d (i,j,b) k+2 , d (i,j,b) k+3 ) =(a (i) 0 + a (i) 1 + b, a (i) 1 + a (i) 2 + a (j) 0 , a (i) 2 + a (j) 1 + b, a (j) 0 + a (j) 2 + b) r (b) i,j =(r (i,j,b) k , r (i,j,b) k+1 , r (i,j,b) k+2 , r (i,j,b) k+3 ) =(a (i) 0 + a (i) 2 + b, a (i) 1 + a (j) 0 + b, a (i) 2 + a (j) 0 + a (j) 1 , a (j) 1 + a (j) 2 + b) (1)

A. Branch Metrics Unit Architecture

From (1), for any pair of parallel paths, d

(i,j,0) k+1 = d (i,j,1) k+1 and r (i,j,0) k+2 = r (i,j,1) k+2
because they are independent of b. Therefore, the channel values corresponding to the bits d k+1 and r k+2 do not affect the choice of the maximum branch metric in the radix-16 ACS unit transition s (i) → s (j) . γ max i,j can be then computed as follows. First, a partial term ξi,j, due to the bits that are different in both paths, for b = 0 is calculated. If ξi,j < 0 then γ (1) i,j is the maximum branch metric and ǫi,j = -ξi,j. Otherwise, the maximum branch metric is γ (0) i,j and ǫi,j = ξi,j. Afterwards, γ max i,j is obtained by adding to ǫi,j the terms corresponding to the bits d k+1 and r k+2 . The proposed BMU architecture is given in Fig. 3. Note that only 16 values ξi,j have to be computed for all the 64 state transitions s (i) → s (j) , i, j = 0, . . . , 7. This BMU only costs 53% of the hardware resources of a conventional implementation. A low complexity adder-sharing BMU is proposed in [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF]. It requires a memory to store 128 branch metrics. With our BMU architecture this memory is reduced to the half (64 γ max i,j values). Moreover, the BMU in [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF] needs additional registers that increase the latency since more cycles are required to share the adders. In our architecture this disadvantage is overcome. branch metrics). Then, four values are selected. These values are sent to the radix-4 CS that has to produce the final state metric value. In order to reduce the ACS unit critical path, we have applied the modulo normalization technique as presented in [START_REF] Benkeser | Design and optimization of an HSDPA turbo decoder ASIC[END_REF]. Thus, state metric normalization blocks have been removed. Besides, we have adopted the radix-4 CS architecture presented in [START_REF] Studer | Design and implementation of a parallel turbo-decoder ASIC for 3GPP-LTE[END_REF]. Indeed, this architecture has been optimized in terms of throughput.

B. Radix-8 ACS Unit Architecture

C. Soft Output Unit Architecture

Since it is not possible to determine a priori which paths are eliminated in each state transition, a static comparison structure cannot be used for the SOU. Note however that we can group the 64 state transitions s (i) → s (j) in eight sets Q leach one having eight elements (pair of states)for l = 0, . . . , 7, such that Q l = {(s (i) , s (j) ) :

d (0) i,j = (c (l) 0 , c (l) 1 , c (l) 2 , c (l)
3 )}, with c (l) n ∈ {0, 1}, for n = 0, 1, 2, 3. Therefore, the eight branch metric values γ max i,j with starting and ending states s (i) and s (j) respectively, such that (s (i) , s (j) ) ∈ Q l , correspond to only two possible systematic bit sequences, according to d

(b) i,j in (1). For instance, if (c (0) 0 , c (0) 1 , c (0) 2 , c (0)
3 ) = (0, 0, 0, 0), then the values of γ max i,j related to the elements of C0 are for the systematic bit sequence (0, 0, 0, 0) or (1, 0, 1, 1). Based on this observation we propose in Fig. 5 an original SOU architecture that contains comparing-routing elements. First, the 64 values L = α k + γ max + β k+4 , corresponding to the non discarded paths, are computed. These values are then sent to eight Max-L blocks. The eight inputs of the l-th Max-L block correspond to the values L for the paths defined by the states (s (i) , s (j) ) ∈ Q l . Each Max-L block processes its inputs in three stages. In the first stage, four Switch-I blocks (Fig. 5(b)) either find in parallel the maximum values of their inputs or directly send the inputs to the outputs. In the two other stages, Switch-II blocks are used (Fig. 5(c)). These blocks find the maximum value or let one of the inputs to pass through them. Thus, at the output of the Max-L blocks the maximum value for each d is found 2 . These values are then compared using a fixed tree structure composed of Switch-II blocks, such that the number of comparators is minimized. Finally, the four extrinsic values are computed. Due to the elimination of paths, values L e k+h , for h = 0, 2, 3, may not be valid. In this case, a multiplexer enables to replace them by p • q k+h , where q k+h = ±1, and (q k+h +1)/2 is the hard decision corresponding to the systematic bit d k+h . An expression for p will be proposed in Section III. On the other hand, the value L e k+1 is always valid since the systematic bit d k+1 does not change in the parallel paths.

III. TURBO DECODER PERFORMANCE COMPARISON

The proposed radix-16 SISO architecture significantly affects the asymptotic gain of the TD. We have observed a penalty around 0.2dB, for code rates R = 1/2 and 1/3, at FER of 10 -6 , with an error floor that remains high compared to the error floor of a radix-2 SISO architecture. For high SNR values (error floor region), a-priori values grow fast during the iterative process, and thus, it is more probable to eliminate all the paths for a specific bit value. For instance, if the a-priori value for d k (L a k ) is high and positive, it is highly probable that all the paths corresponding to d k = 1 are chosen, eliminating all the paths for d k = 0. It means that L e k cannot be computed. Furthermore, this high a-priori value affects the computations of L e k+2 and L e k+3 , and thus, an undesirable correlation may appear between the hard decisions dk , dk+2 and dk+3 during the iterative process. To solve this problem, we propose two techniques: 1) select an appropriate value p when there is not enough information to compute a determinate extrinsic value (section III-A); 2) reduce the mutual interference between the extrinsic values in a same radix-16 trellis transition (section III-B).

A. Expression of p for Unknown Extrinsic Value

We have established an expression for p following a similar approach to the one presented in [START_REF] Adde | Recent simplifications and improvements in Block Turbo Codes[END_REF] for the decoding of Block Turbo Codes. In [START_REF] Adde | Recent simplifications and improvements in Block Turbo Codes[END_REF], when there is not competing codeword in order to compute the reliability value of a certain bit, the soft output is calculated as the sum of the magnitude of a set of channel observations at the input of the decoder. Since there are only four systematic bits for each radix-16 trellis transition, we have modified the summation by a minimum operation. Thus, we avoid too optimistic extrinsic values that are inconvenient during the iterative process performed by the turbo decoder. The value of p corresponds then to the minimum Fig. 6. SISO decoder displacement to avoid interference between symbols in a radix-16 trellis transition. reliability value at the SISO decoder input (absolute value of the systematic plus the a-priori LLR values) between all the four bits in the radix-16 trellis transition as shown by [START_REF] Robertson | A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain[END_REF]. Extensive Monte-Carlo simulations showed the convenience of this expression.

p = min i=0,1,2,3 (|L a k+i + L(d k+i )|) (2)

B. Reduce the Interference of Bits in the Same Trellis Transition

We emphasize that the bit d k+1 affords a higher level of protection compared to the others bits in the radix-16 trellis transitions. Therefore, we propose to apply a shift operation on the frame treated by the SISO decoder, as presented in Fig. 6 for a frame size of 1024 bits. For this example, when no shifting is applied, bits 1, 5, 9, . . . , 1021 are protected. With a shift of one bit, the bits 0, 4, 8, . . . , 1020, 1023 are protected. With a shift of two or three bits, other bits are protected. Note also that the bits that interfere are different, in each radix-16 trellis transition, in function of the shift value. Thus, if we change the number of shifted bits in consecutive turbo decoder iterations, the negative effects introduced by our SISO architecture can be significantly reduced. This technique can be adapted to the sliding window process [START_REF] Benedetto | A Soft-Input Soft-Output Maximum A Posteriori (MAP) Module to Decode Parallel and Serial Concatenated Codes[END_REF], and when multiples SISO decoders are assigned for the decoding of a frame [START_REF] Zhang | Parallel turbo decoding[END_REF].

If the shifting technique is applied, there are some bits that do not belong to the frame for the first and last radix-16 trellis transitions (Fig. 6). Therefore, the SISO decoder must be fed with appropriate a-priori values for these bits outside the frame, such that α and β values at the frame limits are not altered. Moreover, if the tail biting approach is used [START_REF] Berrou | Multiple parallel concatenation of circular recursive convolutional (CRSC) code[END_REF], bits outside the frame of the first (last) transition correspond to bits of the last (first) transition since the frame is circular. Thus, no special consideration should be given in this case.

BER and FER performance after six iterations for the LTE TD with 1024 bits per frame are given in Fig. 7 3 . Two architectures have been considered: one based on a radix-2 SISO decoder and another one based on the proposed radix-16 SISO decoder. Two different code rates R = 1/3 and 1/2 have been studied. The code rate R = 1/2 is achieved by puncturing the original code. 6 and 9 bits have been chosen for the channel and extrinsic values representation, respectively. For the radix-2 SISO decoder, 10 bits are necessary for α and β metrics. For the radix-16 SISO decoder, 12 bits are necessary so that the modulo normalization technique can be applied. When p is given by ( 2), an error floor appears for both code rates. However, when the shifting technique is also applied, the architecture based on the proposed radix-16 SISO decoder exhibits a negligible degradation by comparison with an architecture based on the radix-2 SISO decoder.

IV. SYNTHESIS RESULTS

The proposed radix-16 SISO architecture was synthesized with a STMicrolectronics 90nm CMOS process ASIC target. The hardware complexity in terms of equivalent 2-input (NAND) gate count is given in Table I. In order to compare the SISO decoder hardware complexity as function of NT , radix-2, radix-4 and radix-16 SISO decoders were also designed based on the scheme presented in Fig. 1. For the radix-2 and radix-4 SISO decoders, the ACS units in [START_REF] Benkeser | Design and optimization of an HSDPA turbo decoder ASIC[END_REF] and [START_REF] Studer | Design and implementation of a parallel turbo-decoder ASIC for 3GPP-LTE[END_REF] were adopted, respectively. For the radix-16 SISO decoder, the two-dimensional ACS unit in [START_REF] Tang | A 952ms/s Max-Log MAP decoder chip using radix-4 x 4 ACS architecture[END_REF] was chosen, and a SOU implementing the static comparison tree that minimizes the number of comparisons as presented in [START_REF] Shr | A 6.6pj/bit/iter radix-16 modified log-MAP decoder using two-stage ACS architecture[END_REF] was implemented. All SISO decoder architectures were synthesized for a 200MHz clock frequency. The input and β buffer complexities are not included in the comparison since their values depend on the window size. However, a radix-2 N T ACS unit enables to reduce the β buffer size by a factor of NT .

Thanks to the complexity reduction performed in all the SISO decoder blocks, our radix-16 SISO decoder is about 55% less complex than the considered radix-16 implementation. The proposed ACS unit helps considerably to achieve this result. Furthermore, the proposed BMU and SOU enable to overcome the hardware penalty cost introduced by a radix-16 approach. Compared to radix-2 and radix-4 SISO decoders, the proposed architecture is about 7.8 and 2.9 more complex, respectively. With this additional hardware complexity, the proposed SISO decoder improves by a factor of 4 and 2 the radix-2 and radix-4 SISO decoder throughput, respectively.

V. CONCLUSION

A low complexity radix-16 SISO decoder for the Max-Log-MAP algorithm is presented in this paper. Two complementary techniques have been proposed in order to limit the BER/FER performance degradation introduced by a TD architecture based on the proposed radix-16 SISO decoder. Moreover, an elimination of parallel paths in the trellis diagram allows us to use a radix-8 architecture for the ACS unit. The association of these different contributions enable to design a high speed low complexity radix-16 SISO decoder. The ideas presented in this paper can be applied to design higher radix SISO decoders with an acceptable cost-throughput ratio.

Fig. 1 .

 1 Fig. 1. SISO decoder architecture.

Fig. 2 .

 2 Fig. 2. Radix-16 ACS unit trellis diagram transition.

Figure 1

 1 Figure 1 details the block diagram of the considered SISO decoder. Inputs are the channel informationsystematic L(d k ) and redundant L(r k ) LLRand a-priori (L a k ) values. The Branch Metric Units (BMU) calculate γ values used by the ACS units. The Soft Output Unit (SOU) computes extrinsic values L e k and takes hard decisions dk . To ensure that the critical path is in the ACS unit, pipeline stages

Fig. 3 .

 3 Fig. 3. Proposed radix-16 BMU architecture.

Figure 4

 4 Figure 4 details the proposed radix-8 ACS unit. This unit is composed of four radix-2 ACS units and one radix-4 Compare-Select (CS) unit. First, eight additions are performed (state metrics plus

Fig. 4 .

 4 Fig. 4. Radix-8 ACS unit used.

Fig. 5 .

 5 Fig. 5. (a) Proposed radix-16 SOU. (b) Switch-I circuit. (c) Switch-II circuit.

Fig. 7 .

 7 Fig. 7. Fixed point simulation for the LTE turbo decoder (1024 bits per frame), when radix-2 and the proposed radix-16 SISO are used.

TABLE I HARDWARE

 I AREA IN TERMS OF EQUIVALENT 2-INPUT (NAND) GATE COUNT FOR DIFFERENT SISO DECODERS (INPUT AND β BUFFER NOT INCLUDED). CLOCK FREQUENCY IS SET TO 200MHZ.

										Radix-2			Radix-4	Radix-16 (radix-16 ACS unit)	Radix-16 (proposed)
					BMU					635				3.2k	28.6k	15.3k
		ACS (Area for all the 8 states)	3.4k ( [9], 10 bits)	8.2k ( [5], 11 bits)	34.1k ( [6], 12 bits)	16.6k (12 bits)
					SOU					2.2k				5.3k	23.5k	18.3k
			Total estimated SISO area			10.5k				28.1k	148.9k	82.1k
		1e-01
									R=1/2
		1e-02
			R=1/3
		1e-03
	BER / FER	1e-04
		1e-05
		1e-06
			BER -Radix-2
		1e-07	FER -Radix-2 BER -Radix-16, p
			FER -Radix-16, p
		1e-08	BER -Radix-16, p, Shift SISO FER -Radix-16, p, Shift SISO
		0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2	2.2	2.4	2.6
							E b /N 0 (dB)

Note that although a radix-8 ACS unit is used, the SISO decoder corresponds to a radix-16 architecture (N T = 4).

Note that it is possible that there is not a valid value for a specific d, since all its related paths could have been eliminated in the BMU unit.

These results have been obtained without applying the sliding window technique.


Yangyang Tang

Gopalakrishnan Sundararajan

Chris Winstead

Emmanuel Boutillon

Christophe Jégo

Michel Jézéquel

Techniques and prospects for fault-tolerance in post-CMOS ULSI

This paper presents a survey of fault-masking techniques suitable for tolerating short-duration transient upsets in minimum-scale switching devices. Two types of fault masking are considered. The first type, coded dual-modular redundancy (cDMR), represents a family of parity-checking methods suitable for correcting a low rate of transient upsets. The second type, Restorative Feedback (RFB), is a triple-modular solution suitable for compensating a higher rate of transient upsets. We show that cDMR can be used efficiently for crossbar-style logic, but is not efficient in general for all logic styles. By contrast, RFB offers a fixed redundancy, and can be applied in general to any logic circuit. Finally, we propose novel circuits for ternary Muller C implementation based on carbon nanotube FET devices.

I. INTRODUCTION

When devices are scaled to minimum physical dimensions, very small energy is available for storing and communicating logic levels between gates. Additionally, when devices are integrated at maximum density, the resulting high thermal density may seriously increase transient noise. Logic circuits must therefore operate with reduced signal-to-noise ratios that result in a high rate of momentary logic faults and memory state upsets. Transient signal upsets can be caused by thermal noise, electromagnetic interference (e.g. from switching events in neighbouring devices on the same chip), timing failures in synchronous pipelines, power supply noise, high-energy particle collisions and other sources [START_REF] Kamal | Intermittent faults: A model and a detection procedure[END_REF]- [START_REF] Baumann | Soft errors in advanced semiconductor devices-part i: the three radiation sources[END_REF]. Various techniques are available to mask the occurrence of momentary faults for binary and multiple-valued logic system [START_REF] Shim | Reliable low-power digital signal processing via reduced precision redundancy[END_REF]- [START_REF] Tang | A new single-error correction scheme based on self-diagnosis residue number arithmetic[END_REF].

In this work we present a survey of methods for tolerating transient upsets in nano-CMOS and post-CMOS ULSI systems. We first present a coded dual-modular redundancy (cDMR) technique, which has low redundancy in some cases. The cDMR method is permissible if certain constraints are imposed on the logic synthesis. The required constraints sometimes result in more complex logic synthesis, which may limit the generality of cDMR solutions.

Triple-Modular Redundancy (TMR) [START_REF] Lyons | The use of triple-modular redundancy to improve computer reliability[END_REF] provides a more general solution for protecting black-box logic modules without requiring any constraints on the logic synthesis method. TMR-style solutions have a fixed redundancy of two, whereas the cost of cDMR varies for each synthesized circuit. Hence it may be preferable to use TMR because it provides more certain characteristics during architecture planning. The Restorative Feedback method (RFB) [START_REF] Winstead | An error correction method for binary and multiple-valued logic[END_REF] is a TMRstyle solution which is specialized for correcting transient faults in digital logic. The RFB method was previously shown to be applicable to multiple-valued logic circuits. In this paper, we present novel ternary implementations of the Restorative Feedback (RFB) method, which is comprised of Muller Celements. We present two ternary C-element implementations using carbon nanotube FET (CNTFET) devices, which may be used to realize a multiple-valued RFB solution. A review of alternative implementations and tradeoffs is also presented.

Original Function

f 1 (u) Parity-Mapped Function f 2 (u) ECC s r ŝ r

II. CODED DUAL-MODULAR REDUNDANCY TECHNIQUES

One of the authors (Winstead) has proposed an LDPC-coded Fault Compensation Technique (LFCT) [START_REF] Winstead | Probabilistic LDPC-coded fault compensation technique for reliable nanoscale computing[END_REF] which achieves reliable operation in the presence of transient and permanent defects. The LFCT method uses Gaudet and Rapley's theory [START_REF] Gaudet | Iterative decoding using stochastic computation[END_REF] of stochastic decoding to correct errors that appear at the output of some logic computation.

The LFCT architecture is depicted in Fig. 1. In the proposed LFCT system, a logic function f 1 is subject to transient internal upsets, causing one or more errors to appear on the function's outputs, s. To correct these error, a second function f 2 is introduced which maps the input data u to a vector of parity-check bits r, so that the output [s r] forms a codeword according to a traditional block error correction design. In this system, we assume that the input vector u is error-free. If the number of errors in [s r] is sufficiently small, then they are corrected by the ECC module, resulting in a correct output vector [ŝ r]. Finally the corrected data ŝ can be propagated to other logic modules for additional computations.

If the logic function is constrained to use a crossbar synthesis architecture, as commonly proposed for post-CMOS electronics [START_REF] Dehon | Hybrid cmos/nanoelectronic digital circuits: devices, architectures, and design automation[END_REF], [START_REF] Rao | Logic mapping in crossbar-based nanoarchitectures[END_REF], then the fault statistics are suitable for using the LFCT technique. To explain this constraint, we may contrast the two circuits shown in Fig. 2. Both circuits implement a two-bit adder function. Fig. 2(a) shows a traditional ripple-carry implementation, and Fig. 2(b) shows a crossbar implementation. In the ripple-carry implementation, a single gate error may propagate to the several of the output signals. An example of error propagation is indicated by the ⋆ symbol in Fig. 2(a). In the LFCT system, error-propagation may induce many simultaneous faults in the [s r] codeword, which are not likely to be correctable.

In the crossbar implementation, as shown in Fig. 2(b), logic is implemented by fabrics of AND-logic and OR-logic. The "dots" in Fig. 2(b) indicate the placement of junctions which physically implement the logic operations. In this style of implementation, every operation is associated with only a single output. If a momentary fault occurs at some junction, it will propagate only to a single output. This style of logic guarantees that single-error events are correctable. The major disadvantage of crossbar logic is that the operation counts are not optimal. The crossbar adder in Fig. 2(b), for instance, has 57 separate operations. Crossbar logic does not generally obtain minimized gate complexity, but it offers improved reliability by eliminating error propagations. The added gate complexity could be considered as a form of redundancy. Furthermore, if crossbar-style logic is used with the LFCT method, then the logic function can be duplicated only once to provide error correction.

The authors (Tang, Winstead, Boutillon, Jégo and Jézéquel) have developed a specialized LFCT method, called the LDPC Stochastic Decoding (LSD) architecture [START_REF] Tang | An ldpc decoding method for fault-tolerant digital logic[END_REF], which can be considered as a circuit-level implementation of the LFCT concept. To evaluate the LSD solution, an example system was simulated using a variety of error-correcting codes. All codes are rate 1/2, meaning that there is one redundant parity bit for each systematic bit. The Bit Error Rate (BER) results for the LDPC codes applying LSD algorithm are shown in Fig. 3. In our simulations, the output bits s and r from F (x) and E • F (x), respectively, are assumed to have a uniform independent error probability of α. Internal errors are also modeled in the correction circuit, with a uniform error probability equal to ǫ. Stuck-at faults were also inserted in uniformly random positions in [s r] with a fault rate of β, here, β = 0.001. III. TMR METHOD: RESTORATIVE FEEDBACK TMR methods require duplicating a logic function twice, and require no constraints on how the logic functions are synthesized (i.e. TMR methods can correct cases of errorpropagation). Although cDMR requires only a single duplication, it requires suboptimal synthesis that may significantly increase the operation count. Therefore TMR methods are more attractive to be used with traditional non-crossbar logic styles. A variant of TMR, referred to as Restorative FeedBack (RFB), was previously described by one of the authors (Winstead) [START_REF] Winstead | An error correction method for binary and multiple-valued logic[END_REF]. The RFB method is based on Muller C-element gates, as shown in Fig. 4. The RFB circuit is derived from the theory of stochastic iterative decoding, and achieves better performance than traditional majority-based TMR for correcting momentary upsets. The RFB method is also applicable to M -ary logic systems as well as conventional binary logic circuits. A ternary RFB circuit was previously described using CMOS semifloating gate circuits.

(a)

A 0 A 0 B 0 B 0 A 1 A 1 B 1 B 1 S 0 S 1 C 1 AND fabric OR fabric (b)
✲ ✲ C ✚✙ ✛✘ ❢✲ 1 0 φ ❢ S ✈ ✲ y 2 ✲ ✲ ✲ C ✚✙ ✛✘ ❢✲ φ ❢ S ✈ ✲ y 3 ✲ ✲ ✲ C ✚✙ ✛✘ ❢✲ φ ❢ S ✈ ✲ y 1 ✲ ✈ ✈ ✈ x 1 x 2 x 3
In this section, we consider a post-CMOS implementation based on carbon nanotube FET devices (CNFETs). We first review the basics of ternary logic, and then investigate the CNTFET transistor, its operation and multi V th design in the context of CNTFET devices. Muller C Element is presented afterwards. At last, two proposed circuits for implementing ternary C-element are illustrated, as well the simulation results.

A. CNTFET Transistor

Fig. 5 shows the cross-section of a CNTFET transistor. CNTFET transistors are constructed by replacing the silicon channel of traditional MOSFETs with semiconducting carbon nanotubes (CNT). The CNTs deployed in the channel region can be either metallic or semiconducting depending on the angle of atom arrangement along the tube. Similar to traditional MOSFETs, CNTFETs have four terminals: drain, gate, source and back-gate [START_REF] Appenzeller | Carbon nanotubes for high-performance electronics x2014;progress and prospect[END_REF].

B. Overview of Ternary Logic

In this subsection, we review the basics of ternary logic. Ternary logic functions are defined as those having importance if a third value is introduced into the binary logic value. Ternary Logic values consists of three states (1, 2, 3) representing false, undefined and true respectively [START_REF] Yamamoto | P-functions-ternary logic functions capable of correcting input failures and suitable for treating ambiguities[END_REF]. Table I shows the logic values and the corresponding voltage levels used in this paper to represent ternary logic values.

The inverter is a basic logic gate used in digital design. In ternary logic, there are three types of inverters: positive ternary inverter (PTI), standard ternary inverter (STI) and negative ternary inverter (NTI). The truth table for all the inverters mentioned above is shown in table II.

The current through the CNTFET transistor is controlled by adjusting the device parameters such as gate length, number of nanotubes, chirality vector and the pitch distance [START_REF] Deng | Modeling and analysis of planar-gate electrostatic capacitance of 1-d fet with multiple cylindrical conducting channels[END_REF]. The operation of CNTFETs is very similar to traditional MOSFETs except for differences in the device orientation. In contrast to MOSFETs, the source and drain terminals of CNTFETs transistors are not interchangeable. This is because doping variations are introduced in the source and drain regions during lithography rendering it impossible to interchange them [START_REF] Avouris | Carbon nanotube electronics[END_REF].

For CNTFET circuits, multiple Vth transistors can be achieved by tuning the chirality of each transistor. Even though CNTFET are relatively new, plenty of existing literature has documented the feasibility of adjusting the Vth of transistors by growing different nanotube diameters [START_REF] Li | Growth of single-walled carbon nanotubes from discrete catalytic nanoparticles of various sizes[END_REF].

The V th of the intrinsic CNT channel is given by :

V th ≈ E g 2e = √ 3 3 aV π eD CN T
where a = 2.49 × 10 -10 m, V π = 3.033 ev, and

D CN T = √ 3a 0 π n 2 + m 2 + nm
where a 0 = 0.142 nm is the inter-atomic distance between each carbon atom and its neighbor and (m, n) is called the chirality vector that describes the structure of a carbon nanotube [START_REF] Lin | A new sram cell design using cntfets[END_REF].

C. Muller C Element

In this subsection, we describe the binary and ternary Muller C elements and its operation. The C element has been employed in asynchronous circuits design [START_REF] Shams | A comparison of cmos implementations of an asynchronous circuits primitive: the c-element[END_REF]. Fig. 6 shows the circuit and the schematic of the Muller C element. The

c = c (a + b) + a.b
The ternary Muller C element is similar to its binary counterpart except for the fact that the inputs and the output could take three logic values. Hence there will be an increase in the number input combinations. Table IV shows the truth table of ternary Muller C element. From the table, we find that a new value is latched at output c when the two inputs a and b are equal such that c = a = b. Otherwise, the ternary Muller C retains its old value. The inveter latch consisting of cross coupled inverters are also implemented using static complementary CNTFET design from literature [START_REF] Lin | A novel cntfet-based ternary logic gate design[END_REF].

D. Static Complementary CNTFET Logic Implementation

Static complementary logic design is a widely used design style in MOS technology because of its robustness and low

Input a Input b Output c 0 0 0 0 1 S 0 2 S 1 0 S 1 1 1 1 2 S 2 0 S 2 1 S 2 2 2
The operation of the circuit is as follows. When the input voltages are below 300 mV P-CNTFET transistors T1, T2, T4 and T5 turn on thereby pulling the node cout to a logic high value. This logic value is passed on to cross coupled inverters thereby pulling the output c to a low value. When the input voltages are raised above 300 mV transistors T4, T5 ,T9 and T10 are off and the combination of transistors T1, T2, T7 and T8 along with the diode connected transistors T3 and T6 produce a voltage drop of 0.45 V or logic 1 at node cout. This intermediate logic value is fed to the inverter latch thereby producing a voltage drop of 0.45 V or logic 1 at the output c. As the input voltages are raised above 600 mV, transistors T4 and T5 are off and transistors T9 and T10 are on which pull the node cout to a logic low value. This low logic value is fed to the cross coupled inverters thereby driving the node output to a high value. The Transistors T1, T2, T7 and T8 have diameter of 1.487 nm. Transistors T4, T5, T9 and T10 have diameter of 0.783 nm. Diode connected transistors T3 and T6 have diameter of 1.018 nm. Two supply voltages are used in this implementation. The first supply voltage is 0.45V and the second supply voltage is 0.9V. When the input voltage is below 0.3V, T1, T2, T3 and T4 are on and the remaining transistors are off. A high output voltage resulting from first part is then fed to the inverter latch thereby resulting in low output voltage at the output c. The output voltage is 0V. When the input voltage is between 0.3V and 0.6V T5 and T6 are on and T1, T2 are operating in sub-threshold region. As discussed in [START_REF] Nan | Novel ternary logic design based on cnfet[END_REF] lower threshold CNTFET devices operating in sub-threshold region with forward biasing offers high drive current which is equal in measure compared to the drive current provided by higher threshold CNTFET device operating in super-threshold region. Therefore, the output is the first supply voltage VDD1 which is 0.45V. When the input voltages are above 0.6V then transistors T1, T2, T3 and T4 are off and low voltage value appears at the cout which is fed to the inverter latch resulting in high output at c. The designs were built and evaluated using HSPICE circuit simulator. The technology node used in this paper to evaluate 0.9V a

E. CNTFET Muller C Element with different back gate biased voltages

0.9V b 0V b 0V a -0.9V b -0.9V a 0.9V a 0.9V b V DD2 = 0.9V V DD1 = 0.45V T 3 T 4 T 1 T 2 T 7 T 8 T 5 T 6 cout c c I1 I2

F. Simulation Results

In this subsection we present our simulation results. The static and the resistive load CNTFET designs were built and simulated using HSPICE. Transient simulations were run on the two designs to verify the functionality of the designs. Figure 9 and Figure 10 show the transient simulation results for Complementary CNTFET design. From the transient simulations we observe that when the first input a is held at a steady state and input b is swept from logic low to a high value the output changes state when the two inputs are at the same logic level. When the inputs a and b are at a different logic level the output holds the previous latched logic value. The inputs are swept from logic low to logic high and then changed to logic low in constant steps and the output is observed for a full low to high and a high to low transition. From the analysis, we can conclude that the ternary C element works as intended.

Table V shows the average delay, average power and the Power Delay Product (PDP) for the two proposed CNTFET ternary Muller C designs. From the table, we could see that though back gate design dissipates less power compared to the complementary CNTFET design it has a larger propagation delay which results in a higher power delay product.

IV. CONCLUSION

While electronic circuit is downsized into the nanometer scale, efforts are necessary to jointly optimize robustness and efficiency, while accounting for the probabilistic nature of nanometer devices. Among all available fault-tolerant methods to deal with transient fault occurrence, two latest works, the DMR-based LDPC-coded Fault Compensation Technique (LFCT) and TMR-based Restorative FeedBack (RFB) technique, are first reviewed. Finally, two novel implementations of ternary Muller C-element using carbon nanotube FETS (CNTFETs) are presented for the purpose of error-resilience in multi-valued logic circuit.

Figure 1 :

 1 Figure 1: Architecture of the LFCT method. The original logic function f1 is composed with a block ECC code to create the parity-mapped function f2. Outputs are then corrected by the ECC circuit.

Figure 2 :

 2 Figure 2: Implementations of a two-bit binary adder function, representing a traditional ripple-carry design (a), and a crossbar design suitable for some nanoelectronic device families (b). The '⋆' symbol indicates the occurrence of an error which propagates to multiple signals.

Figure 3 :

 3 Figure 3: Simulation results for rate-1/2 LDPC codes based on LSD architecture with five iterations. The hard-fault rate β is 0.001.

Figure 4 :

 4 Figure 4: The RFB circuit based on Muller C-element gates. The S gate is a storage element.

Figure 5 :

 5 Figure 5: Side view of a CNTFET transistor

Figure 6 :

 6 Figure 6: Binary CMOS Muller C element

Fig. 8 Figure 7 :

 87 Fig. 8 shows the circuit diagram of CNTFET based Muller C element based on different back gate biasing voltages. In this implementation, two transistors are reduced compared to

Table I :

 I Ternary Logic Symbols

	Input X	STI	PTI	NTI
	0	2	2	2
	1	1	2	0
	2	0	0	0

Table II :

 II Ternary Inverter Truth TableMuller C-element is a fundamental circuit element widely used for control synchronization in asynchronous designs. In general, a C-element is a state holding circuit which is transparent when all its inputs are equal, and holds the previous output otherwis. The C element consists of two inputs a, b and one output c. TableIIIshows the truth table of the binary Muller C element. The logic equation describing the behaviour of the Muller C element is described as follows:

Table III :

 III Muller C-element truth table. The S value indicates that the C-element holds its state (i.e. does not change value) for the corresponding pattern of inputs.

Table IV :

 IV Ternary Muller C-element truth table.

Table V :

 V Performance Comparison Of Two Proposed Designs

					First Input -a
	Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
					Time in 10E-8 seconds				x 10 -8
					Second Input -b
	Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
					Time in 10E-8 seconds				x 10 -8
						Output -c
	Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
					Time in 10E-8 seconds				x 10 -8
	Figure 10: Transient Results Of Back-Gate biased Ternary CNTFET
			Muller C Element
				Design		Delay(ps) Power(uW)	PDP(E-
									16J)
		Complementary CNTFET	120.704	3.417	4.12
		Back gate biased CNTFET	359.925	1.548	5.57
											Figure 8: CNTFET Muller C Element with different back gate
												biased voltages
												First Input -a
											Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
												Time in 10E-8 seconds	x 10 -8
												Second Input -b
											Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
												Time in 10E-8 seconds	x 10 -8
												Output -c
											Voltage in volts	0 0 0.2 0.4 0.6 0.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
												Time in 10E-8 seconds	x 10 -8
											Figure 9: Transient Results Of Static Complementary Ternary CNT-
												FET Muller C Element
											the two designs was 32 nm. Stanford CNTFET device models
											were used to design we perform simulations in HSPICE
											using CNTFET parameter models by Stanford Nanoelectronics
											Group in the 32nm technology node [28]. All the designs were
											simulated at supply voltage of 0.9 V and at room temperature
											of T = 27 C.

As long as α < 0.05, the output s from F (x) can be recovered with a significantly reduced error probability. As α is reduced below β, the performance becomes dominated by the gate-level fault probability ǫ. The results show that, in the case of ǫ = 10 -5 , the LSD architecture introduces gains about two orders of magnitude by comparison with uncoded data (output s from F (x)). Consequently, the proposed decoding method is able to suppress the resulting error probability to a level equal to the decoder's internal fault rate.⋆ A 0 B 0 A 1 B 1 C 1 S 1 S 0 ⋆ ⋆ ⋆ ⋆

ACKNOWLEDGEMENT

This work was supported by the US National Science Foundation under awards ECCS-0954747 and CCF-0916105.


Cristopher Hermosilla

STRATIFIED DISCONTINUOUS DIFFERENTIAL EQUATIONS AND SUFFICIENT CONDITIONS FOR ROBUSTNESS

Keywords: 1991 Mathematics Subject Classification. Primary: 34A12, 34A36; Secondary: 34D10 Discontinuous vector fields, Stratified systems, Existence of solutions, Robustness, State-constrained ODEs

This paper is concerned with state-constrained discontinuous ordinary differential equations for which the corresponding vector field has a set of singularities that forms a stratification of the state domain. Existence of solutions and robustness with respect to external perturbations of the righthand term are investigated. Moreover, notions of regularity for stratifications are discussed.

Introduction

One important issue in Control Theory of ODEs is the feedback synthesis, that is, for a control system on R N (1) ẋ = f (x, u), x(t) ∈ K ⊆ R N , u(t) ∈ A ⊆ R m construct a function U : K → A in such a way that all the trajectories of the vector field x → f (x, U (x)) belong to a certain class of curves of the control system; typically, (sub)minimizers of a given cost function.

It is a well-accepted fact that optimal feedback laws are in general discontinuous functions of the state. For instance, the first order necessary conditions of optimality (Pontryagin maximum principle) show that even for linear systems it is likely to occur. Moreover, for Time-Optimal problems, the Brockett Condition represents a topological obstruction for the existence of such feedbacks, and even for suboptimal strategies it remains true; see for instance the discussion in [START_REF] Clarke | Discontinuous feedback and nonlinear systems[END_REF][START_REF] Sontag | Mathematical control theory: deterministic finite dimensional systems[END_REF]. Therefore, the classical theory of ODEs can not be applied to a closed-loop systems arising in this way, and so existence of solution in the sense of Carathéodory and robustness with respect to perturbations on the data are more delicate to treat.

It is also known that usually the singular set of the optimal strategies have a regular structure. This means that it can be decomposed into a locally finite family of submanifolds, in such a way that in each of these sets the feedback is smooth. Indeed, many authors have shown this fact for certain classes of controlled dynamics; see for instance [START_REF] Hájek | Terminal manifolds and switching locus[END_REF][START_REF] Brunovskỳ | Every normal linear system has a regular time-optimal synthesis[END_REF][START_REF] Brunovskỳ | Regular synthesis for the linear-quadratic optimal control problem with linear control constraints[END_REF][START_REF] Sussmann | Regular synthesis for time-optimal control of single-input real analytic systems in the plane[END_REF][START_REF] Meeker | Local time-optimal feedback control of strictly normal two-input linear systems[END_REF][START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF] among others.

The above mentioned contributions make suitable to assume that optimal synthesis may exhibit a stratified structure where exists a partition of the state-space K into a disjoint family of sets {M i } i∈I (each of the M i is called stratum), such that U (x) = U i (x), whenever x ∈ M i and in some of these strata, the ODE ẋ = f (x, U i (x)) admits arcs remaining on M i for at least a small interval of time. In fact, it is likely to have a subfamily of strata where no trajectory of the system can stay for a set of times with positive measure. In this way, trajectories may slide from one stratum to another of bigger dimension. The aim of this work is to study ODEs arising in this context. For this purpose the concept of stratified vector fields (SVF) is introduced. The main issues are the existence of solutions and the robustness of trajectories with respect to external perturbations on the velocity. Particular emphasis is put on the interplay between regularity conditions on the sets M i and pointwise conditions on the vector field in order to ensure the existence of solutions. In particular, the case of ODEs on closed sets with empty interior is treated, and to do this the notion of relatively wedgedness is required.

Let us mention that the notions of SVF and relatively wedged set are already present in the literature, however their definitions are slightly different from the adopted in this work. In particular, the first one was brought to light in the notes [START_REF] Mather | Notes on topological stability[END_REF] in order to prove properties about Whitney stratifications. The principal and most important difference between both definitions is that here a SVF does not need to be defined for all the strata, and in fact, as described above, it is likely that this situation occurs. On the other hand, the second concept was studied in [START_REF] Barnard | Flow invariance on stratified domains. Set-Valued and Variational Analysis[END_REF] with the purpose of proving a similar existence theorem of trajectories of a stratified differential inclusion defined everywhere on the space. In that case, the definition of relatively wedged set was made only for submanifolds that are also proximal smooth sets while here the definition is more general.

The main tool for studying the stability of solutions to stratified ODEs is the modulus outward-pointing. The main feature of this function is that it measures the maximum size of the perturbation and it describes the class singularities allowed in order to make the system stable in the sense described earlier.

Similar works dealing with discontinuous ODEs can be found in the literature. To the best of our knowledge, the closest one is [START_REF] Marigo | Regular syntheses and solutions to discontinuous odes[END_REF] where the author study the properties of a discontinuous ODE starting from an axiomatic definition of stratified solutions. Other papers focused on 3-dimensional piecewise smooth dynamical systems and switching surface are [START_REF] Jeffrey | The two-fold singularity of discontinuous vector fields[END_REF][START_REF] Teixeira | Stability conditions for discontinuous vector fields[END_REF] where a qualitative analysis is made. On the other hand, the literature on generalized solutions is quite large and many authors in the last decades have addressed their attention into the problem; see for instance [START_REF] Brunovskỳ | The closed-loop time-optimal control. I: Optimality[END_REF][START_REF] Brunovskỳ | The closed-loop time optimal control. II: Stability[END_REF][START_REF] Hájek | Discontinuous differential equations, I[END_REF][START_REF] Hájek | Discontinuous differential equations, II[END_REF][START_REF] Honkapohja | Stability with regime switching[END_REF][START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF] and the references therein.

Further works dealing with stratified systems and Hamilton-Jacobi equation can be found in the literature; see for instance [START_REF] Bressan | Optimal control problems on stratified domains[END_REF][START_REF] Barnard | Flow invariance on stratified domains. Set-Valued and Variational Analysis[END_REF][START_REF] Rao | Hamilton-jacobi-bellman equations on multi-domains[END_REF][START_REF] Hermosilla | Infinite horizon problem on stratifiable state-constraints set[END_REF]. This paper is organized as follows. Section 2 describes the model of discontinuous ODE at issue and what is understood as a solution to this. Section 3 deals with the problem of existence of solutions. In Section 4 some regularity notions over the stratification are discussed. In particular, the notions of curvature and relatively wedgedness are considered. In the last section the problem of robustness of the stratified solutions is investigated and the already referred outward-pointing modulus of a vector field is introduced. Finally, in the Appendix, some properties of this last function are discussed. The distance function to S is dist S (x) = inf{|x -y| : y ∈ S}. If the infimum is attained, it is called the projections of x over S and it is denoted by proj S (x). For two compact sets of R N , S 1 and S 2 , the Hausdorff distance between them is given by

d H (S 1 , S 2 ) = max sup x∈S2 dist S1 (x), sup x∈S1 dist S2 (x)
and the distance between two cones of R N , C 1 and C 2 , is defined as follows

D(C 1 , C 2 ) = d H (C 1 ∩ S, C 2 ∩ S). A sequence of cones {C n }, is said to converge to C, another cone of R N , provided D(C n , C) → 0 as n → +∞. It is denoted by C n → C.
For a family {v 1 , ..., v p } of vector in R N , span{v 1 , ..., v p } and cone{v 1 , ..., v p } stand for the space and the cone generated by the family, respectively. Besides, SO(N ) indicates the set of orthonormal matrices of dimension N whose determinant is equal to one.

Let M be a submanifold of R N and let x ∈ M. The tangent and normal spaces to M at x are denoted by T M (x) and N M (x).

Statement of the problem

Before describing the problem, some concepts need to be introduced. In particular, a precise definition of stratification is required and also the basic assumption over this need to be settled.

2.1.

Preliminaries. The elementary objects necessary to define a stratification of a set in R N are the embedded manifolds. The definition is recalled for sake of completeness.

Definition 2.1. A subset M ⊆ R N is an embedded manifold of R N provided for every x ∈ M there exists an open neighborhood Θ ⊆ R N of x such that Θ ∩ M = {x ∈ Θ : h(x) = 0}
where h : Θ → R N -d is a smooth function whose derivative Dh(x) is surjective on Θ and d ∈ {0, . . . N }. In this case, d is called the dimension of the manifold.

In this paper it is always assumed that smooth means at least of class C 2 . Besides, when dealing with stratification it is usual to consider additional regularity assumptions in the way the strata fix together. The basic assumption, in this sense, is the so called Whitney-(a) Condition. The reader is referred to [START_REF] Mather | Notes on topological stability[END_REF] for a further discussion about this condition. Definition 2.2. Let M i and M j be two embedded manifolds of R N . The pair (M i , M j ) satisfies the Whitney-(a) condition at x ∈ M i ∩ M j if for each sequence {x n } ⊆ M j with x n → x the following holds:

T Mj (x n ) → T ⇒ T Mi (x) ⊆ T . (2)
Through this paper the following binary relation is used to set up a hierarchy on a stratification whose strata are exactly {M i } i∈I . ∀i, j ∈ I : i j (or j i) ⇐⇒ M i ⊆ M j .

Definition 2.3. A set K ⊆ R N is said to be (Whitney) stratifiable if there exists a locally finite collection {M i } i∈I of embedded manifolds of R N such that:

• K = i∈I M i and M i ∩ M j = ∅ when i = j,
• Whenever M i ∩ M j = ∅, necessarily i j.

• For any i, j ∈ I, if i j then pair (M i , M j ) satisfies the Whitney-(a) condition at each x ∈ M i .

When a set K admits a stratification in the way described before, ı(x) will stand for the index i ∈ I such that x ∈ M i .

The next figure shows an example of a stratifiable set K in R 2 and a stratification of this set. In Figure 1b, M 0 is a stratum of full dimension (d = 2) (which coincides with int(K)), M 1 , . . . , M 7 are the one dimensional strata and finally, M 8 , M 9 and M 10 are strata of dimension 0, that is, they are isolated points.

K (a) Stratifiable set K M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 (b) Stratification of K Figure 1.
Example of a stratifiable set and its stratification.

Remark 1. An important class of sets that admits a stratification as described above are the polytopes of R N . In fact, these sets can be decomposed into a finite number of open convex polytopes of the form:

P = x ∈ R N η k , x = α k , k = 1, . . . , n, η k , x < α k , k = n + 1, . . . , m where η 1 , . . . , η m ∈ R N .
Furthermore, the class of sets that admits a stratification is quite broad, it includes sub-analytic and semi-algebraic sets and also definable sets of an o-minimal structure; see for instance [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF][START_REF] Kaloshin | A geometric proof of the existence of whitney stratifications[END_REF]. 2.2. Stratified vector fields. As mentioned in the introduction, it is suitable to have some strata where no curve can slide for. Whence, a selection of index where the vector fields are defined need to be involved in the definition, which is outlined in the following definition. Definition 2.4. Let K ⊆ R N be a stratifiable closed set whose stratification is denoted by {M i } i∈I . Let I 0 ⊆ I be a subset of index such that {M i } i∈I0 is dense in K. Then a stratified vector field (SVF) is a family of tangent vector fields

G = {(g i , M i )} i∈I0 such that for each i ∈ I 0 g i (x) ∈ T Mi (x), ∀x ∈ M i . ( 3
)
A SVF is said to be regular if for each i ∈ I 0 , g i is continuous on M i and it can be continuously extended up to M i . In addition, a SVF has linear growth if there exists a constant c > 0 such that

|g i (x)| ≤ c(1 + |x|), ∀i ∈ I 0 , x ∈ M i . (4)
By [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF], the subset of index I 0 is in fact the selection of strata where it is allowed to slide for. Therefore, the manifold corresponding to the index I 0 are usually called sliding strata and the others, bifurcation strata.

Figure 2 shows a SVF defined on K = R 2 . In this case, the stratification of the space consists in M 0 = {(0, 0)}, M 1 , . . . , M 4 the positive and negative semi-axis and M 5 , . . . , M 8 the quadrants of R 2 . Note that in this situation I = {0, . . . , 8} and the vector fields are defined on all the strata except on M 1 and M 2 . Therefore, I 0 = {0, 3, 4, 5, 6, 7, 8} and, M 1 and M 2 are the bifurcation strata of this SVF. On the other hand, as G in general forms a discontinuous map with well determined singularities, it is useful to introduce some notation to identify the surrounding strata where the vector field is defined. This is denoted by

I 0 (i) := {j ∈ I 0 : j i} ∀i ∈ I. (5
(D) ẋ = g i (x) a.e. whenever x ∈ M i x(0) = x 0 .
For all the purposes, the equation (D) will be called the stratified ODE associated with the stratification {M i } i∈I and with the dynamic G = {(g i , M i )} i∈I0 . The attention is centered in Carathéodory solutions to (D), that is, absolutely continuous arcs satisfying

x(t) = x 0 + i∈I0 Ji(t) g i (x(s))ds, ∀t ∈ [0, T ), (6)
where

J i (t) = {s ∈ [0, t) : x(s) ∈ M i }.
In particular, this implies that each J i (T ) is a negligible set whenever i / ∈ I 0 . In order to emphasis the dependence with respect to the stratified structure of the problem, x(•) is called a stratified solution to (D).

Remark 2. Broadly speaking, the set of solutions defined in this way is not closed. In fact, since {M i } i∈I0 is dense in K, the set I 0 (j) = ∅ for any j / ∈ I 0 and so, it may be possible to create an absolutely continuous arc that is the limit of trajectories,

ẋn = g i (x n ), x n (0) = x n 0 ∈ M i , with x n 0 → x 0 ∈ M j with j i.
It could happen that the limiting trajectory lies on M j for a set of times of positive measure. To avoid these situations, some types of singularities have to be ruled out; see Section 5 for more details.

Remark 3. If x(•) is a stratified solution to (D) with G = {(g i , M i )} i∈I0 having linear growth, then ẋ(t) ≤ c(1 + |x(t)|), a.e. t ∈ [0, T ).
and so, the Gronwall estimation holds (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Proposition 4

.1.4]), |x(t) -x(s)| ≤ (e c(t-s) -1) (|x(s)| + 1) , ∀0 ≤ s < t < T. Moreover, sup{|x(s)| : s ∈ [0, T )} < +∞.

Existence of solutions

Recall that a SVF is not necessarily defined everywhere on the state space. For this reason, the analysis is divided into two cases, namely, when it is defined everywhere (I 0 = I) and when it is not (I 0 = I). The first case in simpler and merely need the definition of SVF. The other case, is more delicate to treat and some extra hypothesis are required. (1) The Bouligand cone or Contingent cone to S at x, denoted by T B S (x), is

T B S (x) = v ∈ R N : lim inf t→0 + dist S (x + tv) t ≤ 0 .
(2) The Clarke tangent cone to S at x, denoted by T C S (x), is

T C S (x) = v ∈ R N : lim sup y→x, t→0 + dist S (y + tv) -dist S (y) t ≤ 0 .
Remark 4. Note that T C S (x) is always convex and closed, in particular, ri T C S (x) and rbd T C S (x) are well defined. Furthermore, T C S (x) ⊆ T B S (x). The analysis of the existence of stratified solution is based in the Nagumo Theorem whose proof can be found in [1, Theorem 4.2.2] for instance.

Proposition 1 (Nagumo Theorem). Let S ⊆ R N be a locally compact set and consider f : S → R N a continuous map. Suppose that

f (x) ∈ T B S (x), ∀x ∈ S. (7)
Then for all x 0 ∈ S there exists T > 0 such that the differential equation ẋ = f (x), x(0) = x 0 has a solution lying in S on [0, T ).

3.2.

Case I 0 = I. In this situation, the simpler one, the existence of local solutions is ensured by the continuity and the tangentiality of the vector fields on each stratum.

Theorem 3.2. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a regular SVF denoted by G = {(g i , M i )} i∈I0 and suppose that I 0 = I. Then for every x 0 ∈ K there exist T > 0 and a stratified solution to (D) defined on the interval [0, T ). Moreover, if the SVF has linear growth, then T = +∞.

Proof. Let x 0 ∈ K and set i = ı(x 0 ). Since i ∈ I 0 , the following ODE is well defined

(D i 0 ) ẋ = g i (x), x(0) = x 0 .
Note that M i is locally compact. Since g i is continuous and satisfies condition (3), the Nagumo Theorem implies that (D i 0 ) has at least a solution that remains in M i on an interval [0, T ), for T > 0. This is clearly a stratified solution to (D).

On the other hand, suppose that G has linear growth on K. Let x(•) be a maximal solution (D) defined on [0, T ) and assume that T < +∞. By Remark 3 there exists

a constant C = C(x 0 ) > 0 such that for all [s, t] ⊆ [0, T ) |x(t) -x(s)| ≤ C(e L(t-s) -1).
This means that for any t n ր T , the sequence {x(t n )} is a Cauchy sequence, and so the limit is well defined. Using the same inequality it is possible to see that the limits are independents of the sequences, so the limit x(T ) := lim t→T -x(t) exists and belongs to K, which contradicts the maximality of T . So, T can not be bounded.

3.3.

Case I 0 = I. In presence of bifurcation strata the existence of solutions to a stratified ODE requires additional hypothesis. For example, consider K = R, the stratification M 0 = {0}, M 1 = (-∞, 0) and M 2 = (0, +∞), and the SVF G = {(M i , g i )} i∈{1,2} with g 1 (x) = 1 and g 2 (x) = -1. Clearly, no solution starting from x 0 = 0 exists.

The problem in this case is reduced to study the existence of solution of an ODE on M, an embedded manifold of R N , whose initial condition lives in M \ M. Theorem 3.3. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a regular SVF denoted by G = {(g i , M i )} i∈I0 and suppose

     ∀x ∈ K with ı(x) ∈ I 0 , ∃j ∈ I 0 (ı(x)) and ∃r > 0 such that: i) x + (0, r]B(g j (x), r) ∩ M j ⊆ M j . ii) g j (y) ∈ T B Mj (y) \ {0}, ∀y ∈ B(x, r) ∩ M j . (H 0 )
Then for every x 0 ∈ K there exist T > 0 and a solution to (D) defined on [0, T ). Moreover, if the SVF has linear growth, then T = +∞.

Proof. Let x 0 ∈ K and set i = ı(x 0 ). In view of Theorem 3.2, it is only necessary to consider the case i / ∈ I 0 . To do this, set j ∈ I 0 (i) and r > 0 given by (H 0 ). Note that B(x 0 , r) ∩ M j is locally compact, so by condition (H 0 ) part (ii) and Nagumo Theorem, there exists a curve x(•) associated with the vector field g j starting from x 0 lying in M j ∩ B(x 0 , r) on an interval of time [0, T ). Moreover, g j (x 0 ) = 0 and since g j is continuous on M j , by reducing T if necessary, x(t) ∈ x 0 + (0, r]B(g(x 0 ), r) for every t ∈ [0, T ). Therefore, by (H 0 ) part (i), x(t) ∈ M j on [0, T ) and so, the arc x(•) is a stratified solution to (D).

Finally, if G has linear growth, the same arguments as in the proof of the previous theorem are valid, so the conclusion follows.

Remark 5. Note that if g j (x) ∈ int T C M (x) in (H 0 ), then conditions (i) and (ii) are automatically satisfied; see for instance [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Section 3.6]. In this case, since int T C Mj (x) = ∅, M j is said to be wedged.

Wedgedness and regularity concepts

The main goal of this section is to provide a pointwise criterion so that Hypothesis (H 0 ) holds. The principal motivation for doing this is based in the especial case when dim(M j ) = N ; see Remark 5. The idea is to extend the notion of wedgedness to sets that are the closure of embedded manifolds of R N . In order to do this, some regularity concepts over those sets need to be studied. (1) The Proximal normal cone to S at x, denoted by N P S (x), is the set of all η ∈ R N such that σ|x -y| 2 ≥ η, yx ∀y ∈ S, for some σ = σ(x, η) ≥ 0.

(2) The Limiting normal cone to S at x, denoted by N L S (x), is given by

N L S (x) := lim n→∞ η n : ∃{x n } ⊆ S with x n → x, ∃η n ∈ N P S (x n ) .
(3) The Clarke normal cone to S at x, denoted by N C S (x), is exactly the convex closed hull of the N L S (x). Remark 6. N P S (x) is always convex and possibly N P S (x) = {0}. Besides, N L S (x) is closed, possibly non convex and always contains a nonzero vector. Additionally, its graph gr

N L S (x) is closed in S × R N , where gr N L S (x) = (x, η) : η ∈ N L S (x)
. Note that by definition, the following inclusion is alway held

N P S (x) ⊆ N L S (x) ⊆ N C S (x) ∀x ∈ S. Also, T C S (x) and N C S (x) satisfy the polar relation T C S (x) = v ∈ R N : v, η ≤ 0, ∀η ∈ N C S (x) .
As shown in the next proposition, some interesting properties come to light when studying normal and tangent cones to sets that are the closure of an embedded manifold of R N . Besides, something extra can be said about the structure of the Limiting and the Clarke normal cone. Proposition 2. Let M be an embedded manifold of R N of dimension d then:

(1)

T B M (x) = T C M (x) = T M (x) for any x ∈ M. (2) N P M (x) = N C M (x) = N M (x)
N M (y) = Dh(y) T R N -d , ∀y ∈ M ∩ B(x, r),
where h : B(x, r) → R N -d is smooth and surjective on B(x, r). Let us fix r 0 ∈ (0, r) and set

L 0 = max y∈B(x,r0) |D 2 h(y)|, λ 0 = min y∈B(x,r0)
λ min (Dh(y)Dh(y) T ).

Here, λ min (A) stands for the minimal eigenvalue of a square matrix A. Note that, L 0 < +∞ because h is of class C 2 and λ 0 > 0 since Dh(y) is surjective on B(x, r). Thus, by the Mean Value Theorem

|Dh(x)(y -x)| ≤ L 0 |x -y|, ∀y ∈ M ∩ B(x, r 0 ).
Let η ∈ N M (x), then there exists µ ∈ R N -p such that η = Dh(x) T µ and so λ 0 |µ| 2 ≤ |η| 2 . Hence, by the previous estimation

η, y -x = Dh(x) T µ, y -x ≤ L 0 √ λ 0 |η||y -x| 2 , ∀y ∈ M ∩ B(x, r 0 ).
If y / ∈ M ∩ B(x, r 0 ), then the inequality holds trivially with r0 2 . Therefore,

δ = min r 0 2 , √ λ 0 2L 0 , in case of L 0 = 0, otherwise, δ = r0 2 .
Finally, by an argument of density, the inequality holds up to M.

(3) Take x ∈ M. If x ∈ M or d = N , the conclusion is straightforward and the vectorial subspace coincides with the normal space and with {0}, respectively. On the other hand, suppose that x ∈ M\M, then there exists

{x n } ⊆ M with x n → x ∈ M. Whence, for each n ∈ N, {N C M (x n )
} is a subspace of dimension N -d and it is possible to select an orthonormal base for this subspace, denoted by {η 1 n , . . . . , η N -d n }. Without loss of generality, for every i = 1, . . . , Nd, each sequence η i n → η i . Clearly, {η 1 , . . . . , η N -d } is also an orthonormal family of vectors. Note that for any n ∈ N, η i n ∈ N L M (x n ) and since this cone has closed graph on M, each η i ∈ N L M (x). Note also that -η i ∈ N L M (x). Finally, span{η 1 , . . . , η N -d } ⊆ N C M (x) and the result follows.

Note that δ in Proposition 2 is the radius of a closed ball centered at x + δ |η| η which intersects M only at x. In this sense, it is possible to interpret this quantity as the curvature of M. Definition 4.2. Let M be an embedded manifold of R N .

(1) The radius of curvature of M at x ∈ M, denoted by κ(x), is given by

κ(x) = sup 2 η, y -x |y -x| 2 : η ∈ N M (x) ∩ S, y ∈ M \ {x} .
(2) M is said to have constant curvature if there is a constant κ 0 ∈ R such that κ(x) ≤ κ 0 for any x ∈ M.

The notion of bounded curvature can be extended to stratifiable sets in the following way. Definition 4.3. A stratifiable subset of R N is said to have bounded curvature if each its stratum has bounded curvature.

Next proposition shows the importance of the Whitney-(a) condition for the analysis. Roughly speaking, it provides a hierarchy between the Clarke normal cones of the strata. Proposition 3. Let K be stratifiable and let i, j ∈ I be two index associated with the stratification such that i j.

Then N C Mj (x) ⊆ N Mi (x) for any x ∈ M i ∩ M j . Proof. Note first that if S 1 ⊆ S 2 then N P S2 (x) ⊆ N P S1 (x) for every x ∈ S 1 . Let η ∈ N L Mj (x)
, by definition there exist two sequences {x n } ⊆ M j and {η n } ⊆ R N such that η n ∈ N P Mj (x n ) with x n → x and η n → η. Recall that the stratification is locally finite, so there is only a finite number of indices k ∈ I satisfying

x ∈ M k , M i ⊆ M k and M k ⊆ M j . (8)
Therefore, there exist a stratum M k that satisfies (8) and a subsequence of {x n } that belong to M k . Without loss of generality, assume that the subsequence is the whole sequence and that T M k (x n ) → T .

By the previous remark and by Proposition 2

η n ∈ N P Mj (x n ) ⊆ N P M k (x n ) = N M k (x n )
Therefore, η is orthogonal to T and so, by virtue of the Whitney-(a) condition applied to (M i , M k ), one gets that T Mi (x) ⊆ T and in particular, η ∈ N Mi (x).

Finally, since N L Mj (x) ⊆ N Mi (x) for any x ∈ M i , by taking convex closed hull in the last inclusion, the proof is completed. 4.2. Relatively wedged sets. In Proposition 2 it was shown that N C M (x) can be rewritten as N d × R N -d (after a change of coordinate), where N d ⊆ R d is convex closed cone. Note that this decomposition is not unique, in particular, the Clarke tangent cone may be contained in a affine subspace of dimension strictly lower than d. Nevertheless, if M is wedged (see Remark 5) then this decomposition is unique (identifying R N with R N × {0}) and even more, it is continuous with respect to x. These facts motivate the following definition. Definition 4.4 (Relatively wedged). Let M be an embedded manifold of R N of dimension d. We say that M is relatively wedged around x 0 ∈ M provided there exists a neighborhood Θ of x 0 and a continuous mapping A : M ∩ Θ → SO(N ), x → A x such that:

A x N C M (x) = N d x × R N -d with N d x pointed in R d .
A similar notion of relative wedgedness was also studied in [START_REF] Barnard | Flow invariance on stratified domains. Set-Valued and Variational Analysis[END_REF] for embedded manifold of R N without asking the continuity of the change of coordinates, but requiring an additional regularity property over the closure of the M. Remark 7. Suppose M is relatively wedged around x ∈ M.

(1) If d = N , by taking A x as the identity matrix of dimension N the original notion of wedgedness is recovered. (2) The projection π

x : N C M (x) → N d x × {0} N -d is closed in the following sense: if x n → x and η n → η as n → +∞, where η n ∈ N C M (x n ), then π xn (η n ) → π x (η) as n → +∞.
(3) Using similar arguments as in the wedged case, it is possible to prove that y → N C M (y) is graph-closed at x; see [10, Proposition 3.6.7 and 3.6.8]. The importance of this property and the relation with hypothesis (H 0 ) is summarized in the next proposition. Proposition 4. Let K be a stratifiable set and let M be one of the strata. Suppose that M relatively wedged around x ∈ M \ M.

(1) Then v ∈ ri T C M (x) if and only if ∃σ > 0 such that v, η ≤ -σ|π x (η)| ∀η ∈ N C M (x). (9)
(2) Let g : M → R N continuous with g(y) ∈ T M (y) for every y ∈ M with g(x) ∈ ri T C M (x) , then there exists r > 0 such that g(y) ∈ T C M (y), ∀y ∈ B(x, r) ∩ M.

(3) Suppose that K has constant curvature, then there exists r > 0 such that

(x + (0, r]B(v, r)) ∩ M ⊆ M.
Proof.

(1) For simplicity, assume that

N C M (x) = N d x × R N -d . In particular, T C M (x) = T d
x × {0} N -d and v = (w, 0). Since M relatively wedged at x, v ∈ ri T C M (x) if and only if w ∈ int(T d x ) relative to R d . However, this is equivalent to the existence of σ > 0 such that w, ζ ≤ -σ|ζ| ∀ζ ∈ N d

x . Finally, since v, η = w, π x (η) the equivalence is stated.

(2) Reasoning by contradiction, there are two sequences {x n } ⊆ M and {η n } ∈ S with x n → x and

η n ∈ N C M (x n ) such that g(x n ), η n > 0. Since g(y) ∈ T M (y) for every y ∈ M, g(y), η = 0, ∀y ∈ M, η ∈ N C M (y) such that π y (η) ∈ {0} d × R N -d . Thus, it is always possible to pick η n such that |η n | = |π xn (η n )| = 1
On the other hand, by Remark 7, y → N C M (y) is closed-graph at x, so without loss of generality, η n → η ∈ N C M (x) ∩ S and, as g is continuous, g(x), η ≥ 0. However, this contradicts the first part of this proposition, so the proof is completed.

(3) By contradiction, suppose there exist some sequence {t n } ⊆ (0, +∞) and

{v n } ⊆ R N with t n → 0 and v n → v such that y n := x + t n v n ∈ M \ M.
Since K is stratifiable, without loss of generality, there exists a stratum

M i ⊆ M with x ∈ M i such that {y n } ⊆ M i . Take η n ∈ N L M (y n ) with |η n | = |π yn (η n )| = 1.
By passing to a subsequence if necessary, η n → η with η ∈ N C M (x) and |π x (η)| = 1 By Proposition 3, η n ∈ N Mi (y n ) and since M i has constant curvature, there exists κ 0 > 0 such that κ 0 2 |y n -y| 2 ≥ η n , yy n ∀y ∈ M i , ∀n ∈ N.

Evaluating at y = x v n , η n ≥ - κ 0 t n 2 |v n | 2 ∀n ∈ N.
Letting n → +∞ v, η ≥ 0.

However, |π x (η)| = 1 which contradicts [START_REF] Clarke | Discontinuous feedback and nonlinear systems[END_REF]. So the conclusion follows.

In view of this last proposition, if K has bounded curvature as in Definition 4.3 and

       ∀x ∈ K with ı(x) ∈ I 0 , ∃j ∈ I 0 (ı(x)) such that: i) M j is relatively wedged. ii) g j (x 0 ) ∈ ri T C Mj (x 0 ) . (H ♯ 0 )
holds then (H 0 ) also holds and so Theorem 3.3.

Theorem 4.5. Let K ⊆ R N be a closed stratifiable set that has bounded curvature. Consider a regular SVF denoted by G = {(g i , M i )} i∈I0 and suppose that (H ♯ 0 ) holds. Then for every x 0 ∈ K there exist T > 0 and a solution to (D) defined on [0, T ). Moreover, if the SVF has linear growth, then T = +∞.

Necessary condition for Robustness

This final section has as purpose to study the robustness of a stratified ODE under external perturbations. The principal issue here is to show conditions in order to ensure that the corresponding solutions to that type of discontinuous ODE are still stratified solutions of the original system, even if the velocities are slightly perturbed.

Hereinafter, the stratification associated with the set K is supposed to be relatively wedged, that is, the closure of each stratum M i is relatively wedged in the sense of Definition 4.4.

5.1.

Robustness with respect to external perturbations. Let S be a closed set and g : S → R N a given vector field. Let x 0 ∈ S and consider {x n 0 } ⊆ S and

{ξ n } ⊆ L 1 ([0, T ]) such that x n 0 → x 0 and ξ n → 0 in L 1 ([0, T ]
). Suppose that for each n ∈ N, there exists x n : [0, T ] → R N a Carathéodory solution to the perturbed system:

(10)      ẋ = g(x) + ξ n a.e. on (0, T ), x(t) ∈ S, for any t ∈ [0, T ],
x(0) = x n 0 . Let x(•) be an accumulation point of {x n (•)} in the topology of the uniform convergence on [0, T ] and suppose that this is a solution to (11)

     ẋ = g(x) on (0, T ), x(t) ∈ S, for any t ∈ [0, T ], x(0) = x 0 .
In this case, the map g is said to be robust with respect to external perturbations. It is not difficult to see that if g is continuous (which is not in general the case of a SVF), then the property holds. However, thank to tameness of the singular set of a stratified dynamics, gathering the continuity on each strata, it is possible to state such property by ruling some types singularities out.

Outward-pointing modulus.

In order to study the way how the continuous part of a SVF interact between each other, the outward-pointing modulus of a vector field is introduced. Definition 5.1. Let M be an embedded manifold of R N and g : M → R N . Suppose that M is relatively wedged. For any x ∈ M \ M, the outward-pointing modulus of g at x is given by:

α g (x) = max g(x), η : η ∈ N C M (x) s.t. |η| = |π x (η)| = 1 . The condition |π x (η)| = 1 is
important in the definition, because if this is omitted, one would get α g (x) = 0 whenever dim(M) < N and g(x) ∈ T C M (x), whereas, if dim(M) = N it can be strictly negative. Indeed, α g (x) < 0 if and only if

g(x) ∈ int T C M (x) .
The main characteristic of this function is that provides a generalization to lowerdimension embedded manifolds of the previous remark, where the role of the interior is played by the relative interior of the tangent cone of Clarke. This is recapitulated in the next proposition. Proposition 5. Let M be an embedded manifold of R N for which M is relatively wedged around x ∈ M \ M and let g : M → R N be a vector field. Then

α g (x) < 0 if and only if g(x) ∈ ri T C M (x) . In addition, if g is continuous at x and satisfies g(y) ∈ T M (y), ∀y ∈ M, then (1) α g (x) = 0 if and only if g(x) ∈ rbd T C M (x) . (2) α g (x) > 0 if and only if g(x) / ∈ T C M (x). Proof. By Proposition 4, g(x) ∈ ri T C M (x) if and only if ∃σ > 0 such that g(x), η ≤ -σ|π x (η)| ∀η ∈ N C M (x)
. So, first equivalence comes directly from the definition of the modulus. Moreover, by the previous part and the polar relation between

T C M (•) and N C M (•), if g(x) ∈ rbd T C M (x) then α g (x) = 0, if g(x) / ∈ T C M (x) then α g (x) ≥ 0. if α g (x) > 0 then g(x) / ∈ T C M (x). Note that if g(x) /
∈ T C M (x) and α g (x) = 0, then g(x), η ≤ 0, ∀η ∈ N C M (x) s.t. |π x (η)| = 0. On the other hand, since g(y), η = 0 for any η ∈ N M (y) if g(y) ∈ T M (y), if in addition g is continuous on M, one gets

g(x), η = 0, η ∈ N C M (x) s.t. |π x (η)| = 0. Hence, g(x), η ≤ 0, η ∈ N C M (x), meaning that g(x) ∈ T C M (x) which is not possible. So, if g(x) / ∈ T C M (x
) then necessarily α g (x) > 0 and the proof is completed.

5.3.

The externally perturbed model. Recall that the main characteristic of the ODE models presented in this work is that there may exist some strata where no trajectory can slide for. However, in presence of external perturbations this feature may not be held because for any i / ∈ I 0 and j ∈ I 0 (i), it is possible to construct a continuous perturbation ξ :

M i → R N such that g j (x) + ξ(x) ∈ T Mi (x), ∀x ∈ M i .
Whence, trajectories of an externally perturbed model may sliding for M i , so an equation written as (D) may not have sense.

To avoid this problem, one may replace the initial stratified dynamics by a setvalued map which take into account all the significative directions of a SVF, namely

G E (x) = i∈I0(ı(x)) {g i (x)} ∩ T B Mi (x).
Remark 8. The idea of using a set-valued map corresponding to the essential directions has already been explored in others related works for studying Hamilton-Jacobi equation with discontinuous data; see for instance [START_REF] Barnard | Flow invariance on stratified domains. Set-Valued and Variational Analysis[END_REF][START_REF] Rao | Hamilton-jacobi-bellman equations on multi-domains[END_REF].

In this case, the perturbed model corresponds to

(12) ẋ ∈ G E (x) + σ(x)ξ a.e. t ∈ (0, T ), x(t) ∈ K, for every t ∈ [0, T ],
where

σ : K → [0, +∞) is continuous, ξ : [0, T ] → R N is measurable.
However, the initial model is still of concern and so, it would be interesting to know when a Carathéodory solution to ( 12) is also a solution in the stratified sense, as much as in the unperturbed ODE (D). It turns out that a sort of maximality condition over the choice of the index I 0 is required. To state the hypothesis, let us introduce some notation, set α j (x) = α gj (x) for any x ∈ K and j ∈ I 0 (x).

(H 1 )                For any i ∈ I and j ∈ I 0 (i): i) The sign of α j (•) is constant all along M i . ii) If i / ∈ I 0 , α j (x) = 0, ∀x ∈ M i . iii) If i ∈ I 0 and α j (x) = 0 with g i (x) = 0, ∀x ∈ M i , then g j (x) = g i (x) ∀x ∈ M i .
Remark 9. The first two points in (H 1 ) can be weakened by requiring, for example, that the first one holds only locally and that the second is replaced by a condition that ensures that no j ∈ I 0 (i) will satisfy g j (x) ∈ T Mi (x). However, the third point seems to be an essential assumption. Indeed, if it does not hold taking a suite of initial condition living in M j but converging to x 0 ∈ M i ⊆ M j . Whence, the limit of the those trajectories dwells in M i , because of the continuity of g j and also because g j (x) ∈ T Mi (x) whenever x ∈ M i . If g i = g j , then the limiting arc can not be a stratified trajectory. So in this sense, this hypothesis is a necessary condition for robustness.

Note that in particular, the set of singularities enlisted in [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF] for planar systems with affine control systems as well as the set of singularities found in the construction of the feedback in [START_REF] Hájek | Terminal manifolds and switching locus[END_REF] satisfy this condition. Furthermore, this also implies that, if g i (x) = 0, then

G E (x) ∩ T Mi (x) = {g i (x)} if i ∈ I 0 ∅ if i / ∈ I 0 , ∀x ∈ M i . (13)
Besides, an upper bound for the size of the perturbation is also essential. Before giving this bound, let us introduce more notation.

I +

i (x) := {j ∈ I 0 (ı(x)) : α j > 0 all along M i }, I - i (x) := {j ∈ I 0 (ı(x)) : α j < 0 all along M i }, I 0 i (x) := {j ∈ I 0 (ı(x)) : α j = 0 all along M i }. Thus, the upper bound for the size of the perturbations is given by [START_REF] Hájek | Discontinuous differential equations, II[END_REF] σ(x) ≤ 1 2 min min

j∈I + i (x)
α j (x), min

j∈I - i (x)
-α j (x) , ∀x ∈ M i . Proposition 6. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a regular SVF denoted by G = {(g i , M i )} i∈I0 . Assume that (H 1 ) holds. Let ξ : [0, T ] → B and σ : K → [0, +∞) be two given measurable functions. Suppose σ(•) satisfying ( 14) and x : [0, T ] → K is a Carathéodory solution to [START_REF] Hájek | Terminal manifolds and switching locus[END_REF].

Set J i = {t ∈ [0, T ] : x(t) ∈ M i }, then i) ∀i / ∈ I 0 , meas(J i ) = 0, ii) ∀i ∈ I 0 , ẋ = g i (x) + σ(x)ξ a.e. on J i ∩ {t ∈ [0, T ] : g i (x(t)) = 0}.
Proof. let i ∈ I such that meas(J i ) > 0, then by the Lebesgue Differentiation Theorem there exists Ji ⊆ J i measurable with meas( Ji ) = meas(J i ) such that lim

ε→0 + 1 ε t+ε t ẋ(s)ds = ẋ(t) ∈ G E (x(t)) + σ(x(t))ξ(t), ∀t ∈ Ji .
Assume that the set Ji does not contain isolated points.

On the other hand, since the stratification is locally finite, for any t ∈ Ji there exist j i with j ∈ I 0 , a sequence {t n } ⊆ Ji \ {t} with t n → t such that

x(t n ) -x(t) t n -t → ẋ(t) = g j (x(t)) + σ(x(t))ξ(t). Since, x(t n ) ∈ M i then ẋ(t) ∈ T B Mi (x(t)) = T Mi (x(t)). i) Suppose i /
∈ I 0 and let t ∈ Ji . By Proposition 3, ẋ(t), η = 0, ∀η ∈ N C Mj (x(t)), ∀t ∈ Ji .

In addition, for any η ∈ N C Mj (x(t)) with |η| = |π x (η)| = 1 one has

g j (x(t)), η -σ(x(t)) ≤ ẋ(t), η ≤ α j (x(t)) + σ(x(t)).
In particular, if j ∈ I + i (x(t)), the leftthand side gives a contradiction with [START_REF] Hájek | Discontinuous differential equations, II[END_REF]. If j ∈ I - i (x(t)), the same occurs with righthand. Finally, by (H 1 ), I 0 i (x(t)) = ∅. Hence, meas(J i ) = 0 otherwise a contradiction is obtained. ii) By [START_REF] Hájek | Discontinuous differential equations, I[END_REF], G E (x(t)) = {g i (x(t))} for any t ∈ Ji . So the conclusion follows.

Notably, if (H 1 ) holds and the perturbation are small enough, the perturbed stratified systems (D σ ) ẋ = g i (x) + σ(x)ξ a.e. whenever x ∈ M i x(t) ∈ K, t > 0 makes sense, and in fact, any solution to this model is a Caratheodory solution to [START_REF] Hájek | Terminal manifolds and switching locus[END_REF]. Solutions to this equation have the following form.

x(t) = x 0 + i∈I0 Ji(t)

g i (x(s))ds + t 0 σ(x(s))ξ(s)ds, ∀t ∈ [0, T ],
where J i (t) = {s ∈ [0, t] : x(s) ∈ M i }.

Main result.

The main result and principal motivation to write this paper can be summarized in the following theorem.

Theorem 5.2. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a SVF denoted by G = {(g i , M i )} i∈I0 . Let {ξ n : [0, T ] → B} be a sequence of measurable functions with ξ n → 0 in L 1 ([0, T ]) and consider σ : K → [0, +∞) a given continuous satisfying [START_REF] Hájek | Discontinuous differential equations, II[END_REF]. Suppose that (H 1 ) holds and in addition

(H 2 )
∀i ∈ I 0 , if i j then j ∈ I 0 and α j (x) ≥ 0, ∀x ∈ M i . ∀i / ∈ I 0 , if j ∈ I 0 (j) then α j (x) < 0, ∀x ∈ M i .

Let x n (•) be a stratified solution to

(15)      ẋ = g i (x) + σ(x)ξ n a.e. whenever x ∈ M i x(t) ∈ K, for any t ∈ [0, T ]
x(0) = x n 0 ∈ K. that converges uniformly to x : [0, T ] → K, then x is a stratified solution to (D).

To prove this theorem, some previous lemmas are required. Lemma 5.3. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a SVF denoted by G = {(g i , M i )} i∈I0 . Let ξ : [0, T ] → B be a measurable map and σ : K → [0, +∞) a continuous function that satisfies [START_REF] Hájek | Discontinuous differential equations, II[END_REF]. Let x(•) be a stratified solution to (D σ ) defined on [0, T ]. Let i ∈ I and j ∈ I 0 with i j and dim(M i ) + 1 = dim(M j ). Assume that

α j (x) > 0 ∀x ∈ M i . ( 16
)
Suppose that x(t) ∈ M i ∪M j for every t ∈ [0, T ]. Then, for any x ∈ M i there exists

ρ i > 0 such that if x(0) ∈ M j ∩ B(x, ρ i ) with dist Mi (x(0)) ≤ ρ for some ρ ∈ (0, ρ i ) it is possible to find τ = τ (ρ) ∈ (0, T ] such that x(τ ) ∈ M i . Furthermore, τ (ρ) → 0 as ρ → 0 and if i ∈ I 0 then x(t) ∈ M i on [τ, T ].
Proof. By Proposition 3, N C Mj (x) ⊆ N Mi (x) for any x ∈ M i . Besides, for any x ∈ M i , there exists r > 0 such that for every x ∈ M i ∩ B(x, r), there is an orthonormal family of vectors {η 1 (x), . . . , η p (x)} such that

N Mi (x) = span{η 1 (x), . . . , η p (x)}.
Moreover, the maps x → η n (x) is continuous for every n = 1, . . . , p.

On the other hand, since M j is relatively wedged, N C Mj (x) = N 1 (x) ⊕ N 2 (x) with N 2 (x) a vector subspace of dimension q. Whence, as dim(M i ) + 1 = dim(M j ), p = q + 1 and without loss of generality,

N 1 (x) = cone{η 1 (x)} and N 2 (x) = span{η 2 (x), . . . , η p (x)}.
Taking all this into account, one gets the following formula

α j (x) = g j (x), η 1 (x) , x ∈ M i ∩ B(x, r).
Moreover, it is always possible to assume that η 1 (x) = p n=1 λ n ∇h n (x) with λ n ≥ 0 and h n (y) < 0 for every y ∈ M j ∩ B(x, r).

Let c r > 0 be an upper bound for |g j (x)| + |σ(x)| on B(x, r) and reducing r if necessary, it is assumed that each ∇h n is Lipschitz of modulus L r n > 0 on B(x, r)

and by continuity of g j and σ

max |g j (x) -g j (x)|, |σ(x) -σ(x)|, r 2L r c r ≤ 1 12 α j (x), ∀x ∈ B(x, r). (17)
Let ρ > 0 to be fixed and suppose x(0) ∈ M j ∩B(x, ρ). Set h(t) = p n=1 λ n h n (x(t)) and note that h(0) < 0. Let T r = sup{t ∈ [0, T ] : h(t) < 0 and x(t) ∈ B(x, r)}.

Hence for a.e. t ∈ (0, T r ) one gets

ḣ(t) = p n=1 λ n ∇h n (x(t)), ẋ(t) = p n=1 λ n ∇h n (x(t)), g j (x(t)) + λ n ∇h n (x(t)), σ(x(t))ξ(t) ≥ p n=1 λ n ∇h n (x(t)), g j (x(t)) -σ(x(t)) p n=1 λ n ∇h n (x(t)) Note that p n=1 λ n ∇h n (x(t)), g j (x(t)) = α j (x) + p n=1 λ n (∇h n (x(t)) -∇h n (x)) , g j (x(t)) p n=1 λ n ∇h n (x), g j (x(t)) -g j (x) ≥ α j (x) -L r |x(t) -x||g j (x(t)| -|g j (x(t)) -g j (x)| and p n=1 λ n ∇h n (x(t)) ≤ 1 + L r |x(t) -x|. Whence ḣ(t) ≥ α j (x) -σ(x) -L r c r |x(t) -x| -|g j (x(t)) -g j (x)| -|σ(x(t)) -σ(x)| ≥ 1 4 α j (x).
Where this last inequality comes from ( 14) and [START_REF] Jeffrey | The two-fold singularity of discontinuous vector fields[END_REF]. Therefore, t → h(t) is strictly increasing and

h(t) ≥ h(0) + 1 4 α j (x)t, ∀t ∈ [0, T r ].
Hence, T r ≤ -4h(0) 1 αj (x) and since each h n is locally Lipschitz, there exists a constant ℓ r > 0 such that

p n=1 λ n h n (x) ≤ ℓ r dist Mi (x) ≤ ℓ r ρ, ∀x ∈ M j ∩ B(x, r),
and so, T r ≤ 4ℓrρ αj (x) . Nevertheless, since x(•) is continuous with essentially bounded derivatives, the time needed to escape from the B(x, r), inf{t ∈ [0, T ] : x(t) ∈ M j with |x(t) -x| = r} should increase as long as ρ goes to 0. Thus, there exists ρ i > 0 such that inf{t ∈ [0, T ] : x(t) ∈ M j with |x(t) -x| = r} > T r whenever |x(0) -x| ≤ ρ ≤ ρ i , and so x(τ ) ∈ M i whenever ρ ≤ ρ i with τ = T r .

Assume now that i ∈ I 0 and τ < T , for sake of simplicity, hereinafter τ = 0. Suppose there exists t ∈ (0, T ] such that x(t) ∈ M j . Since M j is relatively open on M i ∪ M j , there exists t 0 ∈ [0, t) such that x(t 0 ) ∈ M i and x(s) ∈ M j for every s ∈ (t 0 , t]. To simply the notation, t 0 is set to be 0 and x = x(0). Let ε ∈ 0, 1 2 α j (x) , then since σ and g j are continuos, there exists δ > 0 such that

|x -x| < δ ⇒ max {|σ(x) -σ(x)|, |g j (x) -g j (x)|} ≤ ε 3 . By reducing t if necessary, x(t) ∈ B(x, δ) and max s∈[0,t] σ(x(s)) ≤ σ(x) + ε 3 (18) max s∈[0,t] |g j (x(s)) -g j (x)| ≤ ε 3 (19) Note that 0 > h(t) = p n=1 λ n ∇h n (x), x(t) -x + o(|x(t) -x| 2 )
So, by reducing again t if necessary,

x(t) - x t , η 1 (x) ≤ 1 3 ε. (20)
On the other hand, where the last inequality is given by ( 18) and [START_REF] Marigo | Regular syntheses and solutions to discontinuous odes[END_REF]. Therefore, by 14 and ( 20)

x(t) - x t , η 1 (x) = 1
ε ≥ α j (x) - 1 2 α j (x) = 1 2 α j (x).
Which contradicts the choice of ε, so such t 0 can not exist.

Remark 10. The fact that dim(M i ) + 1 = dim(M j ) is crucial for the proof of the first part of the previous lemma. Indeed, if dim(M i ) + 1 < dim(M j ) the same conclusion of Lemma 5.3 does not hold, for example, consider the stratification M 0 = {(0, 0}), M 1 = {min{x + y, y} < 0}, M 2 = {min{x + y, y} > 0}, M 3 = {x + y = 0, x < 0} and M 4 = {y = 0, x > 0} with g 1 = (1, 0), g 2 = (0, -1), g 3 = (1, -1) and g 4 = (1, 0). It is not difficult to see that the couples (M 3 , M 1 ), (M 3 , M 2 ), (M 3 , M 2 ) verifies the result state in the lemma. Note also that α 1 (0, 0) = 1 √ 2 > 0. However, no trajectory of the perturbed system starting from (x 0 , y 0 ) of norm arbitrarily small but with x 0 > 0 and y 0 < 0 will reach M 0 .

A similar lemma can be stated when the sign of the outward-pointing modulus is negative. In this case, there is no restriction over the dimensions of the strata. Lemma 5.4. Let K ⊆ R N be a closed stratifiable set and let {M i } i∈I be its strata. Consider a SVF denoted by G = {(g i , M i )} i∈I0 . Let ξ : [0, T ] → B be a measurable map and σ : K → [0, +∞) a continuous function that satisfies [START_REF] Hájek | Discontinuous differential equations, II[END_REF]. Let x(•) be a stratified solution to (D σ ) defined on [0, T ]. Let i ∈ I and j ∈ I 0 with i j. Assume that α j (x) < 0 ∀x ∈ M i . [START_REF] Meeker | Local time-optimal feedback control of strictly normal two-input linear systems[END_REF] Suppose that x(t) ∈ M i ∪ M j for every t ∈ [0, T ] with x(0) ∈ M i . Let τ = inf{t ∈ [0, T ] : x(t) / ∈ M i }.

If τ < T then x(t) ∈ M j on (τ, T ].

Proof. Let x ∈ M i and r > 0 as in the previous lemma such that x(0) ∈ B(x, r). Without loss of generality, τ = 0 and let τ 1 > 0 be the maximal time such that x(t) ∈ M j ∩ B(x, r) for every t ∈ (0, τ 1 ). By contradiction, suppose that τ 1 < T For any η ∈ N C M (x) there exists a representation as

η = p n=1
λ n ∇h n (x).

Setting h(t) = p n=1 λ n h n (x(t)) and using the similar estimations as in Lemma 5.3, one can show that, ḣ(t) < 1 4 α j (x) for almost all t ∈ (0, τ 1 ), and so, t → h(t) is strictly decreasing. Note that h(0) = 0 for any η, so x(τ 1 ) ∈ M j with |x(τ 1 )-x| = r and the conclusion follows easily. Proof (Thm. 5.2). By (H 1 ), the sets of index I + i (x), I - i (x) and I 0 i (x) are independent of x, so for sake of notation the dependence with respect to x is suppressed. Note that the vector field g(x) = g l (x) whenever x ∈ M l is continuous on S i = {M j : j ∈ I 0 i }, whenever i ∈ I 0 .

Note also that S i is locally closed around each x ∈ M i .

To prove the statement of the theorem it is enough to show that for some τ > 0, x| [0,τ ] is a stratified solution. For this purpose, let i = ı(x 0 ) and i n = ı(x n 0 ), where x 0 = x(0). Since the stratification is locally finite, {i n } is compact and so, for simplicity, it is assumed that i n = j ∈ I for any n ∈ N with i j.

Suppose that i ∈ I 0 , then by (H 2 ), j ∈ I 0 as well. Let R > 0 such that M i is the stratum of lower dimension on K ∩ B(x 0 , R) and let τ n := inf t ∈ [0, T ] : x n (t) / ∈ K ∩ B(x 0 , R) .

Note that, since the set of velocities associated with (15) can be bounded uniformly with respect to n on K ∩ B(x 0 , R), there exists τ > 0 such that τ n > τ for any n ∈ N. Recall that x n 0 → x 0 , so it can be as close as wanted of S i . Moreover, it is not difficult to see that by Lemma 5.3 and (H 2 ) any arc x n (•) reaches S i within time t n , and even more, the sequence {t n } converges to zero as long as n goes to infinity. So, without loss of generality, j ∈ I 0 i . Once again, by Lemma 5.3 and (H 2 ), no trajectory of (15) can pass to a stratum of bigger dimension as long as the outward pointing modulus related to this stratum has positive sign. Therefore,

1. 1 .

 1 Notations. Throughout this paper, R denotes the sets of real numbers, |•| the Euclidean norm on R N , B the unit open ball {x ∈ R N : |x| < 1}, B(x, r) = x + rB and S is the unit sphere {x ∈ R N : |x| = 1}. For a set S ⊆ R N , int(S), S and bdry(S) denote its interior, closure and boundary, respectively. If S is convex, ri(S) and rbd(S) stand for its relative interior and boundary, respectively. The Lebesgue measure of a Borel set J ⊆ R is indicated by meas(J).

Figure 2 .

 2 Figure 2. Example of stratified vector field on K = R 2

3. 1 .

 1 Preliminaires. Let us start by recalling some notions of tangent cones which are going to play a fundamental role in the forthcoming sections. Definition 3.1. Let S ⊆ R N be a locally closed set and x ∈ S given.

4. 1 .

 1 Normal cones and embedded manifolds of R N . Let us recall some notions of normal cones which are going to be of utility in the next sections.

Definition 4 . 1 .

 41 Let S ⊆ R N be a locally closed set and x ∈ S given.

  for any x ∈ M. Additionally, for every x ∈ M, there exists δ = δ(x) > 0 such that |η| 2δ |x -y| 2 ≥ η, yx ∀η ∈ N M (x), ∀y ∈ M.(3) for any x ∈ M, N C M (x) contains a vectorial subspace of dimension Nd. Proof.(1) This is direct consequences of[START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] Proposition 2.7.4].[START_REF] Barnard | Flow invariance on stratified domains. Set-Valued and Variational Analysis[END_REF] The first part of the statement is straightforward from[START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] Proposition 2.7.4]. Besides, by definition, there exists r = r(x) > 0 such that

t t 0 gt t 0 (

 00 j (x(s)) + σ(x(s))ξ(s), η 1 (x) ds ≥ α j (x) -1 |g j (x(s))g j (x)| + σ(x(s)))ds ≥ α j (x)σ(x) -

5. 5 .

 5 Proof of Theorem 5.2.

  ) 2.3. Stratified ordinary differential equations. Once the notion of SVF is well stablished, it is possible to formalize what is the central object of investigation in this paper, namely, the discontinuous ODE engendered by this mapping.

This work was supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO.

1

x n (t) ∈ S i for any t ∈ [0, τ ] and any n ∈ N. Recall that g is continuous on each S i , so the conclusion follows by passing into the limit in

Assume that i / ∈ I 0 . Suppose first that j ∈ I 0 and then, using the same arguments as in the previous part applied to j instead of i, it can be shown that there exists τ n > 0 such that x n (t) ∈ S j for any t ∈ [0, τ n ] and any n ∈ N. Now, by (H 2 ), α j (x 0 ) < 0 then by Lemma 5.4, τ n can be uniformly bounded from below for a positive number, so it is possible to pass to the limit and get the conclusion. Now consider the case j / ∈ I 0 , by (H 2 ), I 0 (j) = I - j and I 0 (i) = I - i , and by Lemma 5.3, each trajectory x n (•) can only dwell in strata whose indices belong to I - j . So, by Lemma 5.4 there exists k ∈ I - j such that x n (t) ∈ M k in a maximal interval (t n , τ n ] with τ n > 0. Since by (H 2 ), k ∈ I 0 then by Lemma 5.3 then the sequence {τ n } is uniformly bounded from below by a positive number τ > 0 and using the same argument as before one can pass to the limit and get the desired result. So the proof is completed.

Appendix A. Properties of the outward-pointing modulus

This appendix has as a principal aim to discuss how continuous can be the outward-pointing modulus of a vector field. Thanks to the relatively wedgedness of the closure of the manifold, it can be shown that this is upper semicontinuous. Continuity along a submanifold contained in the frontier of the manifold is also achieved. Recall that a set-valued map Υ : R N ⇒ R N is said to be lower semicontinuous at x if for any x ∈ Υ(x) and ε > 0, there exists δ > 0 such that

Moreover, if each Υ(x) is a closed cone, then it is said locally Lipschitz if for every r > 0 there exists L r > 0 such that

Proof. For sake of simplicity assume that M is a N -dimensional submanifold of R N , otherwise it is enough to replace the condition |η| = 1 by |η| = |π x (η)| = 1.

(1) Let x ∈ M \ M and take {x n } ⊆ M \ M such that x n → x. Then, by compactness, there exists

Without lose of generality, assume that η n → η, with |η| = 1. We know that the multifunction x → N C M (x) has closed graph on M. So η ∈ N C M (x). Thus, lim

(

, we can assume that, passing to a subsequence if necessary, there exists a sequence

So, taking the liminf in the last inequality proof is completed.

(3) Let x 0 ∈ M b and r > 0, consider x, y ∈ B(x 0 , r) ∩ M b . Let L r be the Lipschitz constant of g(•) and N C M (•) on B(x 0 , r)∩M b . Consider too C r > 0 an upper bound for the norm of g on B(x 0 , r) ∩ M b . By compactness, take