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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 ⊂ N n × R m • ❚❤❛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 → ✿ s 0 → s 1 → s 2 → · · · ◆♦✇✱ ✇❡ ❝❛♥ 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 : R n �→ R n x 1 ✐✛ t❤❡r❡ ❡①✐sts ❛ f : R + �→ R n ❛♥❞ δ ≥ 0 s✉❝❤ t❤❛t ❉✐s❝r❡t❡✲t✐♠❡ s❡♠❛♥t✐❝s✿ � x 0 → � x 0 = f (0) � x 1 = f ( δ ) � df ( t ) = F ( f ( t )) dt ❆ 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❡♠ x + v x ≤ − 2     − → dx dx dt = v x dt = v x     dy dy dt = v y dt = v y         dv x dv x = − 1 − v x = 1 − v x     dt dt     dv y dv y     = 1 − v y = 1 − v y dt dt x + v x ≥ − 2 x + v x ≤ 2 ← − x + v x ≥ 2 ❙t❛rt✐♥❣ ❢r♦♠ ❛ r❡❣✐♦♥ − 1 ≤ x ≤ 1 , y = 0 , v x = v y = 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 + v x + 2 ≤ 0 • ❙✇✐t❝❤❡s ❢r♦♠ ▼♦❞❡ ✷ t♦ ▼♦❞❡ ✶ ✇❤❡♥ x + v x − 2 ≥ 0 ❚✇♦ ♣♦ss✐❜❧❡ s✐♠✉❧❛t✐♦♥ tr❛❥❡❝t♦r✐❡s✿ 12 Position x=−3 x=3 Alt Position 10 8 Position y 6 4 2 0 −3 −2 −1 0 1 2 3 Position x ✻

❍②❜r✐❞❙❆▲✿ ▼♦❞❡❧✐♥❣ 10 ❚❤❡ ❣♦❛❧ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ r♦❜♦t r❡♠❛✐♥s ✐♥s✐❞❡ ❙❛❢❡ st❛rt✐♥❣ ❢r♦♠ ■♥✐t ✿ ■♥✐t := ( x ∈ [ − 1 , 1] , y = 0 , v x = 0 , v y = 0) ❙❛❢❡ := ( |x| ≤ 3) −3 0 3 ❚❤❡ r♦❜♦t ❝❛♥ ♠♦✈❡ ✐♥ ✷ ♠♦❞❡s✿ • ▼♦❞❡ ✶✿ ❋♦r❝❡ ❛♣♣❧✐❡❞ ✐♥ (1 , 1) ✲❞✐r❡❝t✐♦♥ dv y dx dv x dy dt = v x , dt = 1 − v x , dt = v y , dt = 1 − v y • ▼♦❞❡ ✷✿ ❋♦r❝❡ ❛♣♣❧✐❡❞ ✐♥ ( − 1 , 1) ✲❞✐r❡❝t✐♦♥ dx dv x dy dv y dt = v x , dt = − 1 − v x , dt = v y , dt = 1 − v y ✼

❊①❛♠♣❧❡✿ ❉r✐✈✐♥❣ ❛ ❘♦❜♦t ❈♦♥s✐❞❡r ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♥tr♦❧❧❡r✿ • ❙✇✐t❝❤ ❢r♦♠ ▼♦❞❡ ✶ t♦ ▼♦❞❡ ✷ ✇❤❡♥ x + v x + 2 ≤ 0 • ❙✇✐t❝❤ ❢r♦♠ ▼♦❞❡ ✷ t♦ ▼♦❞❡ ✶ ✇❤❡♥ x + v x − 2 ≥ 0 ❚✇♦ ♣♦ss✐❜❧❡ s✐♠✉❧❛t✐♦♥ tr❛❥❡❝t♦r✐❡s✿ 12 Position x=−3 x=3 Alt Position 10 8 Position y 6 4 2 0 −3 −2 −1 0 1 2 3 Position x ✽

❍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❛❧✑ ✶✷