<p>Gegeben ist das Zustandssvstem \( \dot{x}=A x \) mit</p>
<p style="padding-left: 40px;">\[<br />A=\left(\begin{array}{cc}-1 &amp; -4 \\ 1 &amp; -2\end{array}\right) <br />\]</p>
<p>Lösen Sie die Lyapunov-Gleichung</p>
<p style="padding-left: 40px;">\[<br />A^{T} P+P A=-Q \quad, \quad P \in \mathbb{R}^{2 \times 2}<br />\]</p>
<p>für \( Q=I \).</p>

<p>Gegeben ist das Zustandssvstem <mjx-container class="MathJax" jax="SVG"><svg alt=" \dot{x}=A x " style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="7.302ex" height="1.921ex" role="img" focusable="false" viewBox="0 -767 3227.6 849" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1585-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-1585-TEX-N-2D9" d="M190 609Q190 637 208 653T252 669Q275 667 292 652T309 609Q309 579 292 564T250 549Q225 549 208 564T190 609Z"></path><path id="MJX-1585-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1585-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-1585-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,-2) translate(-250 0)"><use data-c="2D9" xlink:href="#MJX-1585-TEX-N-2D9"></use></g></g></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3D" xlink:href="#MJX-1585-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1905.6,0)"><use data-c="1D434" xlink:href="#MJX-1585-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(2655.6,0)"><use data-c="1D465" xlink:href="#MJX-1585-TEX-I-1D465"></use></g></g></g></svg></mjx-container> mit</p>
<p style="padding-left: 40px;"><mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
A=\left(\begin{array}{cc}-1 & -4 \\ 1 & -2\end{array}\right) 
" style="vertical-align: -2.149ex" xmlns="http://www.w3.org/2000/svg" width="16.089ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 7111.6 2400" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1586-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-1586-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1586-TEX-S3-28" d="M701 -940Q701 -943 695 -949H664Q662 -947 636 -922T591 -879T537 -818T475 -737T412 -636T350 -511T295 -362T250 -186T221 17T209 251Q209 962 573 1361Q596 1386 616 1405T649 1437T664 1450H695Q701 1444 701 1441Q701 1436 681 1415T629 1356T557 1261T476 1118T400 927T340 675T308 359Q306 321 306 250Q306 -139 400 -430T690 -924Q701 -936 701 -940Z"></path><path id="MJX-1586-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-1586-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1586-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path id="MJX-1586-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1586-TEX-S3-29" d="M34 1438Q34 1446 37 1448T50 1450H56H71Q73 1448 99 1423T144 1380T198 1319T260 1238T323 1137T385 1013T440 864T485 688T514 485T526 251Q526 134 519 53Q472 -519 162 -860Q139 -885 119 -904T86 -936T71 -949H56Q43 -949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-1586-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(1027.8,0)"><use data-c="3D" xlink:href="#MJX-1586-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2083.6,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJX-1586-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736,0)"><g data-mml-node="mtr" transform="translate(0,700)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-1586-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-1586-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-1586-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="34" xlink:href="#MJX-1586-TEX-N-34"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-700)"><g data-mml-node="mtd" transform="translate(389,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-1586-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-1586-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="32" xlink:href="#MJX-1586-TEX-N-32"></use></g></g></g></g><g data-mml-node="mo" transform="translate(4292,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-1586-TEX-S3-29"></use></g></g></g></g></svg></mjx-container></p>
<p>Lösen Sie die Lyapunov-Gleichung</p>
<p style="padding-left: 40px;"><mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
A^{T} P+P A=-Q \quad, \quad P \in \mathbb{R}^{2 \times 2}
" style="vertical-align: -0.439ex" xmlns="http://www.w3.org/2000/svg" width="32.1ex" height="2.456ex" role="img" focusable="false" viewBox="0 -891.7 14188.3 1085.7" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1587-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-1587-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-1587-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-1587-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-1587-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1587-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-1587-TEX-I-1D444" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path id="MJX-1587-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-1587-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-1587-TEX-D-211D" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path id="MJX-1587-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1587-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-1587-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(783,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-1587-TEX-I-1D447"></use></g></g></g><g data-mml-node="mi" transform="translate(1330.8,0)"><use data-c="1D443" xlink:href="#MJX-1587-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(2304,0)"><use data-c="2B" xlink:href="#MJX-1587-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(3304.2,0)"><use data-c="1D443" xlink:href="#MJX-1587-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(4055.2,0)"><use data-c="1D434" xlink:href="#MJX-1587-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(5083,0)"><use data-c="3D" xlink:href="#MJX-1587-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(6138.8,0)"><use data-c="2212" xlink:href="#MJX-1587-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(6916.8,0)"><use data-c="1D444" xlink:href="#MJX-1587-TEX-I-1D444"></use></g><g data-mml-node="mstyle" transform="translate(7707.8,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(8707.8,0)"><use data-c="2C" xlink:href="#MJX-1587-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(8985.8,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(10152.5,0)"><use data-c="1D443" xlink:href="#MJX-1587-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(11181.2,0)"><use data-c="2208" xlink:href="#MJX-1587-TEX-N-2208"></use></g><g data-mml-node="msup" transform="translate(12126,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="211D" xlink:href="#MJX-1587-TEX-D-211D"></use></g></g><g data-mml-node="TeXAtom" transform="translate(755,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-1587-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(500,0)"><use data-c="D7" xlink:href="#MJX-1587-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1278,0)"><use data-c="32" xlink:href="#MJX-1587-TEX-N-32"></use></g></g></g></g></g></svg></mjx-container></p>
<p>für <mjx-container class="MathJax" jax="SVG"><svg alt=" Q=I " style="vertical-align: -0.439ex" xmlns="http://www.w3.org/2000/svg" width="5.947ex" height="2.032ex" role="img" focusable="false" viewBox="0 -704 2628.6 898" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1588-TEX-I-1D444" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path id="MJX-1588-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1588-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D444" xlink:href="#MJX-1588-TEX-I-1D444"></use></g><g data-mml-node="mo" transform="translate(1068.8,0)"><use data-c="3D" xlink:href="#MJX-1588-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2124.6,0)"><use data-c="1D43C" xlink:href="#MJX-1588-TEX-I-1D43C"></use></g></g></g></svg></mjx-container>.</p>

<p>Gegeben ist folgende Matrix</p>
<p>\[<br />A=\left(\begin{array}{ccc}-1 &amp; 0 &amp; -2 \\ 1 &amp; -2 &amp; 0 \\ 0 &amp; 1 &amp; 0\end{array}\right) <br />\]</p>
<p>Zeigen Sie mit Hilfe der Lyapunov-Gleichung, dass das Zustandssystem \( \dot{x}=A x \) asymptotisch stabil ist.</p>

<p>Gegeben ist folgende Matrix</p>
<p><mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
A=\left(\begin{array}{ccc}-1 & 0 & -2 \\ 1 & -2 & 0 \\ 0 & 1 & 0\end{array}\right) 
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<p>Zeigen Sie mit Hilfe der Lyapunov-Gleichung, dass das Zustandssystem <mjx-container class="MathJax" jax="SVG"><svg alt=" \dot{x}=A x " style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="7.302ex" height="1.921ex" role="img" focusable="false" viewBox="0 -767 3227.6 849" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1867-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-1867-TEX-N-2D9" d="M190 609Q190 637 208 653T252 669Q275 667 292 652T309 609Q309 579 292 564T250 549Q225 549 208 564T190 609Z"></path><path id="MJX-1867-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1867-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-1867-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,-2) translate(-250 0)"><use data-c="2D9" xlink:href="#MJX-1867-TEX-N-2D9"></use></g></g></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3D" xlink:href="#MJX-1867-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1905.6,0)"><use data-c="1D434" xlink:href="#MJX-1867-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(2655.6,0)"><use data-c="1D465" xlink:href="#MJX-1867-TEX-I-1D465"></use></g></g></g></svg></mjx-container> asymptotisch stabil ist.</p>

Lyapunov-Stabilität