This is Think Tech Hawaii, Community Matters here. Aloha and bienvenido to Hispanic Hawaii and Richard Concepcion. Today's guest is Carmen Bell. She is a social media consultant and the owner of MLA Hawaii. She's here to help us to understand the good, the bad, and how to protect our children from social media. Carmen, bienvenida al programa. Welcome to the program. Aloha, Richard. Thank you for having me. Alright, it's good to have you here. Let's start the program by asking you to tell me a little bit about yourself, what you do for the community, and why you become a social media consultant. Absolutely. About four years ago, I decided to leave the finance sector and branch over to the marketing consultant or social media marketing consulting. After working with several non-profit organizations here locally, I was able to learn a lot about how to marry a lot of the social media and the different platforms available with a lot of the local businesses and also non-profit organizations. I was fortunate enough to work with several that deal with a lot of issues here locally, and I was able to also help them branch out and give a lot of exposure to their brand. So you also help an organization with Latin connection and Hawaii Latino Feel Festival? Absolutely, yes. I came to Latina in retiring here in Hawaii. I really wanted to give back to my community, not only the Hawaii community, but also the Latino community here in Hawaii. So it was very important for me to work with a lot of these organizations and helping them in any way that I could, especially with what I'm good at, which is through the social media and helping them with their brand. Well, we have a lot of questions for you. Before we start those questions, I want to go over the internet safety and want to learn a little bit about the data that we have there. We have here a presentation that talks about different data. We had 90% of the teams have used some form of social media. 75% have a profile on social networking sites. More than half of the American teams visit social networking sites every day. 17% of the teams say they have been contacted online, but someone that they didn't know in a way that they made them feel scared or uncomfortable. Also, 30% of the teams say they have received online advertisement that was inappropriate for their age, and 39% of the teams admitted to lying about their age to gain access to the website. So with that information, can you tell me a little bit what is good and bad about social media? Absolutely, it is very important. With social media, it is such a wonderful resource. Not only for a brand or a company, but also for their users. So many people today, they are able to have an unlimited amount of exposure through social media. And being able to do it in a very responsible way, it can result very profitable for their business, and it can give them an exposure that they wouldn't be able to just have locally. However, the downside to that is also the exposure. Just not being able to just put so much information out there in the social media as well as the people that you have as part of your brand. A lot of brands today, they have children. There's a lot of promotions for the children. There's a lot of organizations that children are a part of. So as a brand, as a company, as an organization, there's a lot of responsibility to make sure that they are protected. Not only because they're part of the brand, but because they're minors. So it is important to use these resources responsibly. I think it's one of the children who benefit by being able to keep in contact with families and friends. And also, they can share ideas and use their creativity. But what is really bad about social media? To start off, there's a lot of potential for cyberbullying in social media as well as predators or people that are contacting your children, especially minors. So it is very important to keep that in mind, especially if you are trying to use social media as an exposure for your child, if you want your child to be into modeling, if you want your kid to be into acting. It is wonderful. It's a wonderful tool to use. However, it could really be dangerous because you're putting a lot of the information about your child or a child that you represent onto the World Wide Web. And it could be a little scary. So I guess my best advice is just be careful.  You mentioned about cyberbullying. Tell me a little bit about that. Cyberbullying is pretty much bullying on the Internet. It could happen through social media. It can happen just comments. Kids can read very negative comments. Anyone has access to your child when they have a profile online. So cyberbullying, anyone can come and just make them feel uncomfortable. Maybe perhaps there's some stalking for teens. It could be really scary if not monitored. Wow, that's difficult. Let's talk a little bit about the Internet safety on laws. We have here different information. The first thing I want to talk about is what is COPA? It's Children's Online Privacy Partition Act. It protects kids younger than 13 years old when they engage in online activities. Also children personal information without a parent knowing or permission. They cannot sign in or get the ability to use the Internet or be part of the social media. Also parents need to provide consent. And this consent is just not verbally needed to be written. And also the different companies that provide social media they cannot collect more information what they require. So going over the Internet safety law what else can a parent do to protect their children? I absolutely, there's many platforms such as Instagram, Twitter and Facebook. They have a lot of privacy policies and safety policies in place. You can read them looking through your settings of each of them. One of them for example, let's say Facebook. Facebook has a lot of private policies where you could either disclose the information to the public. You could keep it private to the friends that you have. You could keep it private to the followers that you have. There's different things where you can have friends where you can share a little more information with them and you can share different kind of information with your followers. Other platforms such as Twitter, they have policies in place where you can report anyone that you feel is harassing you or they don't necessarily need to be doing it to you. If you feel someone is making comments that can be a little bit discriminating or it's not making people feel comfortable you can always report them and then they will go ahead and do an investigation and perhaps this person's account will be either suspended temporarily or permanently. It's really important to just kind of go back to the settings of all these different platforms and see what policies they have in place that you can use to protect yourself and the account of your child if you have a child who has an account. And it's also very important for the parents to get involved. Talking to one of my viewers, she mentioned that her age was always trying to type Lego and she misspelled and it came out with Lego. And so many different websites that is pretended to be like and also pornography material. How can parents prevent this kind of situation at home? Well, for starters, as far as computers and what your children are able to access online you can always do the settings on your own personal computer at home and make sure that the kids don't have access to certain sites or don't have access to certain social media platforms or pages. Something else that you can do to protect your children is kind of, you know, talk to them. Talk to them about the different things. If you see something like this, click out. Don't click on if someone, you know, if something pops up, don't click on it. Even if it says, hey, you can see, you know, this cartoon for free. Don't click on that, you know, just kind of talk to your kids a little bit, educate them about the different things that could happen online or to come and find you and then you would tell them, hey, that looks fun.  That's a lot for a parent. That's a lot, yeah. All right, let's talk about what is sexting? Well, sexting is pretty much just sending any pornographic pictures through a mobile device. However, through your mobile device nowadays, I mean, you have access to the internet, you have access to Facebook and Instagram and all these different social media. So it could be dangerous because not only you are exposing or this picture that you're sending is not only being sent to the other person that, you know, they can, that you're sending that text to, but that person could take a screenshot of that and they can put it online and they can share it and it would be viral within minutes. So it's really scary. You know, you have to really keep in mind what it is that you're putting out there, even if it's just a simple text to a friend. And that's staying online forever, right? Absolutely, absolutely. And this is one of the big things between teens. I mean, teens can find themselves sexting.  They could do it to a boyfriend or because they like a boy and they want to go ahead and send him something cute, you know, that they feel they will attract the boy to them, pure pressure as well. It's no different than the, you know, the issues that they have at school, you know, with girls, you know, the attention that they want to have, you know, attract boys and everything like that. So do you think people today, they are desensitized to all these different things they see on the website? I think so. I think so, definitely. I mean, when it comes to social media, I mean the possibilities of being able to just expose everything, everything that you have or everything that you know or putting information out there for everyone to see. It's a scary thing. Let's talk about some of the consequence and concern for the parents. Can you tell me some of that?  I mean, they wanted to have, you know, because all the kids were doing it. They wanted to have an Instagram. To get to your pressure, right? Absolutely. They wanted to have an Instagram. They wanted to have a Facebook. For me personally, it was just with everything else when you have a teenager, it's just compromise. Okay, you want to have an Instagram account, I want access to that account.  It's important for a parent to take, you know, this, just like you would when your children are at school and, you know, they want to do certain things. It's just kind of take that initiative to get involved and make sure they're doing it safely. Well, for those parents that they don't know, here's another statistic. There is nine out of 10 teenagers, you know, those photo of themselves. And nine out of 10 provide the real name or the addresses where they live. And also eight out of 10 reveal their birthdays. And seven out of 10 teenagers, they provide the information where school they go to and where grade they in. What is it dangerous about providing all this information? It is dangerous. I mean, it's no different than pretty much just handing someone, you know, your birth certificate if you think about it, you know.    You can't see nothing else on their page because they went through their settings early and they were able to click on exactly what they wanted exposed to anyone that finds them. My personal account that I have not my business, my personal account, you can't find me on Facebook unless you're friends of my friends. You can't find me through my email. You can't find me through my name. That's the way that I wanted my page, my personal page. So there's a lot of policies and there's a lot of guidelines in place that can help you stay protected. Well, that's great because how do you know that something that I learned today that made sure I go through all the settings and make sure that I get myself away from people researching me. That's perfect. So if I want to protect my child and I know that person, my child is 13 years old the regulations say after 13, you can go ahead and try to set an account with the parent permission. How can I monitor that my kids are using this internet at home and I probably don't even know about it? That's unfortunate. You can't find everything that your 13-year-old is doing. You can only do your best to make sure that they don't have access to that or if they have a page that they tell you and hopefully they will tell you that they have a page but it doesn't mean that you can do so much as a parent. They can use a computer at their friend's house and have a Facebook account that you will never know about or an Instagram account that you will never find. I personally, as a parent, I made sure that I spoke to my girls about the dangers of having these pages and the dangers that they were putting themselves in their home and their friends if they were sharing things they weren't supposed to if they were sharing every single location if they were allowing people to see all this information about them I'm here, you know, I'm over there she's over here. I mean it comes with big responsibility I think the best thing that I you can tell a teen is with a social media page it's like getting your first driver's permit a lot of responsibility you need to make sure that you treat it that way you need to make sure that everything that you do everything that you say, everything that you share you think about it and you make sure that you're doing it responsibly so there's gotta be some kind of communication there has to be agreement between you and your kids absolutely well Carmen I want to say thank you so much we're gonna take a quick break and message to you and we're gonna continue talking story after this wonderful message Welcome back to Hispanic Hawaii we're here with Condon Bell talking story and learning about social media the good, the bad and how we can protect our kids alright so we was talking about what parents can do to protect their children using the internet so can you elaborate a little bit more about that yeah absolutely I think the internet is a great resource for the children it's very educational it could really help them showcase any kind of talent that they have in an artistic way they can find friends with common interests also stay connected to family members I mean we live in an island right now a lot of our family members are in the mainland it's a perfect way to stay connected I really don't want people to feel like with everything that's going on in the internet be afraid to use all these tools to use all these resources steps that I would give the parents as far as to make sure that at least the kids stay protected or they can protect their children because a lot of parents they have pages for their children because they want their grandparents to see them growing up and they monitor those pages the first thing that I would say is to go through all these different platforms all these different places that you have these profiles and go over and look through the settings that you want to share Facebook has I'm sorry a setting where you can see what your friends see and what the public sees when they click on your profile I think that's great because it really allows you to see exactly what it is that you need to do to stay protected other points that I would give to parents is tell their children not to allow or not to accept requests from people that they don't know people that they've never met in person a friend that they've never met in school go to their profile see if they're friends with your friends sometimes it's a cousin that you saw in passing or that you met at the mall she just wants to be your friend that's okay but if it's someone that you go to and they're not friends with any of your friends I'd say another red flag that I would say is someone that does not have a lot of followers or not a lot of friends or they have a fairly new account a red flag for me and I speak to a lot of my clients about that they could be spam they could come and heard your account or they can be predators something else that I would say is sometimes kids being kids very innocent they would say something that can hurt someone's feelings or not knowing perhaps they can become the bully you know so I always say make sure you are mindful of what you say to others online if you don't want someone to say it to you don't say it to somewhere else online you know that's a good point absolutely so that's another point in front of my time when you do something on the internet on the website just to think about it what your grandmother would say I agree I think that would be a perfect way to look at it if your grandmother say that's not the right thing to do just don't do it yeah grandma's no best if that's not the right thing to do for sure well I want to ask you about some of the application that is out there when kids are playing game in the computer and they have to provide the address and their location some of the application can tell you your location how can you avoid that well the only way that that can happen is if you have your setting on your phone or your computer where your location is on on that particular setting if you look for parents if you look through the phone of your kid and you see that they're sharing their location under their facebook application turn it off they won't be able to find them if they're doing it through their twitter turn it off or you know that's the perfect way for no one to locate them they won't be able to locate them if their location setting is not on their device so that'll be something to look into for sure let's talk about youtube I've seen a lot of kids in youtube that participate in or the mother or the father use in the kids in youtube to create a channel what is your recommendation for parents that want to do that absolutely I would say be that person that monitors that account you know as a parent is no different than protecting them at home protecting them you know at the park protect them online that you monitor the people that send the messages you monitor the comments that they receive and if it becomes too much you know you're a parent and I'm sure let's say you're managing your child's you know your children's accounts or your children's business or if you know your child is a model or something and you're managing their career hire someone hire someone that will take the responsibility to go in and you know go through these messages the pictures are not being copied and exposed on another different website that is their job that's what you will be paying them to do so what is your recommendation building trust between yourself and your children in order for them to ok I want to trust my mom and dad to look at my website or looking at you know the different pages that I'm visiting any recommendation on that it all depends on the parent how it is addressed in order and the child how it is received my personal experience with my teens it was pretty much being very open in my mind if they want to have a profile then they're want to you know start becoming adults you know they want to take that responsibility so ok be very open and be like this could happen you could be contacted by a predator you could be contacted by a guy that is not really that guy you know just be very open with them and if they're a lot younger monitor their account for them you know and until they're of age and then you can just you know tell them a little bit more but it's no different than anything else that you try to do with your kid you know try to tell them you know how to use it responsibly or how to build that trust you know like I said just make sure you tell them the do's and don'ts and the consequences you also work with many different organizations here in Hawaii and some of these organizations they have to deal with kids and parents and they do marketing for you know for the community what advice do you give to any one of these organizations they deal with those kids and marketing kids within the community? Yeah absolutely for starters I wouldn't want them to feel discouraged with everything that is going on in the social media because it's really important if there's children that are part of your organization it's people would want to see that people want to see that the kids are getting involved in their community what I would tell them is to always make sure that they have the parents permission with everything that they do that you know has to do with minors make sure that the parents are aware that they're using their picture or that the parents are aware that they're using their story always make sure that the parents are giving you some kind of permission it's preferably written using these pictures and that you're promoting something that has to do or they're using your children's information Let's go back to cyberbullying if my child feels like somebody is harassing or is bullying my child how can that child stop that just block that person completely or approaching in a different way calling the police what would it be the first step? the immediate and the quickest step would be blocking that person you tell your child I'm sorry don't get exposed don't expose yourself I'm sorry to this kind of bullying block this person this person obviously doesn't have your best interest in mind the next step if it continues do your research is it a classmate is it a neighbor perhaps come and address with a school or address with the parents if it's someone that you've never met if it's someone that your kid accidentally befriended online then that's when I would say contact the authorities so they can do a thorough investigation let me ask you this is your opinion do you think the school you provide some kind of curriculars or class to teach about kids using the internet I think that would be amazing if the school could do that I think that would be great because it's such a big tool that schools use it you can protect your kid at home online but what if they're using a computer at school so I think it would be important for the school and it would be great if the school could provide that kind of information or that kind of add that to their curriculum for sure alright we're running out of time I want to say thank you so much for coming to Hispanic Hawaii and helping the Latin community to understand you know social media the good the bad and how we can protect our kid and that kind of stuff thank you so much for having me Richard and happy new year oh happy new year to you feliz año nuevo alright I want to say thank you so much for watching Hispanic Hawaii and don't forget that you can rewatch this program at thinkpechawaii.com and many other programs within our community thank you gracias hasta luego


We good? Okay, so as they're over here So what's happening is there's police literally everywhere. There's about 12 walking up there We've seen two big vans and pull a police over there. We're meeting The animal rebellion guys for an action and there are police that there are police at the watch Keeping watch over here. I'm not too sure how they're gonna how they're gonna work There's police have already police have already worked them out They must have intercepted everything police have been intercepting everything animal rebellion are doing but apparently We might be going to a slaughterhouse to disrupt it. So let's go find them Okay, here's everyone here and here's about 25 police walking up. Looks like we've got a little Little guy here whose job it is to tell on us at the corner here and this is a group of animal rebellion people over there. There's already Looks like community officers or something with them. See how this pans out It's up. Yeah Peabody road All right, so we've got the address of the slaughterhouse The police are already trying to find out where it is I don't know how this is going to go the police are going to intercept it before we do anything There's just police everywhere So we're gonna try to make our own way down there See how we go So for more to lose station, we can go straight to farm bra Is that the popo? Maybe I'm just following the crew. I'm not really. I don't really know what's up. Yeah, I haven't been told of anything Thank you, what's your name? Nice to meet you Tom. You too my friend. Thank you. Thank you so A bunch of activists have locked on to a slaughterhouse truck and they seem to be inside the slaughterhouse over here We had a meeting point at 9 a.m. And all the police went there seems like another group had already Made their way up to the slaughterhouse.  How can people look like this individual here lives right next door? Like all I can smell is sheep carcass right now Disgusting and is it look at the way this place is designed with the prison bars There's someone up on the roof too for any problems you might get some peace signs and some song sung to you That's about it. Maybe a flower put in your pocket No one here is violent or aggressive. We just really care about animals That's all we don't like that. There's chopped up animal bodies and they've been over there and It's more it's more about a disrupting the system not the individuals that work here We don't even think jobs like this should be made available to human beings either You know blood and animals trying to struggle for their lives.  Pretty funny joke not Yeah, he goes out can I squash him? Cops just laugh. I don't know. Can you make jokes like that about running over activists? You can't really make jokes like that. Hey, I'm gonna ask her why you let she let him say that excuse me Did he just say can I squash him? Did he just say can I squash him? But he said can I squash him I heard him But why did you let him say that I was just wondering Why did you allow him to say that without warning him?  There's children playing out the back of a school You can hear them the young children as well. How crazy is that? That's why they're brought up normalizing violence So that's why society is like it is because they're brought up thinking that this stuff is normal right outside of a school They can hear the animals screaming from it. So there's a petition to close down the slaughterhouse But that's interesting because the parents will be feeding the slaughtered bodies to the children But they want the slaughterhouse shut down nearby so This is the residents are trying to remove you know ram into the activists Apparently there's a petition to get this place shut down because they don't like the fact that animals are being slaughtered here But they want to eat animal bodies every day, you know like you can't have one and not the other You want to eat animal bodies? There's gonna have to be a slaughterhouse somewhere So we are here for the workers because they're victims too They suffer for this job mental health low-paid not just this little house, but all of them So we are here in solidarity also with the workers Okay, so I think someone's inside this actual slaughterhouse live-streaming from animal rebellion We're gonna see if we can show you the images from the live-stream right now, but apparently they're inside the slaughterhouse This is a bin out the front all the skins and bodies and Guts from the sheep lambs heads Body parts Stocks down to slaughterhouse stuff I Keep on singing Okay, so a couple of activists to superglue themselves to the fence That's another way of being disruptive and Then it creates another barricade for the police to have to work towards moving them off and It's a peaceful way of disrupting the system, I suppose Let's take their hands the fence ouch, so these guys here that we They're basically Documenting the identity of the activists so they know who's involved with the organization. I think so Ed What are your thoughts on this type of civil disobedience direct action? Yeah, I think it's powerful I think when people are willing to put their liberties and freedom at risk I think it sends like a powerful message that this is it This is something that we feel not just passionate about there's something that you know We're willing to risk a lot for and yeah, that's a credibility to what you're fighting against or fighting for So you're a vegan advocate predominantly you do a lot of your advocacy on social media Do you think that this brings another element into it like yeah, yeah Well, I mean what we're up against is like a multi-headed beast Yeah, so it's not as simple as having like one tactic to remove all of those heads So I think we need social media activism. We need just kind of grassroots. We need civil disobedience I did a whole spectrum of things because that's the only way we're actually going to tackle this huge Monster that we're up against because other vegan activists might be put off a little bit by doing activism because this is quite full-on They're chained on and yeah, what would you say to those people like well? You don't have to be chained on everyone out here, you know really and truly like we're here for support We're here to document we're here to share online Those people inside are incredibly humbling to me because they are willing to do all that But you don't have to know the point being here in solidarity is also an incredibly essential part of civil disobedience So doing that to civil disobedience doesn't mean that you're actually necessary putting your freedom in danger No other people will do that and it's up to you to build up the confidence if you want to but just being here being You know feet on the ground and our voiced ads in solidarity is also just as important. No, no you said it very well Thanks so much. You're welcome. Thanks for being here as well Oh, I've got a feeling something might be going down around the other side. I don't know I might move across the other side Maybe this is deterrent for what's going down over come on We support the animals rights Not yet not there yet Yeah, no, no, what are you saying a bit truck drivers beat the animals what he hears a lot He said the owners are not very nice people instead the truck drivers He hears them beating them kicking them scream at them get off the intro Yeah So apparently workers actually live in this little building here right next door here on the property Which is very interesting. I've never Never actually been to a slaughterhouse, but there's a small house on the same property But this is where the workers actually live It's getting ready for the truck. What's up the trucks coming that's I think there's a truck full of animals coming here Might be cows, so I don't know how visible they will be We have to wait till the seat of the truck gets Can I start you all just step back, but I just don't want to get anyone run over by Laurie Can you come down here now All right, so those animals just looks really lovely and scared and innocent and But we're only out the front of this road because you want to eat animal products and animals have to slaughter Yeah, so this is all really innocent and terrified and now there are any young animals and They're often there now to be bolt gun in the head slashed across the throat had their blood drained from them and To be fed to people in this residential area the same residents who complain about this slaughterhouse's existence I'm paying for it to be there and they're also disgruntled because we're standing out in the middle of the road here But you know they're paying for this place to exist and they're paying for those animals to go into those hellholes So this is all legal legal murder. What's happening here? So it's a legal animal abuse what they're doing to these animals in there Any other animals dogs Will defend them these animals have police protecting this business here legally sanctioned a murder That's what it is So I'm just cleaning the animals feces off of my hand like when you think about your food your food shouldn't Defecate on themselves out of fear, you know Plants of food not scared animals not pieces of scared animals bodies that have been hacked up blood and crap all over them I said I was angry when I see that happening, you know Just fucks me off like Poor babies should be getting taken care of not this Like they have done nothing wrong to anyone those animals That is all they did was be born Forced into it to birth as well Forced into birth to be forced down the kill floor The knife forced into their throat We'll get off the road when you stop paying for slaughterhouses Can you see that that's a stock stock standard Council order today. No, this is just standard. This is Australia actually pretty similar to a UK So couch on escape to the couch on escape I Eat dairy dairy products they all get slaughtered for dairy as well They try to escape you ever see your food get made before that's the first step Murder fear struggle death I Never seen any of this before That's inside the slaughterhouse struggle I'm upside down stabbed in the throat Shot in the head It's what no one sees I'm not gonna make any comment on it. You don't have to comment. No, it's okay. I know you can't anyway You have to remain impartial But I know most of the public and most human beings are definitely not Definitely not happy with that.


So, let me sort of recap very quickly what we have talked about in the context of CLFs control Lyapunov function. So, we very conveniently and comfortably use the notation CLF or the terminology CLF. So, please do not get confused it just means control Lyapunov functions ok. We started with a general non-linear system like this alright and this was the definition for the control Lyapunov function alright. The first one the first condition basically required that it be a Lyapunov candidate a valid Lyapunov candidate and the second condition is sort of the nice negativity condition ok. So, you in the absence of the control you had the negative definiteness condition here what all we are saying is that we are saying you can actually find some control u ok such that you can make the derivative negative for every state yeah that is really what is the CLF it says that this function is such that it allows you to find a control such that for that value of control the v dot is negative and this you have to do point wise this is a point wise computation because. So, so the x here is fixed see notice that because you are taking infimum only over the control the x is actually a fixed quantity ok. So, once you fix the state is just some number that you have basically you just have a function of u in here ok and then you are just trying to find an infimum. In fact, you can just think of this as an optimization problem ok it is actually almost an unconstrained optimization problem I mean the only thing is you want the value to be negative you want the value to come out to be negative if it so happens that you run an optimization on this and the value turns out to be positive of the function here of the cost if you may hear then it is not a good it is not a good v at all ok. So, the v is not a CLF ok and of course such v such a CLF is very useful in control design yeah we will look at it because you can see that it allows you to design a control which makes the v dot negative and v dot being negative definite is what you require for asymptotic stability right. So, obviously that is what you want. Now, we then specialize this to the control affine case why did we do that because we realize that well we did not realize anything, but the researchers who have been working who have been working on this for several years they realize that with just a general non-linear system where the control can appear in any form not necessarily in an affine form then it may not be possible for you to design continuous controllers ok virtually impossible so they figured that it is not possible. So, then they specialize to systems that are linear in the control ok and hence called control affine systems ok. So, this is the structure and this structure is the most universal structure in non-linear control here the F0 is the drift vector field FIs are control vector fields these are basically state dependent vectors ok at each value of the state they give you a direction a velocity direction ok and then you have a control scaling right. So, it is almost like saying that I have if you count these I have F0 that is 0 and then I have 1 to m. So, I have m plus 1 such vectors ok I have potentially m plus 1 velocities notice that m is necessarily less than m ok. So, the state space is n dimensional and your number of velocity vectors that you have to play with is m dimensional. So, m is less than n ok. So, m is typically less than n alright which means you have less velocities than the number of dimensions if you think of the state space as a dimension right it may be less than the number of dimensions. So, the idea is can I play around with these velocity vectors so, I can go in the right direction ok. So, if you just think about you know moving on a sphere for example on the surface of a sphere for example ok if my now suppose I want to move on the surface of the sphere this is my requirement ok and I have say vectors in all three directions velocity vectors in all three directions. So, I can potentially if I specify this vector in a bad way ok I can potentially get thrown out of the sphere also right instantaneously I could just be thrown out of the sphere which is not ok with me ok. So, the ideas can I play with these vectors. So, it would be something if I want to draw some picture like this suppose this is the surface of the sphere I will have a vector this way I will have a vector this way I may have a velocity vector this way alright I may have these three velocity vectors. Now, as long as my actual velocity is in this plane I am more or less ok because this plane is the tangential plane to the circle or to the sphere I am more or less ok I will remain on the sphere at least on the surface of the sphere, but if I start doing anything in this direction I get thrown out of the sphere ok. So, what would my control try to do my control is just a scaling all it is doing is it is scaling each of these fail each of these vector velocity directions right. So, you are just in fact you are doing a linear combination of these velocity directions to get the direction you want to go in ok. So, although we never design controllers in this thinking like this honestly speaking nobody designs control like very difficult to do, but this is the logic by which the controllability of a system is defined ok. If you cannot reach all possible directions then you will have some issue with the controllability ok that is the idea alright. So, what we did was we specialized to control affine systems and for that we defined the equivalent version of the control Lyapunov function ok. We already proved equivalent well at least we proved one side the other side was supposed to be our homework which I will assign soon enough. So, this is what is it what does it say the first one is again exactly the same thing is before the second definition changes a little bit ok nothing significant it just says that if the contributions of the control vector fields are 0 then the drift vector field has to push you in the negative direction that is it should make v dot negative ok. If not then again in a sense what we are trying to say is the system is not stable at all ok you cannot make the system go in a good way ok behave well ok. So, that is the whole idea. So, we proved again one side of the equivalence then further we talked about the small control property this was the final sort of property that is required to design continuous control loss ok. What is the small control property it just formalizes the and we saw it with a very nice example right that for a system like this the control becomes larger and larger as you come closer to 0 ok and 0 is a equilibrium of this system ok. So, which is very bad right because if you if you want to try to reach the equilibrium from both sides you are going to give larger and larger control efforts which is sort of ridiculous you do not want to do that. So, this creates a discontinuity at the origin and in order to prevent this you say or you assume that the system has a small control property and what is the small control property it basically says that if you start with small values of the state that is norm x is less than delta then with small values of control that is control is close to the equilibrium control you can make this v dot negative ok. So, basically it says it essentially says what we do not have here yeah essentially says what we do not have here that if you are close to the equilibrium then the small values of control should sort of send you towards the equilibrium ok that is the whole idea here and this is the small control property we already sort of claimed that this small control property is stronger than the second condition of CLF yeah this is a stronger requirement yeah why because if this holds if this holds then you know that if all these terms are zero all the control terms turn out to be zero then this term is still negative ok. So, this is essentially so this implies the previous definition is satisfied ok. So, small control property is a stronger requirement than the CLF property ok all right. So, once we have this small control property this is where we were last time Archtyne and Sontag it is basically their work primarily work by Archtyne and Sontag yeah they were the ones who started talking about the CLF the small control property and then corresponding control again they gave a universal controller ok. The cool one of the coolest things about this result is that unlike a lot of other mathematical results and this result I tell you is very mathematical they actually give a constructive design of the control ok we already saw this last time. So, anyway what it says is if you have a control affine system with the with the control Lyapunov function as per this definition then if the system admits the system admits a small control property if and only if it admits a almost C infinity stabilizer ok and we clearly said what is almost C infinity it means that smooth everywhere in a perforated neighborhood of the origin and continuous at the origin ok. It means that the control that you obtain will be smooth everywhere infinitely differentiable everywhere, but at the origin it is only continuous ok not smooth not smooth at the origin ok it is continuous at the origin. So, this is what you can achieve and remember in specific cases like example that we will look at or we can look at you will find smooth control which are smooth at the origin also ok, but remember this is a very you know as a result which covers all such control affine system. So, it is a very general result ok therefore they are saying in general you cannot clean this that you will always find a controller which is also smooth at the origin ok. So, what you can claim is it is almost C infinity ok, but in the examples that you will see you might find smooth control ok using this formula itself. So, it is not you know not that this covers all cases ok. So, the Einstein sontag. So, this result the proof of this result is based on the Einstein sontag universal formula ok or the universal controller or the universal formula whatever you wish to call it yeah it is defined by first defining these two placeholders AX and BX what is AX? AX is the derivative with respect to the drift vector field and BX is the derivative of V with respect to the control vector fields yeah. So, it is tagged as a vector we already saw what are the dimensions what is the dimension of this gang we discussed this right last time R1 it is a real value ok and BX is this is Rn it is an m vector ok alright great. So, what is the universal formula? This this is the control slightly complicated looking, but this is the control what is it? UX is minus negative of AX plus square root of A square plus norm of B4 BX divided by BX squared if BX is non-zero and if BX is 0 then the control itself is put as 0 ok. So, you can see that B is a vector. So, therefore, we are being careful whenever we take a square of B it is the square of it is the fourth power of the norm of B you are taking the norm of the vector ok and here also we are dividing by norm square. So, this is B ok. So, you see that this whole thing is in the direction of B ok this whole thing is in the direction of B ok because B is actually a vector of dimension Rm right. Notice this is correct why because control is also required to be of dimension Rm ok control itself is Rm we have m controls. So, we have m control vector fields ok. So, this dimension is ok right because B is also dimension Rm we just discussed this alright. So, dimension wise no problem yeah what is the significance of BX being 0 and BX non-being 0 this is the CLF condition right because BX was what this guy BX is del V del X FIX for all I it is stacked column vector and if this is 0 it means that del V del X FIX is 0 for all I yeah I should write this yeah for all I from 1 to m ok. So, when is B 0 if del V del X FIX is 0 for all I ok. So, this is the control Lyapunov condition under such a under such circumstances you know that the drift vector field itself will give a negative V dot ok. So, we put no control because anyway if the contribution of the controls is 0 because here if you put any non-zero value of control it is useless right because the drift the V dot is going to be 0 even if you put non-zero values of control right because you have UIFI ok. So, it is irrelevant. So, we put the control as 0 vector itself and here we give it some particular value alright. In order to verify that this is in fact a stabilizing controller because that is what we want we can just take V as our control Lyapunov function itself and compute a V dot ok. So, V is our candidate Lyapunov function we already know that the control Lyapunov function is a candidate Lyapunov function. So, I take that as my V for the system for doing Lyapunov analysis right and then I take a V dot. What is V dot is exactly this partial of V with respect to X and this for this control FIN system right and then you know that this is actually AX right and this is actually B transpose U is that clear right because B is B is this right B is this guy. So, B transpose U is exactly this multiplied by this yeah ok. So, once I have that all I am going to do is substitute for the control from here. So, again I get two cases depending on whether B is 0 or non-zero alright. So, when B is non-zero you can see what will happen I will have AX plus this guy sorry I have the control here right. So, I will have my AX and then BX transpose of this. So, BX transpose times this ok. So, this is a scalar. So, BX transpose just moves here right BX transpose just moves here and BX transpose BX is what? Norm of BX whole squared. So, this guy will this guy cancel out. So, all I am left with is AX minus AX plus this guy. So, AX minus AX cancel out again. So, I am left with just this much as expected because V is a scalar. So, V dot is also scalar important thing to notice this is strictly negative yeah because BX is not 0. So, this is strictly negative whatever AX is is irrelevant just because BX is not 0 this is strictly negative yeah alright. And if B is BX is actually equal to 0 what happens to the control? Control is just 0. So, I am left with just AX ok, but I already know by my assumption that AX has to be negative because V is a CLF right V is a control Lyapunov function. So, AX which is defined as this has to be negative when these terms do not contribute when BX is 0 A has to be negative. So, when B is 0 this has to be negative ok this is true only when B is 0 by the way when B is non-zero A can be negative positive whatever ok, but when B is 0 A has to be negative. And so what what have I shown that V dot is negative yeah and in fact this is true for all non-zero X remember in entire CLF definition although we did not stress on it much everywhere you see that for X not equal to 0 for X not equal to 0 ok all these assumptions are for non-zero X ok. So, so it works out nicely. So, V dot comes out to be negative definite and this means by our Lyapunov theorems what? What does it mean if V dot turns out to be negative definite asymptotic stability done ok system is asymptotically stable all right.