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Computer Science Laboratory, SRI International

❍②❜r✐❞ ❙②st❡♠s

❆s❤✐s❤ ❚✐✇❛r✐ ❙❘■ ■♥t❡r♥❛t✐♦♥❛❧

❍②❜r✐❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❆ ❤②❜r✐❞ ❞②♥❛♠✐❝❛❧ s②st❡♠ ❝♦♥s✐sts ♦❢

• ❤②❜r✐❞✲s♣❛❝❡✿ X ⊂ Nn × Rm
• ❚❤❛t ✐s✱ s♦♠❡ ✈❛r✐❛❜❧❡s t❛❦❡ ✈❛❧✉❡s ✐♥ ❛ ❞✐s❝r❡t❡ ❞♦♠❛✐♥ N
• ❖t❤❡r ✈❛r✐❛❜❧❡s t❛❦❡ ✈❛❧✉❡s ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ❞♦♠❛✐♥ R

❚❤❡ tr❛❥❡❝t♦r✐❡s ❛r❡ ❞❡✜♥❡❞ ♦✈❡r

• ❤②❜r✐❞✲t✐♠❡✿ T = R × N
• ❚❤❛t ✐s✱ ❛t s♦♠❡ t✐♠❡ ✐♥st❛♥ts t ∈ R✱ t❤❡ s②st❡♠ ♠❛❦❡s n ∈ N ❥✉♠♣s

❯s❡❢✉❧ ❢♦r ♠♦❞❡❧✐♥❣ s②st❡♠s ❤❛✈✐♥❣ ❝♦♠♣❧❡①✱ ♥♦♥❧✐♥❡❛r✱ ♠✉❧t✐♠♦❞❛❧ ❜❡❤❛✈✐♦r ❖r s②st❡♠s ✐♥✈♦❧✈✐♥❣ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ♣❤②s✐❝❛❧ s②st❡♠ ❛♥❞ s♦❢t✇❛r❡

✶

❙♣❡❝✐❢②✐♥❣ t❤❡ ❉②♥❛♠✐❝s

❉②♥❛♠✐❝s ❛r❡ t②♣✐❝❛❧❧② s♣❡❝✐✜❡❞ ✉s✐♥❣ ❧♦❝❛❧ r✉❧❡s ❆ ❞②♥❛♠✐❝❛❧ s②st❡♠ ❝❛♥ ❜❡ s♣❡❝✐✜❡❞ ❛s ❛ t✉♣❧❡ (X, →) ✇❤❡r❡ X ✿ ✈❛r✐❛❜❧❡s ❞❡✜♥✐♥❣ st❛t❡ s♣❛❝❡ ♦❢ t❤❡ s②st❡♠ → ✿ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦✈❡r st❛t❡ s♣❛❝❡ ❞❡✜♥✐♥❣ s②st❡♠ ❞②♥❛♠✐❝s ❆ ✏r✉♥✑ ♦❢ s✉❝❤ ❛ s②st❡♠ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ st❛t❡s r❡❧❛t❡❞ ❜② →✿ s0 → s1 → s2 → · · · ◆♦✇✱ ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❇✉t ✇❤❛t ❛❜♦✉t ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠s❄

✷

❈♦♥t✐♥✉♦✉s ❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❲❡ ❣✐✈❡ s❡♠❛♥t✐❝s t♦ ❝♦♥t✐♥✉♦✉s✲s♣❛❝❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠s ❜② ♠❛♣♣✐♥❣ t❤❡♠ t♦ ❝♦♥t✐♥✉♦✉s✲s♣❛❝❡✱ ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s ❈♦♥t✐♥✉♦✉s ❞②♥❛♠✐❝s ❛r❡ s♣❡❝✐✜❡❞ ✉s✐♥❣ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s d

x dt = F(

x)✱ ✇❤❡r❡ F : Rn → Rn ❉✐s❝r❡t❡✲t✐♠❡ s❡♠❛♥t✐❝s✿ x0 → x1 ✐✛ t❤❡r❡ ❡①✐sts ❛ f : R+ → Rn ❛♥❞ δ ≥ 0 s✉❝❤ t❤❛t

• x0 = f(0)
• x1 = f(δ)

df(t) dt

= F(f(t)) ❆ st❛t❡ ❝❛♥ ❤❛✈❡ ✉♥❝♦✉♥t❛❜❧② ♠❛♥② s✉❝❝❡ss♦rs ◆♦✇ ✇❡ ❝❛♥ ♠❛❦❡ s❡♥s❡ ♦❢ t❡♠♣♦r❛❧ ❧♦❣✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠s

✸

❍②❜r✐❞ ❙②st❡♠s

❋♦r ❤②❜r✐❞ s②st❡♠s

• X ✐♥❝❧✉❞❡s ❇♦♦❧❡❛♥✲ ❛♥❞ ❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s❀ ❤❡♥❝❡✱ ❛ ❤②❜r✐❞ st❛t❡ s♣❛❝❡
• ❡①❡❝✉t✐♦♥s ❛r❡ ✐♥ ❤②❜r✐❞✲t✐♠❡✱ ❤❡♥❝❡ ✐ts s❡♠❛♥t✐❝s → r❡❧❛t❡s ❛ st❛t❡ t♦ ❛❧❧ ✐ts ❤②❜r✐❞✲t✐♠❡

s✉❝❝❡ss♦rs → := →disc ∪ →cont

✹

❊①❛♠♣❧❡ ♦❢ ❛ ❍②❜r✐❞ ❙②st❡♠

      

dx dt = vx dy dt = vy dvx dt

= −1 − vx

dvy dt

= 1 − vy x + vx ≥ −2        x + vx ≤ −2 − → ← − x + vx ≥ 2       

dx dt = vx dy dt = vy dvx dt

= 1 − vx

dvy dt

= 1 − vy x + vx ≤ 2        ❙t❛rt✐♥❣ ❢r♦♠ ❛ r❡❣✐♦♥ −1 ≤ x ≤ 1, y = 0, vx = vy = 0✱ ❤♦✇ t♦ ♣r♦✈❡ G(−3 ≤ x ≤ 3) ❢♦r t❤✐s s②st❡♠❄

✺

❊①❛♠♣❧❡✿ ❙✐♠✉❧❛t✐♦♥s ♦❢ t❤❡ ❘♦❜♦t

❚❤❡ ❝♦♥tr♦❧❧❡r ✐s ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✿

• ❙✇✐t❝❤❡s ❢r♦♠ ▼♦❞❡ ✶ t♦ ▼♦❞❡ ✷ ✇❤❡♥ x + vx + 2 ≤ 0
• ❙✇✐t❝❤❡s ❢r♦♠ ▼♦❞❡ ✷ t♦ ▼♦❞❡ ✶ ✇❤❡♥ x + vx − 2 ≥ 0

❚✇♦ ♣♦ss✐❜❧❡ s✐♠✉❧❛t✐♦♥ tr❛❥❡❝t♦r✐❡s✿

−3 −2 −1 1 2 3 2 4 6 8 10 12 Position x Position y Position x=−3 x=3 Alt Position

✻

❍②❜r✐❞❙❆▲✿ ▼♦❞❡❧✐♥❣

3 −3 10

❚❤❡ ❣♦❛❧ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ r♦❜♦t r❡♠❛✐♥s ✐♥s✐❞❡ ❙❛❢❡ st❛rt✐♥❣ ❢r♦♠ ■♥✐t✿ ■♥✐t := (x ∈ [−1, 1], y = 0, vx = 0, vy = 0) ❙❛❢❡ := (|x| ≤ 3) ❚❤❡ r♦❜♦t ❝❛♥ ♠♦✈❡ ✐♥ ✷ ♠♦❞❡s✿

• ▼♦❞❡ ✶✿ ❋♦r❝❡ ❛♣♣❧✐❡❞ ✐♥ (1, 1)✲❞✐r❡❝t✐♦♥

dx dt = vx, dvx dt = 1 − vx, dy dt = vy, dvy dt = 1 − vy

• ▼♦❞❡ ✷✿ ❋♦r❝❡ ❛♣♣❧✐❡❞ ✐♥ (−1, 1)✲❞✐r❡❝t✐♦♥

dx dt = vx, dvx dt = −1 − vx, dy dt = vy, dvy dt = 1 − vy

✼

❊①❛♠♣❧❡✿ ❉r✐✈✐♥❣ ❛ ❘♦❜♦t

❈♦♥s✐❞❡r ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♥tr♦❧❧❡r✿

• ❙✇✐t❝❤ ❢r♦♠ ▼♦❞❡ ✶ t♦ ▼♦❞❡ ✷ ✇❤❡♥ x + vx + 2 ≤ 0
• ❙✇✐t❝❤ ❢r♦♠ ▼♦❞❡ ✷ t♦ ▼♦❞❡ ✶ ✇❤❡♥ x + vx − 2 ≥ 0

❚✇♦ ♣♦ss✐❜❧❡ s✐♠✉❧❛t✐♦♥ tr❛❥❡❝t♦r✐❡s✿

−3 −2 −1 1 2 3 2 4 6 8 10 12 Position x Position y Position x=−3 x=3 Alt Position

✽

❍②❜r✐❞❙❆▲ ▼♦❞❡❧ ♦❢ ❘♦❜♦t

r♦❜♦t✿❈❖◆❚❊❳❚ ❂ ❇❊●■◆ s②st❡♠✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ▲❖❈❆▲ ❞✐r❡❝t✐♦♥ ✿ ❇❖❖▲❊❆◆ ✪ ♠♦✈✐♥❣ ❧❡❢t✴r✐❣❤t ▲❖❈❆▲ ①✱ ✈①✱ ②✱ ✈② ✿ ❘❊❆▲ ▲❖❈❆▲ ①❞♦t✱ ✈①❞♦t✱ ②❞♦t✱ ✈②❞♦t ✿ ❘❊❆▲ ■◆❱❆❘■❆◆❚ ❚❘❯❊ ■◆■❚❋❖❘▼❯▲❆ ✲✶ ❁❂ ① ❆◆❉ ① ❁❂ ✶ ❆◆❉ ✈① ❂ ✵ ❆◆❉ ✈② ❂ ✵ ❆◆❉ ② ❂ ✵ ✳✳✳

✾

❍②❜r✐❞❙❆▲ ▼♦❞❡❧ ♦❢ ❘♦❜♦t

❚❘❆◆❙■❚■❖◆ ❬ ❞✐r❡❝t✐♦♥ ❂ ❚❘❯❊ ❆◆❉ ① ✰ ✈① ❃❂ ✲✷ ✲✲❃ ①❞♦t✬ ❂ ✈①❀ ✈①❞♦t✬ ❂ ✲✶ ✲ ✈①❀ ②❞♦t✬ ❂ ✈②❀ ✈②❞♦t✬ ❂ ✶ ✲ ✈② ❬❪ ❞✐r❡❝t✐♦♥ ❂ ❋❆▲❙❊ ❆◆❉ ① ✰ ✈① ❁❂ ✷ ✲✲❃ ①❞♦t✬ ❂ ✈①❀ ✈①❞♦t✬ ❂ ✶ ✲ ✈①❀ ②❞♦t✬ ❂ ✈②❀ ✈②❞♦t✬ ❂ ✶ ✲ ✈② ❬❪ ❞✐r❡❝t✐♦♥ ❂ ❚❘❯❊ ❆◆❉ ① ✰ ✈① ❁❂ ✲✷ ✲✲❃ ❞✐r❡❝t✐♦♥✬ ❂ ❋❆▲❙❊ ❬❪ ❞✐r❡❝t✐♦♥ ❂ ❋❆▲❙❊ ❆◆❉ ① ✰ ✈① ❃❂ ✷ ✲✲❃ ❞✐r❡❝t✐♦♥✬ ❂ ❚❘❯❊ ❪ ❊◆❉❀ ✳✳✳

✶✵

❍②❜r✐❞❙❆▲ ▼♦❞❡❧ ♦❢ ❘♦❜♦t

r♦❜♦t✿ ❈❖◆❚❊❳❚ ❇❊●■◆ s②st❡♠✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ▲❖❈❆▲ ✳✳✳ ■◆❱❆❘■❆◆❚ ✳✳✳ ■◆■❚❋❖❘▼❯▲❆ ✳✳✳ ❚❘❆◆❙■❚■❖◆ ❬ ✳✳✳ ❬❪ ✳✳✳ ❬❪ ✳✳✳ ❪ ❊◆❉❀ ❝♦rr❡❝t✿ ❚❍❊❖❘❊▼ s②st❡♠ ⑤✲ ●✭ ✵ ❁❂ ①✰✸ ❆◆❉ ① ❁❂ ✸ ✮❀ ❊◆❉

✶✶

❍②❜r✐❞❙❆▲ ❆♥❛❧②s✐s

❍②❜r✐❞❙❆▲ ♣r♦✈✐❞❡s ❛♥ ❛❜str❛❝t♦r t❤❛t t❛❦❡s ❛ ❍②❜r✐❞❙❆▲ ♠♦❞❡❧ ❛♥❞ ♦✉t♣✉ts ❛ ✜♥✐t❡ st❛t❡ ❙❆▲ ♠♦❞❡❧ ❍②❜r✐❞❙❆▲ ✐s ✇r✐tt❡♥ ✐♥ ▲✐s♣ ❤❛s ❛ ❝♦♠♠❛♥❞✲❧✐♥❡ ✐♥t❡r❢❛❝❡✿ ♠❧✐s♣ ✭❧♦❛❞ ✧❧♦❛❞✳❧✐s♣✧✮ ✭✐♥✲♣❛❝❦❛❣❡ ✿s❛❧✮ ✭❛❜str❛❝t ✧r♦❜♦t✧ ✬s②st❡♠ ✿♣r♦♣❡rt② ✬❝♦rr❡❝t✮ ❚❤✐s ❝r❡❛t❡s ❛ ✜❧❡ ✏r♦❜♦t❆❇❙✳s❛❧✑

✶✷

❆❜str❛❝t ▼♦❞❡❧ ✐♥ ❙❆▲

✪✪ ❆❜str❛❝t ✈❛r✐❛❜❧❡ t♦ P♦❧②♥♦♠✐❛❧ ▼❛♣♣✐♥❣✿ ✪✪ ❣✶✶ ✲✲❃ ✲✶✯① ✲ ✸ ✪✪ ❣✶✵ ✲✲❃ ① ✲ ✸ ✪✪ ❣✾ ✲✲❃ ✲✶✯① ✲ ✶ ✪✪ ❣✽ ✲✲❃ ① ✲ ✶ ✪✪ ❣✼ ✲✲❃ ✈① ✪✪ ❣✻ ✲✲❃ ✈② ✪✪ ❣✺ ✲✲❃ ② ✪✪ ❣✹ ✲✲❃ ① ✰ ✈① ✰ ✷ ✪✪ ❣✸ ✲✲❃ ① ✰ ✈① ✲ ✷ ✪✪ ❣✷ ✲✲❃ ✲✶✯✈② ✰ ✶ ✪✪ ❣✶ ✲✲❃ ✲✶✯✈① ✲ ✶ ✪✪ ❣✵ ✲✲❃ ✲✶✯✈① ✰ ✶ ✳✳✳

✶✸

❆❜str❛❝t ▼♦❞❡❧ ✐♥ ❙❆▲

r♦❜♦t❆❇❙✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ❙■●◆✿ ❚❨P❊ ❂ ④♣♦s✱ ♥❡❣✱ ③❡r♦⑥❀ ❆❙❙❱P✭①✵✿ ❙■●◆✱ ①✶✿ ❙■●◆✮✿ ❬❙■●◆ ✲❃ ❇❖❖▲❊❆◆❪ ❂ ✳✳✳ ❆❙❙❱◆✭①✵✿ ❙■●◆✱ ①✶✿ ❙■●◆✮✿ ❬❙■●◆ ✲❃ ❇❖❖▲❊❆◆❪ ❂ ✳✳✳ ■◆❱✶✷✭❣✶✶✿ ❙■●◆✱ ✳✳✳✱ ❣✵✿ ❙■●◆✮✿ ❇❖❖▲❊❆◆ ❂ ✳✳✳ s②st❡♠✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆

• ▲❖❇❆▲ ❣✵✱ ✳✳✳✱ ❣✶✶✿ ❙■●◆

▲❖❈❆▲ ❞✐r❡❝t✐♦♥✿ ❇❖❖▲❊❆◆ ■◆■❚■❆▲■❩❆❚■❖◆ ❣✶✶ ❂ ♥❡❣❀ ✳✳✳ ❀ ❣✵ ❂ ♣♦s ✳✳✳

✶✹

❆❜str❛❝t ▼♦❞❡❧ ✐♥ ❙❆▲

❚❘❆◆❙■❚■❖◆ ❬✭❞✐r❡❝t✐♦♥ ❂ ❚❘❯❊ ❆◆❉ ✭❣✹ ❂ ♣♦s ❖❘ ❣✹ ❂ ③❡r♦✮✮ ❆◆❉ ■◆❱✶✷✭❣✶✶✬✱ ✳✳✳✱ ❣✵✬✮ ❆◆❉ ✭❣✹✬ ❂ ♣♦s ❖❘ ❣✹✬ ❂ ③❡r♦✮ ✲✲❃ ❣✶✶✬ ■◆ ❆❙❙❱◆✭❣✶✶✱ ❣✼✮❀ ✳✳✳❀ ❣✵✬ ■◆ ❆❙❙❱◆✭❣✵✱ ❣✶✮ ❬❪ ✭❞✐r❡❝t✐♦♥ ❂ ❋❆▲❙❊ ❆◆❉ ✭❣✸ ❂ ♥❡❣ ❖❘ ❣✸ ❂ ③❡r♦✮✮ ❆◆❉ ■◆❱✶✷✭❣✶✶✬✱ ✳✳✳✱ ❣✵✬✮ ❆◆❉ ✭❣✸✬ ❂ ♥❡❣ ❖❘ ❣✸✬ ❂ ③❡r♦✮ ✲✲❃ ❣✶✶✬ ■◆ ❆❙❙❱◆✭❣✶✶✱ ❣✼✮❀ ✳✳✳❀ ❣✵✬ ■◆ ❆❙❙❱◆✭❣✵✱ ❣✵✮ ❬❪ ✭❞✐r❡❝t✐♦♥ ❂ ❚❘❯❊ ❆◆❉ ✭❣✹ ❂ ♥❡❣ ❖❘ ❣✹ ❂ ③❡r♦✮✮ ❆◆❉ ■◆❱✶✷✭❣✶✶✬✱ ✳✳✳✱ ❣✵✬✮ ✲✲❃ ❞✐r❡❝t✐♦♥✬ ❂ ❋❆▲❙❊ ❬❪ ✭❞✐r❡❝t✐♦♥ ❂ ❋❆▲❙❊ ❆◆❉ ✭❣✸ ❂ ♣♦s ❖❘ ❣✸ ❂ ③❡r♦✮✮ ❆◆❉ ■◆❱✶✷✭❣✶✶✬✱ ✳✳✳✱ ❣✵✬✮ ✲✲❃ ❞✐r❡❝t✐♦♥✬ ❂ ❚❘❯❊ ❪

✶✺

❆❜str❛❝t ▼♦❞❡❧ ✐♥ ❙❆▲

r♦❜♦t❆❇❙✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ✳✳✳ s②st❡♠✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❝♦rr❡❝t✿ ❚❍❊❖❘❊▼ s②st❡♠ ⑤✲ ●✭✭❣✶✶ ❂ ♥❡❣ ❖❘ ❣✶✶ ❂ ③❡r♦✮ ❆◆❉ ✭❣✶✵ ❂ ♥❡❣ ❖❘ ❣✶✵ ❂ ③❡r♦✮✮❀ ❊◆❉

✶✻

▼♦❞❡❧ ❈❤❡❝❦ t❤❡ ❆❜str❛❝t ▼♦❞❡❧

■❢ ❙❆▲ ✐s ✐♥st❛❧❧❡❞✱ t❤❡♥ ✇❡ ❝❛♥ ❛♥❛❧②③❡ t❤❡ ❛❜str❛❝t ❙❆▲ ♠♦❞❡❧ s❛❧✲❞❡❛❞❧♦❝❦✲❝❤❡❝❦❡r r♦❜♦t❆❇❙ s②st❡♠ s❛❧✲s♠❝ ✲✈ ✸ r♦❜♦t❆❇❙ ❝♦rr❡❝t ❲❡ ❝❛♥ t❤✉s ✈❡r✐❢② t❤❡ s❛❢❡t② ♣r♦♣❡rt② ♦❢ t❤❡ ❤②❜r✐❞ r♦❜♦t ♠♦❞❡❧✳ ■❢ ♣r♦♣❡rt② ✐s ♥♦t tr✉❡ ♦❢ ❛❜str❛❝t ♠♦❞❡❧✱ t❤❡♥ ✇❡ ❣❡t ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡ ✐♥ t❤❡ ❛❜str❛❝t ❲❤✐❝❤ ♠❛② ❜❡ s♣✉r✐♦✉s

✶✼

❍②❜r✐❞❙❆▲✿ ❉✐s❝✉ss✐♦♥

• Pr❡❞✐❝❛t❡s ❢♦r ❛❜str❛❝t✐♦♥ ❛r❡ ❝❤♦s❡♥ ❛✉t♦♠❛t✐❝❛❧❧②
• ❚❤✐s ❝❤♦✐❝❡ ✐s ❝r✉❝✐❛❧✱ ❛♥❞ ❝❛♥ ❜❡ ✐♥✢✉❡♥❝❡❞ ❜② ❝♦♠♠❛♥❞✲❧✐♥❡ ✐♥♣✉t
• ❚❤❡ ❛❜str❛❝t✐♦♥ ♣r♦❝❡ss ✐s ❝♦♠♣❧❡t❡❧② ❛✉t♦♠❛t✐❝✱ ❜✉t ✐t ❝❛♥ t❛❦❡ ❧♦♥❣
• ❚♦♦❧ ✐s st✐❧❧ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✿ s❡✈❡r❛❧ ❢❡❛t✉r❡s ♦❢ ❙❆▲ ❛r❡ ♥♦t s✉♣♣♦rt❡❞ ✐♥ ❍②❜r✐❞❙❆▲
• ❙✉❝❤ ❛s ❝♦♠♣♦s✐t✐♦♥❛❧ ❛❜str❛❝t✐♦♥
• ❤tt♣✿✴✴s❛❧✳❝s❧✳sr✐✳❝♦♠✴❤②❜r✐❞s❛❧✴

✶✽