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

<div>Synchronization path states <svg height="12.5794pt" id="M319" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 29.5125 12.5794" width="29.5125pt" 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="M805 658H719L589 415C537 319 349 15 237 15C214 15 208 28 208 43C208 59 219 86 247 130L433 417C464 464 471 506 471 541C471 638 384 687 292 687C164 687 36 597 36 474C36 406 83 360 151 360C238 360 296 442 309 509H289C265 440 225 390 150 390C103 390 77 426 77 474C77 582 175 652 283 652C340 652 404 619 404 568C404 541 397 512 368 467L223 240C170 155 153 111 153 73C153 18 186 -15 234 -15C290 -15 367 41 439 138L442 137C427 106 425 83 425 66C425 18 463 -12 511 -12C575 -12 638 42 714 134L693 147C621 63 565 18 525 18C506 18 492 28 492 53C492 60 493 77 509 108L805 658Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.946,3.132)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.39,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.824,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> vs. <svg height="12.5794pt" id="M320" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 29.5125 12.5794" width="29.5125pt" 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="M805 658H719L589 415C537 319 349 15 237 15C214 15 208 28 208 43C208 59 219 86 247 130L433 417C464 464 471 506 471 541C471 638 384 687 292 687C164 687 36 597 36 474C36 406 83 360 151 360C238 360 296 442 309 509H289C265 440 225 390 150 390C103 390 77 426 77 474C77 582 175 652 283 652C340 652 404 619 404 568C404 541 397 512 368 467L223 240C170 155 153 111 153 73C153 18 186 -15 234 -15C290 -15 367 41 439 138L442 137C427 106 425 83 425 66C425 18 463 -12 511 -12C575 -12 638 42 714 134L693 147C621 63 565 18 525 18C506 18 492 28 492 53C492 60 493 77 509 108L805 658Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.946,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.39,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.824,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> and its error <svg height="12.5794pt" id="M321" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 28.2206 12.5794" width="28.2206pt" 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="M397 143C384 252 309 299 234 299C151 299 67 242 67 142C67 29 167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 15 129 67 129 137C129 212 192 256 241 256C296 256 345 244 377 139L397 143Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.659,3.132)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.605,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.103,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.537,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> with control inputs.</div>

Mathematical Problems in Engineering

fig14

Figure 14

Figure 14: A Lyapunov–Krasovskii Functional Approach to Stability and Linear Feedback Synchronization Control for Nonlinear Multi-Agent Systems with Mixed Time Delays 

Figure 14 | A Lyapunov–Krasovskii Functional Approach to Stability and Linear Feedback Synchronization Control for Nonlinear Multi-Agent Systems with Mixed Time Delays 