Why is the Bay Area known to be so bad for dating especially if you're a guy and is there a possible solution to it? Let's delve into it. Dude. There's hella 49ers, bro We got to talk about this because Andrew the Asian male internet is Surprisingly agreeing with this UC Berkeley professor who is under fire for telling guys to get out of artillery range of the range between San Francisco and San Jose because apparently for some reason that people enlisted off a thousand of them It's really bad for dating for guys, right? And this was a white computer science professor at UC Berkeley I believe he's married to an Asian woman from Asia But anyways regardless he came under fire because it was like an inappropriate thing to say and like the the student discord But a lot of people agree because this has been talked about for decades guys Have you heard of 49ers the fours that act like Niners? I listen I'm not saying how true it is, but there's a bunch of reasons why people say it's true We're about to break it down.  I've heard the same thing about Seattle I believe it is primarily driven Andrew from the tech industry Which brings a lot more heavy volume distribution of men versus women, right as a result I guess the women get really picky and The guys are just like man. It's just not a fun time to have Also people were talking about how there is a lot of ABG's in the Bay Area But of course if you are not a top ranked Kevin win ABB You have no appeal to the ABG's and that actually hurts it for I guess the nerdy guys who have been there or even like an average guy Another big reason is traffic So I heard this and I've seen this from friends and people with multiple people I've told me this and this is listed that SF residents really only date within San Francisco And that goes the same for Oakland and Berkeley residents too because the traffic can get so bad Is that you you don't want to be 30 45 to five minutes an hour from each other So you're just dating within essentially your borough a lot of people also said that Asian engineers are seen as less Cool than engineers of other races and then a lot of Asian guys dress poorly here because they're engineers And that just confirms and perpetuates pre-existing stereotypes even though in LA That's where all the stylish Asian guys move or New York where that stereotype is more broken in real life Yeah, and they also say there's an imbalance where Nearly 60% of the people in the Bay Area are men meaning that women get to be more picky And if they're more if women get more choices then they're obviously gonna have even a higher standard and you know Gonna get to exercise that and take advantage of that situation right right right It's very interesting to see places Andrew go famous for being bad for dating Seattle the Bay Area And then people actually like lose their careers because they stand on business so bad like this white guy Trying to like say it in the computer science chat Not like by the way, that's an inappropriate place to make that comment right even if the comment was true I'll tell you this I spent a lot of time in the Bay Andrew LA, New York, Seattle, Houston, Texas Dallas Fort Worth DFW and Seattle reminds me of a Bay Area light by the way And I just agree with this guy's breakdown In generality, but I will say that you know, you could optimize it different ways Like if you were to become a club promoter Obviously be a bartender work at a bar You would be trying to optimize for like the pings and ultimately an overall not that optimal fishbowl Right, right. You can optimize. I'm saying for your own individual positioning even within A game that's like not the best game to be playing right Also, I'll say this it's just like colleges. You know how there's party colleges hyper intellectual colleges And there's dry campus colleges that completely have no party seen in America Cities are actually kind of the same way We went to the University of Washington agree with me or not either not famous for partying No, not not a huge party school Yeah, I would say that most people rank UW like a 5.5 out of 10 But yeah, if you were like the best looking guy in the best frat For sure, you probably still had a crazy time. You know what I mean? So there's a lot of individual movement within uh macro narratives Well guys, I'm shocked that when you got a bunch of computer coders and nerds together that it was started to uh Be hard to date because essentially you're meeting The the what is it the Nash equilibrium like not everybody can get the hottest girl If everybody goes for the hottest girl, then nobody gets anything. Um, one thing that I will say is this is that uh His solution Was to just increase your radius So let's say you're using a dating app and you're just looking at like a 20 mile radius in the bay area This guy says the only way to break this dating curse in the bay area is open up your radius to 100 Miles therefore you're matching with girls and women from sam uh from las vegas and la la as well right and la essentially or like All of northern california is no so you have to mong chik extend it To 100 miles is what he said david is that an actual viable solution for most people if they meet somebody 90 miles away I guess If you are gonna stay in the bay area because your parents are there and you just feel so many allegiances that you can't break Then I can see how that's like optimizing for within a situation that's never gonna be great Like I said, um, everybody just said sf sucks for so many reasons san jose to san francisco They call it man jose apparently because of the of the gender disparity Somebody said that you know for me I was able to get abg's once I got really buff and really ripped and I had some tattoos But then I realized they were just liking me for the look because the abg girls wanted to be with a rough buff Tattooed bodybuilder guy, but ultimately my personality was more with the fob girls the international students.  And then so if the bay area is a big place that is imbalanced where there's more women than men I mean more men than women Yeah, more men than women than what cities have more women than men new york city and or fashion marketing Lotta lot art art is here Hey, listen guys, stereotypical or not those you look at the stats of those fields There's fields that are heavily men weighted heavily female weighted. It just is what it is still in 2024 Um, somebody said uh, the problem with a lot of nerds are is that they like abgs But abgs are looking for a abb kevin wind type that is high level And they super do not fit what the abgs are looking for but then you got a computer programming guy or a guy Who's like marketing at yelp and wants the abg?  So please Shut up about that and keep it moving somebody said man I feel like sometimes in the bay area specifically sf white nerds are more sought after than even a slightly cool asian guy Uh, and then people obviously posted the photo of the guy who even said that I will say this that is true Especially in the past. I actually think it's less true I think a white nerd in sf andrew is actually even starting to struggle because like we said andrew not every white nerd Is fred again or diplo like those guys are like kind of nerds because they make music And you know I gotta do all the stuff on the computer Um, somebody just said it's because the nerds mostly chinese guys confirm the stereotypes enough to perpetuate it The uh classic and 70 said nope.  I don't care if it's free from these dynamics. That's like the last place I would possibly want to live as an asian guy and they went on to name the pros and cons of every region Andrew this guy said if you want to be an asian guy you move to texas houston You got to be sporty and masculine. Would you agree with that? I agree.  No asian food. No asian restaurants. No asian communities Somebody said arizona might be underrated the southwest. You like it. Andrew. You've been to arizona a few times phoenix Arizona's not bad And then somebody just said the west coast asian enclaves tend to mimic the worst parts of asia when it comes to materialism And uh, even though they are pretty nice and then the east coast asian conclaves and who feel like ghetto parts of east asia like flushing All right, that's hilarious by the way We always say man you gotta check out the 626 in gabriel valley if you want that easy breezy chinese american life Um, i'll tell you this I think the bay area if you're rich and you're raising a family probably some good systems But if you're a single guy you want to go on five dates a week A bad place to be well, yeah I don't know why you would go to the bay take a high level tech job that's very demanding and think that your dating opportunities are gonna be like Extremely vast like just you have to understand what game you're entering you're entering like coder land Like there's gonna be a lot of smart people. They're doing a lot of great things. They're advancing There's a lot of ai guys You're gonna have a lot of great conversations about everything from crypto to ai to ride sharing to all these different to even a dating app That's based in the bay area. How funny is that? You know, but actually tinder came out of la so that kind of makes sense Right, right, right more of the fun apps come out of la. Um, I would say this Also, there is a lot of different types of people in the bay area But I notice more of the azn like abg abz, you know daily city san osa Whatever you want to call east san osa type they more like stay within their world It's not going to be like, oh, they just met you because you got a uh Degree from northwestern or northeast or they don't care about that Yeah, I would say this if you are a tech guy and you're in the bay and you have a good job And you can do it remotely then obviously spending time in different cities is going to benefit you So then you're still part of the bay area company, but you're actually enjoying the life somewhere else You're saying take advantage of work from home or remote work Yes, but as far as solutions on how to beat the game Extend your radius to a hundred miles is what he said I I couldn't even imagine doing that to be honest for myself But I don't know maybe it works for other people some if you're a good enough product Maybe a girl will drive in from central california. I'll say this a lot of life is about fishbowl positioning Okay, which fishbowl are you putting yourself in or what pool of fish? Are you fishing in here's the thing though? You got to make sure your lures are strong if you want to use the pool analogy And if we're going to be fish living in different fishballs Andrew You still got to fish max and be the strongest fish you can be Regardless of the I mean it's like it's twofold It's the fishbowl positioning, but you got to be a strong fish yourself Whatever that looks like whether that's just optimizing being cooler getting better Stuff that we talk about on this channel and stuff that we review Of course family considerations do will come into play lifestyle friends priority. It really depends It's just all an algorithm with different equation splits.  Just do the math make the decisions execute All right, everybody, let us know in the comments down below Is the bay area really that bad for dating and what are some suggestions? Specifically for bay area dating. This is really interesting because uh, I heard Manhattan has more women than men Also, Seattle, I throw Seattle in there as is as suboptimal unless you're born there Oh, you mean as a not a great place to date a great place if you want to date A hundred women in your life before settling down Don't go to the bay or don't go to Seattle unless you're fred again Anyway guys, let us know what you think in the comment section below until next time. We the hop hop boys be out. Peace


Smalled on that byline on the city right, touches it forward to Halliday, inside to Scott Banks, city looking to build. Banks returns that ball to Halliday, into the final third now, into the area, to the goal, Bradford City! Andy Cooke turning, right-footed strike, the goalkeeper had no chance, and Andy Cooke with his seventh goal of the season, give the Bantams the lead. Brilliant return ball to Brad Halliday, who under pressure got the cross into the area, and Andy Cooke from around seven, eight yards out, well turning on his right foot. He doesn't miss those, Andy Cooke, he doesn't need to be asked twice. Deserved, maybe not at the moment, but Mark Hughes and those Bradford City players and those fans across Valley Parade and further afield, they won't care one-jolt because they've persevered, they've weathered the storm, Bradford City, and they've got their reward. Four minutes remaining of this first half, is Bradford City on? Walsall now. Well that's what I meant, Louis, about the opportunity was there for Bradford, there was holes in certain areas of the field, and the way Walsall were playing, they're so attacking, they're so aggressive in their press, there was opportunities there to be taken, and Bradford weren't taking them. In that instance, honestly if you take that goal back, that was an unbelievable team goal because Smallwood's movement with Levi Sutton in the build-up to that goal was fantastic. Scott Banks comes in off the line, Halliday gets round him, literally textbook football, it was a beautiful, beautiful goal.


My name is Michel Mosec, and I'm a level designer at Quantic Dream Montreal. Even though we are in the Montreal studio, we are very much one team, just on a different side of the world. I build 3D spaces for the game. Level design kind of touches all the different aspects, the storytellers, the artists, the animators, the game designers. We're kind of like the bottleneck that everyone comes through because we build the scene, so that's where all the narrative happens. I'm making sure that we properly support all the scenes being this Swiss army knife. So there's a lot of communication between literally all the departments of the studio. I actually went to school. I did two formations, level design and 3D modeling. I'm just curious like that. I like to touch a bit more things. But yeah, I got a very solid theoretical formation at Campus at the End. Ended up being the basis for me building a good portfolio to eventually get a job in the industry. I think Spark was like when I was 8 years old. I played Legend Zelda Ocarina of Time for the first time and ever since then, life was never the same again. I did do film studies at one point in college, but I realized that I was missing something. So I kind of abandoned that for a while and I ended up working in a video game store actually. So it really started then that I realized like, oh my God, this is where I want to be. We're doing a lot of things at Quantic Dream and it hasn't done yet. So this is also a lot of time to figure out how do we bring our knowledge and imbue it into the Quantic Dream DNA. So it's a lot of challenges but it's an interesting one. Honestly, just building the layout. Like once I know what I want, more or less know. Like even if I don't have all the answers, I just put on my headphones, listen to music and I get in the flow and I just build stuff. I really enjoy that part. Not to say that it's easy of course. But there's a certain sense of flow that you get into that it's just a pure joy. Regardless if I think that if you go to school or if you decide to go the self-taught route, the best thing is to get as much feedback as you can even from non-gamers sometimes. Like how would they read the space? Like would they identify the thing that you want them to identify right away? In my past experiences, one of my favorite things that would happen is working as someone who's way smarter than me. You know, I realize I'm like, oh my god.  It elevates you exponentially. Unlimited potential. We can do whatever we please, really. I find that super exciting.


Alrighty, so let's get this going here. You mentioned that you work on the first 200 frames addressing the notes for hands feet and the step over so that is section till here and then You still working on the hands from 150 to 190 Yep, I can see that. All right, and I worked on the hands on 200 until the end here Actually just looking at this before I guess I miss it later at this bit of a pop in this area here Right there. It's a bit of a jiggle Right there in your spacing. So watch out there. And it still has been a feel of Opening all the fingers at the same time. I will probably give the pinky and the The ring finger a bit of a delay and bring that thumb in maybe a bit lower So it's a bit in a bit more relaxed state as it gets out You can see it and then it goes into a bit of a wider Post the way you have here. They can bring up the thumb and that way it gets a bit of a contrasty shape during a good silhouette Get some good contact here It's maybe a bit soft. It's meaning there's a bit of The slide is maybe too much I will probably rest and then give this super minimal slide and then as he goes forward and pushes Pass and there's more pressure and he moves forward then the hand could continue to slide Where it's kind of like a roll off that edge the way you have it That's cool. Okay, this here you might go a bit far here Just I think it's a bit pushed where your fingers are kind of really bending upwards And I think that is a bit strong there Because it's not like he is Resting and pushing his animal is just kind of resting in and taking me the way and now it feels almost like he is really Pushing and then taking it off. I don't I don't feel like there will be that much pressure on it It's kind of you know gently putting his hand on there to go forward And then it's fine on the other hand Same thing again, you might still lay the pinky and ring a bit more kind of start with the ring and index Could be subtle, you know, it doesn't have to be crazy But I do like how the rest drops into this here good offset and a good nice pose at the end Do a simpler shape there. So that's cool. I like all that Looking at this here, so you are mentioning that you meant you worked on the hands I still feel like this is a bit soft just that area So as it comes in here She let me do this here give this just a bit of That's all cool I Am slightly worried about just that look of this should have mentioned before but I'll probably bring in The angle of the thumb to something like that. So meaning that it's going to be slightly hidden Until it kind of come forward here, but it just has this light feeling of it's kind of your hand It's kind of curved this way, but then the thumb is really straight up and I know it's resting on the door like this There's just this weird 90-degree angle on that hand now. So if I could pick he Probably take that down just a tad and I think you could tone down the amount of movements Just a little bit on all of them and then just them a bit more on On the pinky and the ring right now It feels like you're doing like a very exaggerated Grab and if you just reduce that a bit and keep it mostly in just these two fingers kind of It kind of does that so you have a bit more here Maybe a bit more of an extension a bit more movement through here But then these two the ring and a pinky just kind of go with it to kind of grab Just so it doesn't feel like this big fanning out movements All the fingers That's all cool how it comes off That's all cool here, then you're working on that Now the one that pushes against the door. Yeah, there's something through here where I Feel like all fingers are moving at the same time and there's not not really a Three through here a connection Where it feels like he's going around and then you feel that push and then getting a bit of that in there You can feel the flatness on the fingers with the push Get out that the door is super heavy but a little bit and then no sliding from the fingers on the door So you feel that nice pushing another connection point is solid and then at the end It might be a bit of a spreading because you're closing the door shut and then comes off And changes the hand pose and I feel like that is not quite reading All of this here the contact just feels pretty soft oops there now body wise Well hesitation in the head there Yeah Cool stuff there in the feet. I remember there's some pivot point stuff I think you could still push or something about as this foot comes down here You can see how that's the angle of the leg, but then the foot is straight down So at this point I would have the foot angled this way It might be too much, but just as an idea and then it pivots off that side of the shoe And then as he goes on there rolls off so that the foot kind of Like the feet but the foot feels like it's always flat and then coming down as opposed to a little bit like this coming on the edge and then rolling Then as you go over here See I can scrub through here and I don't feel like it's kind of rolling pivoting off this side of the of the shoe Just that contact feels a bit too locked where it's like an IK foot nothing's moving and the rest of the leg is moving Picky-picky, but I think that would be something cool to put in I'm sure how close you get it seems like you're intersecting watch out on this foot the door Not too bad. You're starting to pivot there early. That's cool. That's cool there Uh, let me whoa Oh Not zoom in That much but bring it up. Let's see the feet It's cool You'll probably I know that rake has it then again. It's not like he's Barefoot it is a shoe. I see a Toes finger toe shoes. He's just going to say as he puts his foot down Give it a spreading in the toes, but that would look weird But maybe we have a bit of a squish on the shoe for super subtle stuff, but In terms of Rotations and pivots Not too bad. I still feel as you do something like this. There's a lot of movement I don't have to rig supports it But if you have anything through here in the end, they're just the ankle part Kind of rotate over and you can feel that that squishiness as the foot You know as the whole leg goes like this So you want to feel something in this in this area like that's the ankle You'll feel a little bit of movement how it follows the shin to kind of some compression and then the shoe rolls off over Picky thing and it's in there. I would just push it a bit more But in general the steps are cool It's all cool The side step just a bit much how far back you go this way with the root root here right there That little drift probably cut that in half the rest Yeah, it's just a bit a bit much. She goes over and then kind of drifts Rip to the left So go over here and then just kind of hold a bit and go a little bit to the left for keep alive But that's about it This is cool Not sure if you did anything on the face. I'm going to wait with some more notes I mentioned I don't think you've done that just yet. You know the eyebrows and stuff is all very very locked Much in there I turned some cool here though. I think that was a lot better. So yeah, so picky picky picky things and feet Um, but that is about it as always if you have any questions, let me know and email me If you need clarification on things. All right. Thank you All right, there's an email you can sign up you can start whenever you want you can submit whatever you want You get 16 submissions either way a like and subscribe would be awesome. All right. Thank you


From Bed-Life Stadium in East Rutherford, New Jersey, we're underway. Regardless, you know, this is the NFL. Your numbers call, you gotta go out there and make plays. Tana Hill fires for Batson. Touchdown Titans, Henry. It's a long season, and we have a lot of football fans. Touchdown Titans!  I think we learned from it, watch the film, grow as a player, grow as a team, and do for it. At Nissan Stadium, I'm Lance Smith, but better. This is Mark Mariani right here. Good to be here. It's good to have you back. It's good to see you.  Well, thank you. And we need it coming off of this game. You're right. After watching that film, you know, today early, I need a little beam of positivity myself. And guess what? The Jacksonville Jaguars are the perfect opponent. Wow! Get a little confidence.  Get our swag back. Right, we'll get to that game. Okay, yes. We do have to talk about this game a little bit. We'll get into that. But how have you been, sir? Oh my goodness. Life is good. I got two little nuggets now, growing those.  There he is. Oh, he's watching? What's up, little man? He just keeps me busy every day. We got little girl, Romy, there. She's, well, look it. She's like, I love my little brother. And he's like, this alien's in my bed now. I'm not so sure. So they're keeping me busy. And so life is very good. I'm a very blessed man. And now that it's football season, that's pretty much kids and football. They consume me on a daily basis. Now I don't feel like me when you watch from home, like I scream at my TV. My kids, like I'm a very easygoing guy all the time, but they don't know who I am when I watch a game. Yeah, well, my little man knows when he sees that logo on the helmet, it's Go Titans every time. Yeah, yeah. So he knows. He knows. He's screaming right now. I'm certain of it. All right, what's up, little man? Yeah. All right, well, let's get into it. Let's do a little Go Titans. Let's do some hot takes with Mark and we'll talk about the game. Because look, yeah, we lost, but there were good things that happened in the game. And right off the bat, guys, I mean, we're watching history in the making. Derek Henry is a legend that is currently playing. The guy had 33 carries, 157 yards, a TD, two catches for 20. He is on pace for 2,100 yards. This guy's a man amongst boys, man. He is a freak. He's like a superhero out there. So when we get things, even in a bad game, he can go off for 33 for 157, a five yard average almost. When we get things really clicking, he will not be stopped. There is not a single human on this planet that wants to try to tackle this guy one-on-one. So I look forward to this season. He's an absolute beast.  Well, he is. And it's one thing. I mean, they said that without Julio and AJ would be all Henry. And I know they looked like they were trying to limit his carries early, but towards the second half of the game, when we needed him and we got him, and they got a load of him. So he's still so fun to watch. The next, it's something to point out, one of the great things. I mean, we didn't quite get there. Zach Wilson made one too many plays scrambling out, but the pressures are there. That front seven's working and it's getting better. Yeah, I take any chance I get to highlight what Harold Landry's been doing this year and he deserves it, man. We've been asking for it and he's been delivering.  And again, he and Danico Autry in the front seven this weekend without Bud Dupreev and we're getting after Zach Wilson.  Harold Landry gets me going. So anyway, I love watching him this year. He's really getting after it. He's really coming into his own. And again, once we get everybody healthy and back after it, I can't wait to see what this front seven does because there's a lot of football left to be played. Absolutely. We're, you know, we're two and two. We've got this thing. We're in control of the South right now. Also to, you know, Jeffrey Simmons, maybe not getting the numbers or his name called, but he's the reason Harold Landry and Danico Autry are able to do their thing. So it's a complimentary effort up front. Can't block them all. And our man leading the team, Tannehill is doing all that he can. It's still without his number one receivers spreading the ball around, throws for almost just shot 300 yards, gets a touchdown. Doesn't turn the ball over this game. You know, he's the leader we want and need. He's, he's become one of the most consistent players on this football team. An unbelievable leader, tough as nails. And this is a second game in a row where he's had a completion to nine different receivers. That is really hard to do, especially when your two guys are down. So love what he's doing to respect him. And again, even though we lost again, you put his hand, you give him the ball with a, with a needing a touchdown in the fourth quarter with two minutes left. He delivers again.  I got all the faith in the world and I know he's going to lead these troops. Yeah. Biggest key got to keep him up right though. Oh man. None of the turf 17 times. It's too many this early, right? It's too many. You can't be at beat him up. He is tough as nails, like I said, but the number one thing to do with it, you got to protect this guy. And it's a combination. We talked about it. Receivers got to get open. Line has to block. You know, running backs have to pick up the blitz. So we got to take care of business. There's a lot to correct, but moving forward, we got some big opportunities in front of us.  I really, we had pulled out this win because without AJ, without Julio, you still got guys like Jeremy Nichols stepping up, Chester Rogers, Josh Reynolds, there's between those guys alone, that's 19 receptions right there. So next man up mentality. Always man. I was always one of those guys. I was a depth player in my career. So when you get your number called, you go make plays. And that's what those guys are doing. They play with a chip on their shoulder. It's fun to watch. I love seeing them make plays out there. So, you know, we're always in good hands. We got enough horses in the stable to get things done. Just more consistency probably, but it's good to see these numbers. Those guys are putting up and hopefully, hopefully the big dogs are back out there. But if not, we're in good hands. Hey, hey, hey, hey, Vrable has a superb history of coming off of losses. So I'm looking forward to this game coming up. All right. We've got more of Titans Blitz to talk about. More Titans football to talk about right now. Here's some Nissan memorable moments. Stick around. Titans Blitz will be right back. Picks up a block to the 45. Turns the corner to the 50, to the 45, to the 40, to the 35. And he's finally pushed out of bounds by Carter. Guess what? That's enough for a first down. And Jeremy McNichols with a smooth move. Late bank. Wilson with a deep drop. Throwing. Intercepted Fulton. Dead ball looked like it was thrown right to Christian Fulton. Let's come to you live from Nissan Stadium. Lance and Mark, back with you. This game this weekend, watched it like everybody else, or a lot of people at home on the couch, taking it in with the family. My daughter and I, we watch football together. You got kids? I got kids. Yeah, yeah. My daughter, Maggie and I, we watch these games together. Now she's almost four, so she's just kind of learning the ropes. There's Maggie right there. My pride and joy. Nice. Look at her. Well, watching the pre-game show though, I got a little worked up because we're going to this game against the Jets. We're coming. We're hot. Two wins in a rope. Seattle, Colts. Got this thing. And it fell today. Bill Cower, every before the games, always picks his upset of the week, and his upset of the week was Jets over Titans, to which I tweeted this, because I was feeling froggy. I said, Bill Cower, just picked the Jets over the Titans in an upset. I will rename my youngest daughter, Bill Cower, if this happens. Notice the time is 1251 p.m. This is Eastern, so this is just before kickoff. Mark, yeah, I said that. We all know what happened. But, you know, I would like to be a man of my word and I had to let Maggie know. I'm gonna do something like this. I want to be Gecko. Okay. Thank you, Bill. I'm not Bill. You're not? No.  What is that? No, I made a promise, honey, and I have to call you Bill. So. Why? Why don't you call me Maggie?  See, Twitter, I can't do that. It hurts my heart. Well, anyway. That is a man of his word right there, okay? So I've been calling her little Bill Cower. It's the whole name. She hates it still. But, you know, I'm a man of my word. However, the real Bill Cower got word of it and he wants to chat with Maggie and I this week. So maybe next week or next week, she'll give you a little update, let you know how it goes. So be careful writing checks that your children have to catch. And I hate that we made him write. Yeah, you don't want to get them involved. Right. Anyway, I love you, Maggie. Moving on, keeping in the theme of kids, one of my favorite segments here on the show, we asked a certain local child artists to draw their favorite Titans players to see if our Titans could see who it is. Okay. So from six-year-old Beckett, here is this week's vague van go. All right, this is Beckett. Beckett drew this one. Well, I'm gonna tell you something.  You know, he's gonna be one of the top, one of the best ever in the world. So keep scribing. Sunday afternoon, you know, Sunday at his highest point. Blue eyes, he's holding football, nice shoes, football in his hand. He's throwing it. Everybody drawin' that can get saved five, six years old. You got a lot of talent, man. You got a lot of talent. That's Ryan Tannehill. Titans jersey, pants, cleats. Gotta be Tannehill. Throwin' a football. Ryan Tannehill. Looks like he's throwin' a ball. I see brown hair, blue eyes. I feel like this is number 17. Gotta be me. Told you, I know my quarterback. All right, Beckett, what do you got? It is, after all, me. Thanks, Beckett. Appreciate it. Thank you so much, Beckett. You know, I hadn't seen it yet either myself, and I know you hadn't, and instantly, Mark's like, it's Tannehill. Cause he's throwin' the ball. It was such great artwork. It was so, I mean, it was so beautifully done. Looks like Tannehill. I saw the football it gettin' thrown. I mean, it was Woodside, Matt Barclay maybe, but then the brown hair, you know. I think that Tannehill's people should call the football card companies and just have that image replaced with his football card. We're makin' t-shirts. Yeah. All right, guys. Hey, stick around, we got more Titans boots on their side, including the Titans, Trivia Gauntlet. Play along at home. Winston prizes, Premier Kitchell. Blitz, you feelin' lucky? Yeah. Yeah, oh, they're nervous. It is time now for my new favorite installment on this show. We're here live at Nissan Stadium. It's time to play the Titans, Trivia Gauntlet. Roll it. It sounds spooky, and that's just, that has nothing to do with Halloween, just cause this game is terrifying. Right. Let's start over here to my right, Shuri. How you doin'? I'm doing great, how are you? I'm excellent. You say you're from Nashville? Yes, born and raised. Season ticket member? Yes, sir. Yeah, you gotta, are you wearing your favorite Titan of all time, or do you have a favorite Titan of all time? I just, my favorite Titan of all time was Baroness. Oh, we're Rob Baroness? Oh, man, we miss Rob Baroness. Absolutely. I love him. Absolutely. He was a great man. Gave us on and off the field. Great memories, yes, on and off the field as well. Thank you for being here, Shuri. Absolutely, thank you. Good luck to you, and Erica. I asked Erica where she was from in Tennessee. She said Tennessee, and she said in the country, and you know what that means. All right, I got it, I got it. But you're a season ticket member as well, right? I am. Yeah, you got a favorite Titan player? Steve McNair was. McNair. Totally. Yeah, wow, great man. You got a favorite Titan moment? No. Oh, you gotta say this. I love him. I love him. I know. You know what, the favorite Titan, when we win the Super Bowl this year. All right, let's do this thing. It's time to play the Titan tribute gauntlet. Erica won our coin toss that happened five seconds before we came back from commercial break.  Prizes up for grabs. Here we go, it's multiple choice. Don't get nervous, Erica. First questions for you. Which head coach has the most wins in Titans Oilers franchise history? Is that Jerry Glanville, Jeff Fisher, Abum Phillips or Jack Party? Jeff Fisher for the win. You Jeff Fisher for the win, she says, yes. Question number one goes to you. All right, see how this goes. So you got one point. Cherie, next to you. What position other than quarterback did Ryan Tannehill play at Texas A&M? Tight end, wide receiver, safety or place kicker? Tight end. He unfortunately was not a tight end. He was a wide receiver. You know what, if he played basketball and was a tight end, that's all you'd ever hear about when they watch games. All right, moving on, you're up a score here, Erica. Try to add to the lead. Your question is what officiating signal would include placing a hand on each hip? Is that illegal motion, delay of game, offside or pass interference? Illegal motion, delay of game, offside. Offside, man, we could use you in a game, I think. We've had some rough calls this year. Two points to you, got to catch up, Cherie. Here we go. I need an easy one. You want an easy one? See how easy this is. Do some math. Jersey math, Julio Jones' jersey number, plus Nick Westbrook, Akina's jersey number equals what player's jersey number? Oh, Jeepers. So is that Ryan Tannehill, a defense player, or a defense player? So is that Ryan Tannehill, Derek Henry, Kevin Byard, or Bud Dupree? Do you know their jersey numbers and what they add up to it? I don't know Akina's numbers. It's tough, right? You want to show off, don't you? I want to show off. You don't want to show off. All right, it's actually, so you're looking at two plus Westbrook, Akina's 15, that adds up to 17, that adds up to Ryan Tannehill. Sorry, all right, moving on, moving on. Last question. Final question, this is worth two. This is worth two. You have to go first, though. Here we go. Erica, what? Oh, I'm sorry. In 2016, which opponent did Derek Henry score his first NFL touchdown against? His first NFL touchdown was that the Jaguars, the Ravens, the Colts, or the Lions? Ravens. Ravens, what say you? What say you, Cherie? I'll say the Lions. You can say the Lions, the answer is actually upcoming opponent, the Jaguars, but Erica, you are our winner for the tightest trivia gauntlet. Congratulations, won Mini-Humming Sun by AJ Brown. New line from Nissan Stadium. Wrapping up the show here, but we do have a big game coming up Sunday in Jacksonville on the road again. You know, hey, some news to point out. AJ practiced today, budget pre-practice today. Darren's and Evan's practiced today, guys. Getting back into it. Hey, we need all of our weapons, man. We got a great, very, very talented roster. Some of the most superstars that I've been around since I've been around this organization. So getting everybody healthy is pivotal.  And to me, this game has nothing to do with the Jacksonville Jaguars. This game has everything to do with the Tennessee Titans. You prepare and show us what kind of team you're gonna be in 2021. And I think they got some film to watch. They have some corrections to make, but this opponent, someone we should take care of, it's all about how our mentality is when we get off the bus. And so that's what I wanna see out of this group. And I personally, because I am the driver of the fan bus, I think we're gonna take care of business. But this is about the Tennessee Titans. We got beat down on Sunday against the Jets. We gotta go take care of business in Jacksonville. Well, talk about that from your mindset as a Tennessee Titan. So when you're playing, you're in the pros.  Do you play that much harder? And do we know that? I mean, look, the Jags are in a PR nightmare right now and they're not winning. So, but I mean, look, when the game starts, you know, 11 men across from you, those guys wanna play. Let's be very clear about this. This is the NFL. It is extremely hard to get a W in the NFL. These are the most competitive dudes on the planet, the best in the world at what they do. It's very hard to do. But I'll tell you what you do with an opponent like this as you get off the bus. Like what we used to say, you punch them in the mouth.  Cause what we did against the Jets is we let them linger. We let them sit around to the fourth quarter. They make a couple, you know, on the run plays to beat us. With the Jacksonville Jaguars, all the stuff that they're going through, us coming off a loss, they're backed up in a corner.   That's what you do to an opponent like this. Looking at another rookie QB, but the franchise and Trevor Lawrence just keep the pressure up, right? I mean, we're getting there, but we got to close the deal. Yeah, exactly. This guy is a winner.  He will beat you.  Get after him. If he wants to tuck that ball and run, you hit him every single time. He's very dangerous with his legs. He'll throw you the ball like we're seeing right here. But you got to hit this guy all day. He's down a couple of his weapons as well, right? He's got some issues in their locker room. You got to make his day miserable. You know, I'd like to see us do a little rebuttal here and get seven sacks on Trevor Lawrence this weekend. So let's send him to the ice tub early. Okay, right. Look, the guy's got talent. He's going to be in the league a long time. Every time that he's on the field throwing an interception, they say, well, you know, Peyton did this a hundred times his first season. That doesn't matter. It doesn't matter. He's a rookie or he's played 15 years. Yep. And like I said, all that matters to me is the Tennessee Titans mindset. And what is our identity going to be in the 2021 season? We got to go win this game. We got to win this football game. It's a must win this weekend. Hey, Gretel, he's for hire, for motivational speaker in the locker room. Mark, Mary, Annie, thank you for being here, sir. Thank you all for being here. Keep it loud and proud. We will see you next week right here. Same place, same time. See you next week. Tighten up, baby, tighten up.


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discuss the Hamiltonian formulation for the solution of optimal control problem and then we will consider some numerical example. So, this is formulation is Hamiltonian formulation for solution of optimal control problem of course, using calculus of variation. So, if we recollect why we are going for Hamiltonian formulation, suppose the system dynamics or system plant is described in state based form means any system can be described with a n dimensional vector. So, n dimensional what is any system can be described with a n differential equation in first order differential equation, if you represent the system in state based form it is convenient to deal with Hamiltonian function rather than the Lagrangian function. So, let us recall our earlier problem consider the plant or system x dot is equal to f of x t u of t and t and this dimension is n cross 1, this is n cross 1 and this state number of states is n cross 1 and number of inputs to the plant is n cross 1. You can think of it we have a plant or system plant or system this one input is u of t whose dimension is n cross 1 and this state you can say output is the state you can say x of t whose dimension is n cross 1. So, this plant is described in general we have a dynamic system is there we can describe with a nth order differential equation which can be converted into a what is called n first order differential equation. So, once you convert into a state based form then our problem is here and the corresponding performing index j if you recollect this one we have written x of the terminal cost is a function of x t and t at t is equal to t f final time plus integration of t 0 to t f final time v x of t u of t and t d t. So, our problem is to find out the control law u such that this performing index is minimized not only that subject to the constraint that equation let us call this is the equation number 1 that is the equation number 2. So, our problem is to find out the control law u of t which in turn to find out the optimal trajectory x of t such that this performing index is minimized subject to this constraint equation number 1 that is what we discuss less plus and correspondingly we got what is called the necessary condition. If you say our necessary condition along with the boundary condition we got it that del l this x of t minus d of d t del l dot function of l is a Lagrangian function which is a function of x t u t lambda t and comma t. So, that differentiate with respect to x dot of t whole that you solve this one along the because you have to find out star means you have to solve this one whatever the solution you will get it that is the optimal trajectory or optimal control law from which in turn it will give you the optimal trajectory x star of t. So, that equation you have to solve. So, we have a in turn we have a n cross 1 differential equation is there. So, that differential equation in general it is a non-linear differential equation. So, that is let us call equation number 3 and not only this another x condition we got del u del t this one is equal to 0. So, this is equation number 4 corresponding to our this problem and we have a boundary condition that boundary condition if you see this one that we obtain boundary condition we obtain l Lagrangian function minus del l of with respect to x of t that since x is a vector. So, l is a square quantity that will be a vector column vector. So, we have to take transpose of that one and then multiplied by x dot of t. So, this you along the trajectory we x star then t is equal to t f this is the boundary condition we got it when t f plus then del l dot del x dot of t whole star t is equal to t f and delta x f is equal to 0. So, let us call this equation is 5. So, and we know what is the l the l expression we have written in terms of what is called Hamiltonian functions where l dot which is a function of x t comma u t comma lambda t comma t. So, this is a function of that Lagrangian function is split up into a what is called Hamiltonian function and then Hamiltonian function free form x dot. So, this is the Hamiltonian function and that function is called Hamiltonian function. So, Lagrangian function this Lagrangian function is nothing but a if you see what we have consider this that we will write it in this plus del s dot plus del x of t whole transpose x dot of t. So, this is a function of what is called v this is the that function if you see integral part of this integral 1 v plus Hamiltonian function plus lambda transpose of f that one. That means this two terms if you see this term and this term is free form x dot and that function we consider as a Hamiltonian function because why you have express that necessary condition what we got it here necessary condition equation number 3 4 and boundary condition 5. If you express this thing into Hamiltonian function then we will see if the system is described in a state based form. If you express this in place of Lagrangian function if you replace by a what is called Hamiltonian function a set of equation what will get it in a simpler form and it will be easier to solve what is called if the description is state based form. So, that we will see later of this one. This is our equation number let us call 6 now we replacing the you can say using 6 in 4 to using 6 in using 6 in equation 3. That is the 3 equation not 4 3 4 and 5 3 2 3 2 3 2 5 comma we get what let us see. So, our first form 3 we can write form 3 is del 3 you see del l del x. So, in place of del l I will just write it in terms of h that whole expression I will write it. So, del l del x in place of l I am writing h x 2 del x. x of t u of t lambda of t comma t this one plus delta s dot of that x of t whole transpose that dot of t plus del s this del t minus lambda transpose of t x dot of t. So, this whole thing if you just see the whole thing from here and including this one is nothing but our l Lagrangian function. So, I have just written the first term of our equation this is that one then second term if you see this one I am writing is minus d of d t. Then del l del l again this whole quantity differentiate this with respect to x dot. So, what is the del l I am just writing h x of t u of t lambda of t comma this plus del s dot del x of t whole transpose x dot of t plus del s dot del t this one minus lambda transpose of t into x dot of t. So, that is our we have written l this whole thing just same as this one same as this one we have written it here then this is differentiating with respect to x dot. Now, look at this expression that one chain rule we have used frequently that chain rule is like this way. Suppose, we have a function x is a function of x t t is the parameter and y t and z of t this is the function of this. Now, if you want to find out the differentiation of x with respect to d t this is nothing but a partial differentiation of f with respect to x then x dot then del f del y then another variable del y of t then y dot. So, what is this if you have a function x of function x of t y of t function of x y t which is a function of time each then I can differentiate on of x with respect to time t is nothing but a partial differentiation of x partial differentiation of f with respect to x multiplied by x dot of t. Similarly, partial differentiation of f with respect to y t is y dot of t and z dot of t. So, this things we will write it here. Now, see this one s is a function of what this and this I can write it s is a function of x t and t this is a function of if you see this is the function of x t and t this one. So, now I am writing it this one that this and this term and this term commonly I can write it. So, look this term by using the chain rule this term and this term commonly I can write it that del d s of del y of t. So, this is what we can write it this and this commonly I can write. So, rewrite this equation what we write it this one just say del l del x of t then h I am not writing h dot means is a function of x t u t lambda t and t this h plus d s dot d t agree. And now this what is left this and this d s of d t and lambda transpose minus lambda transpose of the x dot of t agree. This is the first part which we have simplified then second part of the Lagrange equation d of d t see this one that whole thing this quantity this quantity you are differentiating with respect to x dot h this h is not a function of x dot. So, if you partial differentiation if you do with respect to x dot this term will not be there agree the first term will vanish then this and this is a function of s is a function of x t and t. So, this is a function of x dot. So, we can take this is a constant because it is not a function of x dot t. So, if you do the differentiation of this one del s this del x del s partial differential s with respect to x t we will get it that that one agree and this is not a function of x t minus x dot t. So, this will not come into the picture. So, and now it is a function of x t. So, only this term and this term will be remain in the differentiation of the Lagrangian function with respect to x dot when Lagrangian function express in Hamiltonian form. So, this term ultimately it will come if you see is coming like this way if you just do it then it will come and if you recollect once again that if you have a function if you are differentiating that function with respect to x x is a vector let us call say and you are getting a transpose x a is a row vector a is a row vector of dimension n cross this is the row vector of dimension 1 cross n and this is n cross 1. So, it is a scalar quantity. So, that is differentiating with respect to a vector that results is a. So, now if you just use that one the differentiation of this is you can consider is a scalar this is you can consider is a scalar multiplied by a what is called differentiation this thing by a what is called a vector. So, that results will be a row vector and you are differentiating with respect to this. So, results will come the differentiation of this with respect to x dot that term. So, I will ultimately I will get differentiation of s with respect to x of t and this term similarly this is minus lambda transpose x dot differential x dot is one and this will be lambda of t minus lambda of t. So, this whole thing if you see this one you can put it that whole thing is in bracket of that you if you just do it here this is not l this is of that that one is star you can put it you like you can put it in star that one. Now, this equal to 0 because what from where you are getting that first equation of from equation 3 this equal to. So, right hand side of this one equal to 0 this one so from 3 I am writing from 3 this is 0 and I am getting that one. Now, see look at this expression differentiation of partial differentiation of that quantity with respect to x of t and this order of differentiation you can change it. So, if you change this one this is plus this is minus this this is cancelled only is left over is this one del h del because that is the function of you have to differentiate with respect to x t. So, that will be vanish that there is this is lambda t lambda transpose t x dot of t. So, you have to differentiate with respect to x ultimately it is a del h this del h. So, del x of t whole star this bracket this star is here this and this is also star from here to here. So, this equal to this plus this minus this minus plus equal to your this term is what differentiation with respect to t that. So, it will be coming minus minus plus if you take right hand side it will be lambda dot of t. So, let us call this equation we have used the equation number up to 6 let us call this is equation number 7. So, now see this one when you use the Lagrangian equation the del l del l the Lagrangian function you differentiate partial differentiation with respect to x minus d of d t del l del x dot is equal to 0. So, this expression is boils down to what is called a simple Hamiltonian function form this is and this function is free form x dot is a function of x you t lambda t and t. So, next is so this is the this equation and what is this equation another equation is that one from 4 what we can write from 4.  So, it is a function of your differentiating partial differential with respect to u there is no u is here. So, del h del u must be 0. So, from 4 we can write del h dot del u of we can write del h dot del u of t whole star is equal to 0. Let us call this is equation number 8. Now, you see this one another expression we call this equation is called what is called costate equation number 7 is called costate equation costate equation. So, this is the costate equation. So, another equation one can derive like this way del h dot del lambda del h del lambda is equal to x dot of t. Now, see the our objective function if you see if you refer to your last class note if you see we have written the what is the Lagrangian function we have written the objective function plus the lambda transpose constant that is the Lagrangian function. So, del l del one of the what is called KKT condition del l del lambda must be equal to 0. So, we can get this equation from Lagrange expression one can get this expression get this expression from Lagrange function or expression. So, this is also if you like you can write star this star this one. So, let us call this equation is equation number this is the equation number 9. So, this is called is optimal state equation and you see the costate equation dimension because lambda dimension is if you say n cross 1 same dimension of the state vector. So, that is why it is called costate equation this equation and this equation nature of this equation and this equation is there are costate one of costate of another. So, this is called costate of a state equation vector. So, now we have that this is the equation number 3 is now when you express Lagrangian function in terms of Hamiltonian matrix this expression simplified form you have got in del h that corresponding expression that one will get in simpler form del h del x equal to minus lambda dot of t. Another expression you have to in place of del l del u partial differential Lagrangian function with respect to u when l is expressed in terms of what is called that Hamiltonian matrix then we will get del h del u and the state equation that what we have equation is del h del lambda is equal to x dot of t. So, now see the boundary conditions if you see the boundary condition of that one then what we are getting it will suggest see the boundary condition now. So, note and the boundary condition what we will do we will replace the Lagrangian function in terms of Hamiltonian function that is all what is the final boundary condition we will get that we will see it. So, the boundary condition and our original problem a boundary condition is expression is equation 5. So, boundary condition 5 is now replaced by in the form of boundary. So, in the form of Hamiltonian function. So, boundary condition is now replace Lagrangian function in the form of now you replace Lagrangian function in the boundary conditions Lagrangian function in the form of Hamiltonian function now replace the Lagrangian function Lagrangian function in the form of Hamiltonian function. If you replace this one now you see first what will write it for this one l that equation number 4 if you see here. So, this is h you will get it that is l differentiate of this that is l differentiate of this with respect to x whole transpose x dot putting t is equal to t f that one and then del l del x dot t is equal to f del x t f. So, let us see what we can write for this one. So, we can write it for this one if I just put it here that total let us call rewrite 5 equation 5. If I rewrite this one l dot minus l dot of this x x dot of t whole transpose x dot of t.  Now, see this one our expression for that that one what we can write it l that one what we can write it l is a what you can write it this l l is the our expression like this function expression h plus. So, if I write it this one you just see what we are writing l l is nothing but a h dot del s del x of t whole transpose x dot delta dot of t plus del s of t del t this minus lambda transpose this is lambda transpose of t x dot of t. So, this I have written in place of l now this l is f see that one l is now we are differentiate with respect to x. So, if you do this one then what will get it this when we are differentiating with respect to x dot then it will be a del x this we are whole thing we are differentiate with x dot then it will be a if you do differentiate with respect to x dot l the here is x this. So, minus see this one I am differentiate this with respect to x dot. So, you have a minus x this. So, this will be del s del x whole minus there is a x dot here this is l I am differentiating this thing with respect to x dot. So, then what will be this one this will be a del s del x t minus lambda t minus lambda t of that one and this is the bracket that thing will be bracket because it is a minus sign is here. So, the whole thing is a bracket. So, I have written if you just this one l del l del x dot transpose x dot what is this value that I have written it that quantity then this star t is the equal to t f delta t f plus again plus again this term this equal to 0 because our boundary conditions from 5 this is equal to 0. So, next I am writing similarly that one will be delta s delta x of t there is differentiation with respect to x dot. If you are differentiating with respect to x dot that is x delta of x t this minus delta of t plus term and that you are making star t is equal to t f and delta x f is equal to 0. So, now see what is it this is l and this is the delta l you are doing that differentiation of that one with respect to x dot agree we got it that that one then again it is a differentiation of this with a x dot we will get it this one. Now, see this one what simplification we can do it here because now delta s and there I missed it here that your what is called x dot x dot you see this expression that this is the x dot is there that x dot is missed here. So, delta s this transpose also this delta x dot this x dot this this cancel now this is plus this is minus.  So, that and this is our like the boundary equation is this. Now, see what we did it I just replace once again I just what I did it this is the necessary condition when you express in language and function and when language and function is expressed with a what is called emulation function and some other terms we replace it l by this and do simplification we got that expression condition del h del lambda del h del lambda is equal to x dot that is what we got it.  Similarly, the boundary condition what we did it here when we just replace that expressed boundary with in terms of Hamiltonian we got it this function that is x dot this expression. So, our simplified form not this expression that is just now we have calculated this expression in terms of Hamiltonian. So, this is the important boundary condition in terms of Hamiltonian functions. So, now we will summarize this point. So, if you see this one this is the basic three equation we need to solve now this is the equation 7 8 del h del x is equal to minus lambda dot t when thus the system dynamics is expressed in terms of what is called state phase form. Then we can solve this one by using the Hamiltonian function where the Lagrangian function is expressed in Hamiltonian function form if we express in that form then del h del x star is equal to x dot what is called x dot lambda dot of t that one expression. Then del h del u is star is equal to 0 then another expression. So, this dimension is m cross 1 this dimension is n cross 1 the next equation what we got it here that equation 4 5 page number 5. So, this is another equation we got it the del h del lambda is equal to x dot of the state equation of this that one can get it from Lagrangian equation of this one then this is the boundary condition. So, this is the equation number 9 this is the equation number 10 let us call. So, if you solve the equation 7 to 10 then you will get the trajectory of control input as well as which in turn it will give you the trajectory of the state trajectory path in optimal path of this one which will minimize the our objective function or performing index. So, we will summarize the results like this way. So, if you see this our problem algorithmic steps now algorithmic steps. So, if you see our problem is like this way given the function our problem is x dot is equal to plant is given A x plant is given x of t u of t. And t and performing index is j the terminal cost is x of f t f t or if this or you can write it x t comma t t f is equal to t is equal to t f this plus t 0 to t f v x of t u of t of t this d t. So, what you form it you form first step is form the Hamiltonian matrix Hamiltonian matrix x of t u of t lambda of t and t what is the Hamiltonian matrix this integral plus lambda transpose of t into f of x of t u of t and t. And this dimension n cross 1 and the whole dimension is 1 cross n 1 cross n. So, this is the scalar. So, this is free form x dot this Hamiltonian function once you know the Hamiltonian function then our what is called step is there step 1 compute del h del u is assigned to 0. Let us call this is equation number once I know the Hamiltonian function from the performing index given n the constraint this is the constraint our problem is finally, we have to find out u of u t or u star of t such that this performing index is minimized subject to the constraint equality constraint. And that equality constraint may be linear may be non-linear dynamic equations. So, this is the first equation second equation the state equation that we have written it del del h dot del lambda of t this if you like you can put it in this equation. So, put it because you have to solve it and what is the solution you give that is the optimal star means optimal solution you get or you can omit this star it does not matter because you have to solve this equation. So, that equal to this equal to x dot of t. So, this is also n cross 1 let us call this is equation number 2. And this is the call this is state equation then third is lambda del h del lambda del h del x differential h with respect to that means what is called Hamiltonian function you differentiate with respect to x which is equal to minus lambda dot star. This is if you give star that one and this is the your costate vector see this one what we got it from this from the linear costate vector that expression is the third. So, this is the equation number 3 and then you have to solve this 3 equation by using what is called boundary condition that our boundary condition is del h sorry h Hamiltonian function del s which is function of x t and t del t whole star t is equal to t f delta t f plus del s dot del x dot of t sorry x of t this minus lambda of t whole star t is equal to t f into delta x f is equal to 0. So, this is a vector so transpose is there that. So, this transpose do not forget to give transpose that one. So, this is the boundary condition and this is the boundary condition in terms of Hamiltonian function boundary condition. So, if we just recall once again that given the plant dynamics this x dot is equal to f of x or performing index is that way this is the terminal cost and this is the integral part of what is called cost function. Then our first step is find out the Hamiltonian function that Hamiltonian function is nothing but the integral part of this integral this one plus the Lagrangian multiply into f of x. That is we have we have seen it how we have converted a constant optimization problem into unconstraining optimization problem using what is called Lagrange multiplier technique. So, this is the that Hamiltonian function once you find out the Hamiltonian function then differentiate that partial differential h with respect to u because h is a function of x t u t and lambda t t. So, next is once you find out then differentiate that h with respect to lambda t that will be x dot of t that we have derived this one. Next is you differentiate this with respect to x t partial differentiation of s with respect to x t that will be given lambda dot of t. So, this will be giving you the what is called boundary condition of this one that. So, and this is the four equation you have to solve simultaneously. So, next question is let us call we have taken we take a simple example and before that what is that that after solving this one what is the guarantee that objective function or the what is called your that performance index will give you the minimum value of the functional or maximum value of the functional. So, that is called sufficiency condition sufficient condition sufficient condition. So, that sufficient condition if you see this one is is provides to determine it provides or in order to determine in order to determine the nature of nature of optimization either functional value is minimum or maximum this is the nature we must confirm this one that can be done by using sufficient condition. We have already derived what is the sufficient condition is there in terms of Lagrangian function and if you replace that l in terms of h and some other terms we have shown it you replace l by Hamiltonian function plus some other terms then you will get it the second variation of the functional this equal to t 0 to t have delta x of t transpose then delta square h of this delta x square capital S. Let us call it right capital because we have started with the capital X vector this one this star and delta x of t plus twice delta delta x of t plus delta x of t plus delta x of t delta transpose then delta square h delta x of t delta x of t then delta u of t multiplied by delta that whole thing compute at along the trajectory optimal trajectory del delta x of t del u of t then plus del u of t whole transpose del square l dot del u square of t whole this del u of t that whole bracket and that you differentiate with respect to d t. So, we know this one the delta square of this must be positive if the functional value is function will be minimum if that value will be negative delta square will be negative if functional value is this functional value is this integration value is negative if the integration value is negative then function value is maximum. So, this you can write it one can write this thing into a matrix and vector form t 0 t f del x t and this is what this matrices this is a Hessian matrix symmetric matrix again this is a symmetric matrix you will get it. So, that we can write it now del u of t transpose this whole thing is a scalar quantity that we can write it in quadratic form. So, you are writing del h square dot del x square of t del second derivative of h with respect to del x del u the order can be changed del u del x this does not matter the results are same. So, del x del u of t then del square h del u square of t that whole thing along the optimal trajectory u t extra multiplied by delta x t and delta u of t this and then differentiating with respect to this. So, delta square the second variation of second variation of functional second variation of the functional if it is a negative that second variation is a scalar quantity if it is a negative indicates that this integral part that matrix whose dimension is this matrix dimension is n plus m into n plus m matrix dimension and that is a symmetric matrix and that matrix must be a if it is greater than 0 that matrix must be a greater than 0 means positive definite. If it is a positive definite matrix then we will call that our what is called functional value is minimum and at what point along the trajectory along the trajectory u star and x star and that u star solution and x star which you got once you find out u star x star also you can find out and that is the optimal trajectory that we obtain by solving equation just now we have mentioned it by solving equation that algorithmic steps if you see the algorithm by solving the equation that our that is just a minute that means just our equation is del h del u that means that 3 equation you have to solve it del h del u is equal to 0 that means if you write del h del u is equal to 0 then del h del lambda is equal to 0 that is equal to x dot of t that you have to solve it and del h del x is equal to again this one is equal to lambda minus lambda dot star this your lambda dot star the star you can write it this 3 equation along with the boundary condition that just now we have written the boundary condition and this boundary condition 4 equation number that means equation number 1 equation number 1 del h del u equation number 2 del h del lambda a x dot and equation number 3 del h del x is equal to minus lambda x dot and equation number 4 is necessary to solve the set of non-linear equation that boundary condition is required. So, that way we can solve it this one and once you solve the optimal trajectory in order to check whether the functional is a optimum means minimum or maximum to show this one nature of the optimization that this matrix that is this matrix I am now writing this matrix the Hessian matrix del h del dot del x square of t del h dot del x of t del u of t del square h dot del x of t del u of t del square h del x del u square of t. So, this matrix you can write this matrix if it is greater than 0 means positive definite matrix agree and this matrix is a symmetric matrix this matrix is symmetric matrix. If it is greater than 0 it implies that it implies that functional value is minimum that j star is minimum functional value is minimum if it is less than equal to 0 the negative definite this matrix is negative definite along the trajectory optimal trajectory what we got by solving just now mention equation number 1 2 3 and boundary condition that one if you and put it here in this matrix if you get a negative definite this implies this implies that j star is maximum functional value is maximum. So, this is the necessary and sufficient condition you have to necessary condition is required to find the trajectory of the what is called trajectory of control u of t and x of t once you get it optimal trajectory of u star of t control in afford u star of t and x star of t optimal trajectory of the state variables. Then next question is to know the nature of this optimality whether it is a objective whether the objective function or functional value is minimum or maximum then you have to test with this matrix and that matrix dimension you see n is the number of states m is the number of inputs. So, it is n plus m into n plus m that matrix you have to check it if it is positive definite j star is minimum if it is a negative definite j star is maximum. So, this is the basic theorem of this one. So, we will take an example and see that how to solve a problem a practical problem of this one and you can see this one we can just consider the three cases as a case a similarly as we discussed earlier fixed time fixed final time and fixed final state this is case one and fixed final state this is the case one. Then case two we can see the fixed final time and you see that case b free final time and fixed final state similarly case c we can say the case c is the fixed final time fixed final time and free final state and last one and last one case d both are free case d, but free final time final time and final state and free final state. So, this whatever the we have considered the boundary condition all this thing that is one of these cases will be a special case of that general boundary condition. So, we will stop it here next class we will just take an example and show how to solve this problem by using the Hamiltonian function method to get the optimal value of the functionals to solve the control problems agree. So, we will stop it here.


This study proposes a new kinematic chain for the mobilization of the metacarpophilangial, MCP, joint of the long fingers, which can be integrated into an exoskeletal device for rehabilitative purposes. The proposed mechanism has been designed to be compatible with the human joint, allowing for natural movement while minimizing parasitic torques. Additionally, it has been shown to accurately estimate the MCP joint angle and track torque, making it suitable for use in a clinical setting. This article was authored by Emanuele Peperoni, Stefano Lazlo Capitani, Tommaso Fumalbi, and others.