A Lyapunov–Krasovskii Functional Approach to Stability and Linear Feedback Synchronization Control for Nonlinear Multi-Agent Systems with Mixed Time Delays

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td class="align_left">Theorem <a href="https://static.hindawi.com/articles/mpe/volume-2021/6616857/figures/#thm2" target="_blank">2</a></td><td class="align_center"><span style="width: 15.2763ptpx;"><svg height="9.14241pt" id="M259" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -5.96091 15.2763 9.14241" width="15.2763pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.192,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg></span></td><td class="align_center"><span style="width: 15.2763ptpx;"><svg height="9.14241pt" id="M260" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -5.96091 15.2763 9.14241" width="15.2763pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.192,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg></span></td><td class="align_center"><span style="width: 7.30254ptpx;"><svg height="9.49473pt" id="M261" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g></svg></span></td><td class="align_center"><span style="width: 6.32387ptpx;"><svg height="9.39034pt" id="M262" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 6.32387 9.39034" width="6.32387pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M449 290C449 369 411 446 324 446C271 446 212 414 173 378C76 289 55 151 55 19C55 -41 73 -100 116 -142C162 -187 228 -202 291 -217C320 -224 348 -238 371 -257C376 -227 375 -219 362 -192C349 -168 330 -145 280 -137C228 -128 179 -110 134 -82C100 -61 96 -17 96 24C121 1 154 -12 188 -12C213 -12 256 1 298 24C374 64 449 162 449 290ZM367 278C366 125 292 56 262 40C244 31 227 27 212 27C172 27 134 50 135 87C142 261 201 360 246 390C260 400 278 403 292 403C340 403 367 348 367 278Z"></path></g></svg></span></td><td class="align_center">Solution</td></tr><tr><td class="align_left">1</td><td class="align_center">0.2</td><td class="align_center">1.8</td><td class="align_center">1.8</td><td class="align_center">1.4</td><td class="align_center">Feasible</td></tr><tr><td class="align_left">2</td><td class="align_center">0.2</td><td class="align_center">2.0</td><td class="align_center">1.8</td><td class="align_center">1.4</td><td class="align_center">Feasible</td></tr><tr><td class="align_left">3</td><td class="align_center">0.2</td><td class="align_center">2.5</td><td class="align_center">1.8</td><td class="align_center">1.4</td><td class="align_center">Infeasible</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Comparison table of lower and upper bounds of Theorem <a href="https://static.hindawi.com/articles/mpe/volume-2021/6616857/figures/#thm1" target="_blank">1</a>.</div>

Mathematical Problems in Engineering

tab1

Table 1

Table 1: A Lyapunov–Krasovskii Functional Approach to Stability and Linear Feedback Synchronization Control for Nonlinear Multi-Agent Systems with Mixed Time Delays 

Table 1 | A Lyapunov–Krasovskii Functional Approach to Stability and Linear Feedback Synchronization Control for Nonlinear Multi-Agent Systems with Mixed Time Delays 