Some mathematical aspects of RNA velocity Loc D EMEULENAERE Universit - - PowerPoint PPT Presentation

Some mathematical aspects of RNA velocity

Loïc DEMEULENAERE

Université de Liège – GIGA-Genomics

Liège, January 9, 2019

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

P✉r♣♦s❡✿ s✉♠♠❛r② ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❛s♣❡❝ts ♦❢ t❤❡ ♣❛♣❡r ✏❘◆❆ ✈❡❧♦❝✐t② ♦❢ s✐♥❣❧❡ ❝❡❧❧s✑ ✭❬✶❪✮

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

✵ ✵ ✵ ✵

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

t0+t t

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

t0+t t

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

t0+t t

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

t0+t t

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡ ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

▲❡t f ❜❡ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t✵ ❛ ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t)

• t0

df dt (t✵) := lim

t→✵

f (t✵ + t) − f (t✵) t

✭✐❢ t❤✐s ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ✜♥✐t❡✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼

✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ ✮ ✐s

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼

✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ ✮ ✐s

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼

✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ ✮ ✐s

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼ f (t) = x(t)✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀

t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ ✮ ✐s

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼ f (t) = x(t)✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡

♣❛rt✐❝❧❡ ✭❛t t✐♠❡ t✮ ✐s v(t) := dx dt (t)

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

df dt (t✵) := lim

t→✵

❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡

• f (t✵ + t) − f (t✵)

t

• ■♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ✈❛r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡

❊①❛♠♣❧❡s ✭P❤②s✐❝s✮

❼ f (t) = x(t)✿ ♠♦t✐♦♥ ✭✶❉✮ ♦❢ ❛ ♣❛rt✐❝❧❡❀ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡

♣❛rt✐❝❧❡ ✭❛t t✐♠❡ t✮ ✐s v(t) := dx dt (t)

❼ ■♥ ✸❉✿ ❧✐❦❡✇✐s❡✦ ■❢

x(t) := (x(t), y(t), z(t)) ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ s♣❛❝❡✱ t❤❡♥ ✐ts ✈❡❧♦❝✐t② ✐s

• v(t) := d

x dt (t) := dx dt (t), dy dt (t), dz dt (t)

• .

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② ❙♣❧✐❝✐♥❣ ❘❛t❡

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② ❙♣❧✐❝✐♥❣ ❘❛t❡

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② u(t) ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② u(t) ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s(t) ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② u(t) ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s(t) ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ(t)

• 

      du dt (t) = α(t) − β(t)u(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛

■♥ ♦♥❡ ❝❡❧❧✱ ❢♦r ♦♥❡ ❣❡♥❡✳✳✳ ❉◆❆ ❚r❛♥s❝r✐♣t✐♦♥ ❘❛t❡ α(t)

❯♥s♣❧✐❝❡❞ ❘◆❆

◗✉❛♥t✐t② u(t) ❙♣❧✐❝✐♥❣ ❘❛t❡ β(t)

• ∅

❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s(t) ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ(t)

• 

      du dt (t) = α(t) − β(t)u(t) ds dt (t) = β(t)u(t) − γ(t)s(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

■♥ t❤✐s ❝♦♥t❡①t

❼ u(t) ❛♥❞ s(t) ❛r❡ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ♥✉♠❜❡rs ♦❢

♠♦❧❡❝✉❧❡s ♦❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t✮

❼ ❘❡❛❧ ♥✉♠❜❡rs ♦❢ ♠♦❧❡❝✉❧❡s ✭❛t t✐♠❡ ✮ ❤❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♦✐ss♦♥

❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ ❛♥❞ ✳

❖✉r ❡q✉❛t✐♦♥s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

■♥ t❤✐s ❝♦♥t❡①t

❼ u(t) ❛♥❞ s(t) ❛r❡ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ♥✉♠❜❡rs ♦❢

♠♦❧❡❝✉❧❡s ♦❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t✮

❼ ❘❡❛❧ ♥✉♠❜❡rs ♦❢ ♠♦❧❡❝✉❧❡s ✭❛t t✐♠❡ t✮ ❤❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♦✐ss♦♥

❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ u(t) ❛♥❞ s(t)✳

❖✉r ❡q✉❛t✐♦♥s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

■♥ t❤✐s ❝♦♥t❡①t

❼ u(t) ❛♥❞ s(t) ❛r❡ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ♥✉♠❜❡rs ♦❢

♠♦❧❡❝✉❧❡s ♦❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t✮

❼ ❘❡❛❧ ♥✉♠❜❡rs ♦❢ ♠♦❧❡❝✉❧❡s ✭❛t t✐♠❡ t✮ ❤❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♦✐ss♦♥

❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ u(t) ❛♥❞ s(t)✳

❖✉r ❡q✉❛t✐♦♥s

       du dt (t) = α(t) − β(t)u(t) ds dt (t) = β(t)u(t) − γ(t)s(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ r❛t❡s α✱ β✱ γ ❛r❡ ❝♦♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❼

✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ✱ ✐✳❡✳ ❡✈❡r②t❤✐♥❣ ❞✐✈✐❞❡❞ ❜② ✮✳

❋✐♥❛❧ ❡q✉❛t✐♦♥s

✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✑

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ r❛t❡s α✱ β✱ γ ❛r❡ ❝♦♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❼ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ β✱ ✐✳❡✳ ❡✈❡r②t❤✐♥❣ ❞✐✈✐❞❡❞

❜② β✮✳

❋✐♥❛❧ ❡q✉❛t✐♦♥s

✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✑

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ r❛t❡s α✱ β✱ γ ❛r❡ ❝♦♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❼ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ β✱ ✐✳❡✳ ❡✈❡r②t❤✐♥❣ ❞✐✈✐❞❡❞

❜② β✮✳

❋✐♥❛❧ ❡q✉❛t✐♦♥s

       du dt (t) = α − u(t) ds dt (t) = u(t) − γs(t) ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✑

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ r❛t❡s α✱ β✱ γ ❛r❡ ❝♦♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❼ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ β✱ ✐✳❡✳ ❡✈❡r②t❤✐♥❣ ❞✐✈✐❞❡❞

❜② β✮✳

❋✐♥❛❧ ❡q✉❛t✐♦♥s

       du dt (t) = α − u(t) ds dt (t) = u(t) − γs(t) ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✑

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥

du dt (t) = α − u(t)

❙♦❧✉t✐♦♥

■❢

✵

✵ ✱

✵ ✵ ✵

■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥

du dt (t) = α − u(t)

❙♦❧✉t✐♦♥

■❢ u✵ := u(✵)✱ u(t) = α + (u✵ − α)e−t.

✵ ✵

■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥

du dt (t) = α − u(t)

❙♦❧✉t✐♦♥

■❢ u✵ := u(✵)✱ u(t) = α + (u✵ − α)e−t.

✵ ✵

■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥

du dt (t) = α − u(t)

❙♦❧✉t✐♦♥

■❢ u✵ := u(✵)✱ u(t) = α + (u✵ − α)e−t. u✵ < α

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α

u✵ > α

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α

■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥

du dt (t) = α − u(t)

❙♦❧✉t✐♦♥

■❢ u✵ := u(✵)✱ u(t) = α + (u✵ − α)e−t. u✵ < α

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α

u✵ > α

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α

■♥ ❛❧❧ ❝❛s❡s✱ limt→∞ u(t) = α✱ ✐✳❡✳ u(t) ≈ α ✐❢ t ≫ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢

✵

✵ ❛♥❞

✵

✵ ✱

✵

✶

✵ ✵

✶ ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) = α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt.

▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) = α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt.

▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) =      α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt

✐❢ γ = ✶ α + [(u✵ − α)t + s✵ − α] e−t ✐❢ γ = ✶(= β). ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) =      α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt

✐❢ γ = ✶ α + [(u✵ − α)t + s✵ − α] e−t ✐❢ γ = ✶(= β). ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) =      α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt

✐❢ γ = ✶ α + [(u✵ − α)t + s✵ − α] e−t ✐❢ γ = ✶(= β). ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐✳❡✳ ✐❢ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥

ds dt (t) = u(t) − γs(t)

❙♦❧✉t✐♦♥

■❢ u✵ = u(✵) ❛♥❞ s✵ = s(✵)✱ s(t) =      α γ + u✵ − α γ − ✶ e−t +

• s✵ + α − u✵

γ − ✶ − α γ

• e−γt

✐❢ γ = ✶ α + [(u✵ − α)t + s✵ − α] e−t ✐❢ γ = ✶(= β). ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ lim

t→∞ s(t) = α

γ , ✐✳❡✳ s(t) ≈ α γ ✐❢ t ≫ ✵✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✿ ❣r❛♣❤✐❝❛❧ ❡①❛♠♣❧❡s

u✵ = s✵ = ✵, α = ✵.✷✺, γ = ✵.✼✺

2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t s(t) α/γ

u✵ = ✸, s✵ = ✶, α = ✵.✷✺, γ = ✵.✼✺

2 4 6 8 10 0.5 1.0 1.5 t s(t) α/γ

u✵ = ✸, s✵ = ✹, α = ✸✵, γ = ✺

2 4 6 8 10 2 3 4 5 6 t s(t) α/γ

u✵ = ✹, s✵ = ✻, α = ✶, γ = ✶

2 4 6 8 10 1 2 3 4 5 6 t s(t) α/γ

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r②

❚❤❡r❡ ❡①✐st s♦❧✉t✐♦♥s u✱ s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ u✵✱ s✵ ✭✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✮ ❛♥❞ ♦♥ α✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❤ lim

t→∞ u(t) = α

❛♥❞ lim

t→∞ s(t) = α

γ . ■♥ ♣❛rt✐❝✉❧❛r✱

❙t❡❛❞② st❛t❡

❲❤❡♥ ✵✱ t❤❡ s②st❡♠ r❡❛❝❤❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❤ ❛♥❞

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r②

❚❤❡r❡ ❡①✐st s♦❧✉t✐♦♥s u✱ s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ u✵✱ s✵ ✭✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✮ ❛♥❞ ♦♥ α✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❤ lim

t→∞ u(t) = α

❛♥❞ lim

t→∞ s(t) = α

γ . ■♥ ♣❛rt✐❝✉❧❛r✱ lim

t→∞

u(t) s(t) = γ.

❙t❡❛❞② st❛t❡

❲❤❡♥ ✵✱ t❤❡ s②st❡♠ r❡❛❝❤❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❤ ❛♥❞

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r②

❚❤❡r❡ ❡①✐st s♦❧✉t✐♦♥s u✱ s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ u✵✱ s✵ ✭✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✮ ❛♥❞ ♦♥ α✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❤ lim

t→∞ u(t) = α

❛♥❞ lim

t→∞ s(t) = α

γ . ■♥ ♣❛rt✐❝✉❧❛r✱ lim

t→∞

u(t) s(t) = γ.

❙t❡❛❞② st❛t❡

❲❤❡♥ t ≫ ✵✱ t❤❡ s②st❡♠ r❡❛❝❤❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❤ u(t) ≈ α, s(t) ≈ α γ , ❛♥❞ u(t) ≈ γs(t).

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

P❤❛s❡ ♣♦rtr❛✐t

• r❛♣❤✐❝ ✧s♣❧✐❝❡❞ ✈s✳ ✉♥s♣❧✐❝❡❞✧

u✵ = ✵, s✵ = ✵, α = ✸, γ = ✵.✼✺

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

u ≥ γs u✵ = ✸, s✵ = ✹, α = ✵, γ = ✵.✼✺

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

u ≤ γs

❚❤❡ s②st❡♠ r❡❛❝❤❡s t❤❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❤❡ str❛✐❣❤t ❧✐♥❡ u = γs✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

P❤❛s❡ ♣♦rtr❛✐t

• r❛♣❤✐❝ ✧s♣❧✐❝❡❞ ✈s✳ ✉♥s♣❧✐❝❡❞✧

u✵ = ✵, s✵ = ✵, α = ✸, γ = ✵.✼✺

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

u ≥ γs

= ⇒

u✵ = ✸, s✵ = ✹, α = ✵, γ = ✵.✼✺

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

u ≤ γs

❚❤❡ s②st❡♠ r❡❛❝❤❡s t❤❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❤❡ str❛✐❣❤t ❧✐♥❡ u = γs✳

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ✈❡❧♦❝✐t②

❈♦♥t❡①t

❼ ❍❡r❡✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❡ ❝❡❧❧✱ ✇✐t❤ p ❣❡♥❡s✳ ❼ ▲❡t

❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ t❤ ❣❡♥❡ ✭❛t t✐♠❡ ✮✳

❼ ❊❛❝❤

✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥ ♣❛r❛♠❡t❡rs ✵✱ ✶✱ ❛♥❞ ✵✳

❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦

✶ ❢♦r ❛❧❧ ✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦

❉❡✜♥✐t✐♦♥

❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s

✶

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ✈❡❧♦❝✐t②

❈♦♥t❡①t

❼ ❍❡r❡✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❡ ❝❡❧❧✱ ✇✐t❤ p ❣❡♥❡s✳ ❼ ▲❡t sj(t) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞

❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ jt❤ ❣❡♥❡ ✭❛t t✐♠❡ t✮✳

❼ ❊❛❝❤

✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥ ♣❛r❛♠❡t❡rs ✵✱ ✶✱ ❛♥❞ ✵✳

❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦

✶ ❢♦r ❛❧❧ ✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦

❉❡✜♥✐t✐♦♥

❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s

✶

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ✈❡❧♦❝✐t②

❈♦♥t❡①t

❼ ❍❡r❡✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❡ ❝❡❧❧✱ ✇✐t❤ p ❣❡♥❡s✳ ❼ ▲❡t sj(t) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞

❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ jt❤ ❣❡♥❡ ✭❛t t✐♠❡ t✮✳

❼ ❊❛❝❤ sj(t) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥

♣❛r❛♠❡t❡rs αj ≥ ✵✱ βj = ✶✱ ❛♥❞ γj > ✵✳

❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦

✶ ❢♦r ❛❧❧ ✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦

❉❡✜♥✐t✐♦♥

❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s

✶

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ✈❡❧♦❝✐t②

❈♦♥t❡①t

❼ ❍❡r❡✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❡ ❝❡❧❧✱ ✇✐t❤ p ❣❡♥❡s✳ ❼ ▲❡t sj(t) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞

❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ jt❤ ❣❡♥❡ ✭❛t t✐♠❡ t✮✳

❼ ❊❛❝❤ sj(t) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥

♣❛r❛♠❡t❡rs αj ≥ ✵✱ βj = ✶✱ ❛♥❞ γj > ✵✳

❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦

βj = ✶ ❢♦r ❛❧❧ j✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦

❉❡✜♥✐t✐♦♥

❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s

✶

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘◆❆ ✈❡❧♦❝✐t②

❈♦♥t❡①t

❼ ❍❡r❡✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❡ ❝❡❧❧✱ ✇✐t❤ p ❣❡♥❡s✳ ❼ ▲❡t sj(t) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞

❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ jt❤ ❣❡♥❡ ✭❛t t✐♠❡ t✮✳

❼ ❊❛❝❤ sj(t) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥

♣❛r❛♠❡t❡rs αj ≥ ✵✱ βj = ✶✱ ❛♥❞ γj > ✵✳

❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦

βj = ✶ ❢♦r ❛❧❧ j✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦

❉❡✜♥✐t✐♦♥

❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ t✮ ✐s d s dt (t) := ds✶ dt (t), ..., dsp dt (t)

• .

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶ ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭

✶ ✷

✮✳ ❆rr♦✇s✿ ❘◆❆ ✈❡❧♦❝✐t② ❘❡❞ ♣♦✐♥t✿ st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

• ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

• ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

❼ ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥✿ ❛♥ ✉♥r❡❛❧ ✇♦r❧❞✳✳✳

❼ ❆ ❝❡❧❧ ✇✐t❤ ✷ ❣❡♥❡s✳✳✳ ❼ α✶ = ✷✱ γ✶ = ✵.✺❀ α✷ = ✸✱ γ✷ = ✶

1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s1 s2

• ❼ ●r❡② ❝✉r✈❡✿

tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❡❧❧ ✭(s✶(t), s✷(t))✮✳

❼ ❆rr♦✇s✿

❘◆❆ ✈❡❧♦❝✐t②

❼ ❘❡❞ ♣♦✐♥t✿

st❡❛❞② st❛t❡ ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❘◆❆ ✈❡❧♦❝✐t②

❆♥❞ ✐❢ p > ✸❄

❼ Pr✐♥❝✐♣❧❡ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♦❥❡❝t✐♦♥ ♦♥

P✳❈✳❀

❼ t✲❙◆❊❄ P♦ss✐❜❧❡✱ ❜✉t ♠♦r❡ tr✐❝❦②✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❘◆❆ ✈❡❧♦❝✐t②

❆♥❞ ✐❢ p > ✸❄

❼ Pr✐♥❝✐♣❧❡ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♦❥❡❝t✐♦♥ ♦♥

P✳❈✳❀

❼ t✲❙◆❊❄ P♦ss✐❜❧❡✱ ❜✉t ♠♦r❡ tr✐❝❦②✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❘◆❆ ✈❡❧♦❝✐t②

❆♥❞ ✐❢ p > ✸❄

❼ Pr✐♥❝✐♣❧❡ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♦❥❡❝t✐♦♥ ♦♥

P✳❈✳❀

❼ t✲❙◆❊❄ P♦ss✐❜❧❡✱ ❜✉t ♠♦r❡ tr✐❝❦②✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❘◆❆ ✈❡❧♦❝✐t②

❊①❛♠♣❧❡✿ ❙❝❤✇❛♥♥ ❝❡❧❧ ♣r❡❝✉rs♦rs ✭❝♦♠✐♥❣ ❢r♦♠ ❬✶❪✮

P❈❆ t✲❙◆❊

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❘◆❆ ✈❡❧♦❝✐t②

❊①❛♠♣❧❡✿ ❙❝❤✇❛♥♥ ❝❡❧❧ ♣r❡❝✉rs♦rs ✭❝♦♠✐♥❣ ❢r♦♠ ❬✶❪✮

P❈❆ t✲❙◆❊

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ γ

❲❡ st✉❞② ♦♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❤r♦✉❣❤ ❛ s❛♠♣❧❡ ♦❢ s❡✈❡r❛❧ ❝❡❧❧s✳

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ s❛♠♣❧❡ ♦❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♦ ❝♦✈❡r ❛❧❧ t❤❡ ✏❘◆❆

❝②❝❧❡✑ ✭❢r♦♠ ❜❡❣✐♥♥✐♥❣ ♦❢ ♣r♦❞✉❝t✐♦♥ t♦ st❡❛❞② st❛t❡✮✳

❼ ❚❤❡ r❛t❡ ♦❢ ❞❡❣r❛❞❛t✐♦♥

♦❢ t❤❡ ❣❡♥❡ ✐s t❤❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s✳ ❊st✐♠❛t✐♦♥ ♦❢ ✇✐t❤ ♣❤❛s❡ ♣♦rtr❛✐ts✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ γ

❲❡ st✉❞② ♦♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❤r♦✉❣❤ ❛ s❛♠♣❧❡ ♦❢ s❡✈❡r❛❧ ❝❡❧❧s✳

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ s❛♠♣❧❡ ♦❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♦ ❝♦✈❡r ❛❧❧ t❤❡ ✏❘◆❆

❝②❝❧❡✑ ✭❢r♦♠ ❜❡❣✐♥♥✐♥❣ ♦❢ ♣r♦❞✉❝t✐♦♥ t♦ st❡❛❞② st❛t❡✮✳

❼ ❚❤❡ r❛t❡ ♦❢ ❞❡❣r❛❞❛t✐♦♥ γ ♦❢ t❤❡ ❣❡♥❡ ✐s t❤❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s✳

❊st✐♠❛t✐♦♥ ♦❢ ✇✐t❤ ♣❤❛s❡ ♣♦rtr❛✐ts✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ γ

❲❡ st✉❞② ♦♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❤r♦✉❣❤ ❛ s❛♠♣❧❡ ♦❢ s❡✈❡r❛❧ ❝❡❧❧s✳

❆ss✉♠♣t✐♦♥s

❼ ❚❤❡ s❛♠♣❧❡ ♦❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♦ ❝♦✈❡r ❛❧❧ t❤❡ ✏❘◆❆

❝②❝❧❡✑ ✭❢r♦♠ ❜❡❣✐♥♥✐♥❣ ♦❢ ♣r♦❞✉❝t✐♦♥ t♦ st❡❛❞② st❛t❡✮✳

❼ ❚❤❡ r❛t❡ ♦❢ ❞❡❣r❛❞❛t✐♦♥ γ ♦❢ t❤❡ ❣❡♥❡ ✐s t❤❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s✳

❊st✐♠❛t✐♦♥ ♦❢ γ ✇✐t❤ ♣❤❛s❡ ♣♦rtr❛✐ts✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

❼ α = ✸, γ = ✵.✼✺ ❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✹✵✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳

Pr♦❝❡ss

❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❤❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♦♥ ♦❢ γ ✭s❧♦♣❡✮✿ ✵✳✼✸✹✾✸

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✹✵✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳

Pr♦❝❡ss

✶✳ ❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❤❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♦♥ ♦❢ γ ✭s❧♦♣❡✮✿ ✵✳✼✸✹✾✸

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✹✵✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳

Pr♦❝❡ss

✶✳ ❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❤❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ✷✳ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♦♥ ♦❢ ✭s❧♦♣❡✮✿ ✵✳✼✸✹✾✸

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✹✵✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳

Pr♦❝❡ss

✶✳ ❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❤❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ✷✳ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♦♥ ♦❢ γ ✭s❧♦♣❡✮✿ ✵✳✼✸✹✾✸

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❉✐✣❝✉❧t✐❡s ♦❢ ❡st✐♠❛t✐♦♥ ❢♦r γ✳✳✳

❆ss✉♠♣t✐♦♥ ✶ ♥♦t r❡s♣❡❝t❡❞✳✳✳

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

❼ α = ✸, γ = ✵.✼✺ ❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✶✺✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳ ❊st✐♠❛t✐♦♥ ♦❢ γ✿ ✵✳✾✼✹✸✸✳✳✳

❈♦rr❡❝t✐♦♥s❄

❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ❝❧✉st❡rs ♦❢ ✈❡r② ❝♦rr❡❧❛t❡❞ ❝❡❧❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❉✐✣❝✉❧t✐❡s ♦❢ ❡st✐♠❛t✐♦♥ ❢♦r γ✳✳✳

❆ss✉♠♣t✐♦♥ ✶ ♥♦t r❡s♣❡❝t❡❞✳✳✳

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✶✺✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳ ❊st✐♠❛t✐♦♥ ♦❢ ✿ ✵✳✾✼✹✸✸✳✳✳

❈♦rr❡❝t✐♦♥s❄

❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♦♠❡ ❝❡❧❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❉✐✣❝✉❧t✐❡s ♦❢ ❡st✐♠❛t✐♦♥ ❢♦r γ✳✳✳

❆ss✉♠♣t✐♦♥ ✶ ♥♦t r❡s♣❡❝t❡❞✳✳✳

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✶✺✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳ ❊st✐♠❛t✐♦♥ ♦❢ γ✿ ✵✳✾✼✹✸✸✳✳✳

❈♦rr❡❝t✐♦♥s❄

❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♦♠❡ ❝❡❧❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❉✐✣❝✉❧t✐❡s ♦❢ ❡st✐♠❛t✐♦♥ ❢♦r γ✳✳✳

❆ss✉♠♣t✐♦♥ ✶ ♥♦t r❡s♣❡❝t❡❞✳✳✳

• 1

2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

s u

• ❼ α = ✸, γ = ✵.✼✺

❼ ❙t❡❛❞② st❛t❡✿

u = γs

❼ ✶✺✵ ❝❡❧❧s✱

✉♥✐❢♦r♠❧② ❣❡♥❡r❛t❡❞ ✐♥ t✐♠❡✳ ❊st✐♠❛t✐♦♥ ♦❢ γ✿ ✵✳✾✼✹✸✸✳✳✳

❈♦rr❡❝t✐♦♥s❄

❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♦♠❡ ❝❡❧❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

▼✉❧t✐♣❧❡ s♣❧✐❝✐♥❣

❼ ■♥ ❬✶❪✱ ± ✽✾ ✪ ♦❢ st✉❞✐❡❞ ❣❡♥❡s s❤♦✇❡❞ ❛ ✉♥✐q✉❡ ❞❡❣r❛❞❛t✐♦♥

r❛t❡ γ✳✳✳ ❜✉t ✶✶ ✪ s❤♦✇❡❞ s❡✈❡r❛❧ ❞❡❣r❛❞❛t✐♦♥ r❛t❡s✦

❼ ❊①❛♠♣❧❡ ❢r♦♠ ❬✶❪✿

◆tr❦✷

❼ ❚❤❡♥ t❤❡ ♠♦❞❡❧ ❢❛✐❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

▼✉❧t✐♣❧❡ s♣❧✐❝✐♥❣

❼ ■♥ ❬✶❪✱ ± ✽✾ ✪ ♦❢ st✉❞✐❡❞ ❣❡♥❡s s❤♦✇❡❞ ❛ ✉♥✐q✉❡ ❞❡❣r❛❞❛t✐♦♥

r❛t❡ γ✳✳✳ ❜✉t ✶✶ ✪ s❤♦✇❡❞ s❡✈❡r❛❧ ❞❡❣r❛❞❛t✐♦♥ r❛t❡s✦

❼ ❊①❛♠♣❧❡ ❢r♦♠ ❬✶❪✿

◆tr❦✷

❼ ❚❤❡♥ t❤❡ ♠♦❞❡❧ ❢❛✐❧s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ α

❆❝❝♦r❞✐♥❣ t♦ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♦ ❡st✐♠❛t❡ α✳✳✳ ❚✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✿

❼ ▼♦❞❡❧ ■✿

✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱

✵

✇✐t❤

✵ ✵✳

❼ ▼♦❞❡❧ ■■✿

✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱

✵ ✵ ✵

❚❤❡s❡ t✇♦ ♠♦❞❡❧s ❛r❡ ❝♦rr❡❝t ✐♥ t❤❡ s❤♦rt t❡r♠❀ t❤❡② ❤❛✈❡ t♦ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♦ ♣r❡❞✐❝t t❤❡ ❢✉t✉r❡ ✭▼❛r❦♦✈ ♣r♦❝❡ss✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ α

❆❝❝♦r❞✐♥❣ t♦ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♦ ❡st✐♠❛t❡ α✳✳✳ ❚✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✿

❼ ▼♦❞❡❧ ■✿ v := ds

dt ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱ s(t) = vt + s✵, ✇✐t❤ v := u✵ − γs✵✳

❼ ▼♦❞❡❧ ■■✿

✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱

✵ ✵ ✵

❚❤❡s❡ t✇♦ ♠♦❞❡❧s ❛r❡ ❝♦rr❡❝t ✐♥ t❤❡ s❤♦rt t❡r♠❀ t❤❡② ❤❛✈❡ t♦ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♦ ♣r❡❞✐❝t t❤❡ ❢✉t✉r❡ ✭▼❛r❦♦✈ ♣r♦❝❡ss✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ α

❆❝❝♦r❞✐♥❣ t♦ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♦ ❡st✐♠❛t❡ α✳✳✳ ❚✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✿

❼ ▼♦❞❡❧ ■✿ v := ds

dt ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱ s(t) = vt + s✵, ✇✐t❤ v := u✵ − γs✵✳

❼ ▼♦❞❡❧ ■■✿ u ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱

s(t) = u✵ γ +

• s✵ − u✵

γ

• e−γt.

❚❤❡s❡ t✇♦ ♠♦❞❡❧s ❛r❡ ❝♦rr❡❝t ✐♥ t❤❡ s❤♦rt t❡r♠❀ t❤❡② ❤❛✈❡ t♦ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♦ ♣r❡❞✐❝t t❤❡ ❢✉t✉r❡ ✭▼❛r❦♦✈ ♣r♦❝❡ss✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❊st✐♠❛t✐♦♥ ♦❢ α

❆❝❝♦r❞✐♥❣ t♦ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♦ ❡st✐♠❛t❡ α✳✳✳ ❚✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✿

❼ ▼♦❞❡❧ ■✿ v := ds

dt ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱ s(t) = vt + s✵, ✇✐t❤ v := u✵ − γs✵✳

❼ ▼♦❞❡❧ ■■✿ u ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱

s(t) = u✵ γ +

• s✵ − u✵

γ

• e−γt.

❚❤❡s❡ t✇♦ ♠♦❞❡❧s ❛r❡ ❝♦rr❡❝t ✐♥ t❤❡ s❤♦rt t❡r♠❀ t❤❡② ❤❛✈❡ t♦ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♦ ♣r❡❞✐❝t t❤❡ ❢✉t✉r❡ ✭▼❛r❦♦✈ ♣r♦❝❡ss✮✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❆♥❞ ✐❢ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ♥♦♥✲❝♦♥st❛♥t❄

▼✉❝❤ ♠♦r❡ ❝♦♠♣❧❡①✳✳✳

❊①❛♠♣❧❡

❆ss✉♠❡ t❤❛t α(t) = ✶ − cos(t)✱ β = ✶✱ ❛♥❞ γ > ✵ ✐s ❝♦♥st❛♥t✳

5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 t 1 − cos(t)

❘◆❆ ❡q✉❛t✐♦♥s

✶

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❆♥❞ ✐❢ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ♥♦♥✲❝♦♥st❛♥t❄

▼✉❝❤ ♠♦r❡ ❝♦♠♣❧❡①✳✳✳

❊①❛♠♣❧❡

❆ss✉♠❡ t❤❛t α(t) = ✶ − cos(t)✱ β = ✶✱ ❛♥❞ γ > ✵ ✐s ❝♦♥st❛♥t✳

5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 t 1 − cos(t)

❘◆❆ ❡q✉❛t✐♦♥s

      du dt (t) = [✶ − cos(t)] − u(t) ds dt (t) = u(t) − γs(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥s

❯♥s♣❧✐❝❡❞ ❘◆❆

u(t) = ✶ − ✶ ✷ (cos(t) + sin(t)) +

• u✵ − ✶

✷

• e−t.

u✵ = ✵

5 10 15 20 25 0.0 0.5 1.0 1.5 t u(t)

u✵ = ✸

5 10 15 20 25 0.5 1.0 1.5 2.0 2.5 3.0 t u(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥s

❙♣❧✐❝❡❞ ❘◆❆

■❢ γ = ✶✱ s(t) = ✶ γ − ✶ ✷(✶ + γ✷) ((γ − ✶) cos(t) + (γ + ✶) sin(t)) + u✵ − ✶/✷ γ − ✶ e−t +

• s✵ − ✶

γ + ✶/✷ − u✵ γ − ✶ + γ − ✶ ✷(✶ + γ✷)

• e−γt

❛♥❞✱ ✐❢ γ = ✶✱ s(t) = ✶ − ✶ ✷ sin(t) +

• u✵ − ✶

✷

• t + s✵ − ✶
• e−t.

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❙♦❧✉t✐♦♥s

u✵ = s✵ = ✵, γ = ✵.✷

5 10 15 20 25 1 2 3 4 5 t s(t)

u✵ = ✼, s✵ = ✺, γ = ✵.✷

5 10 15 20 25 5 6 7 8 t s(t)

u✵ = ✶, s✵ = ✷, γ = ✷.✺

5 10 15 20 25 0.5 1.0 1.5 2.0 t s(t)

u✵ = ✷✵, s✵ = ✺, γ = ✶

5 10 15 20 25 2 4 6 8 t s(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

P❤❛s❡ ♣♦rtr❛✐ts

u✵ = s✵ = ✵, γ = ✵.✷

1 2 3 4 5 0.0 0.5 1.0 1.5 s(t) u(t)

u✵ = ✼, s✵ = ✺, γ = ✵.✷

5 6 7 8 1 2 3 4 5 6 7 s(t) u(t)

u✵ = ✶, s✵ = ✷, γ = ✷.✺

0.5 1.0 1.5 2.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 s(t) u(t)

u✵ = ✷✵, s✵ = ✺, γ = ✶

2 4 6 8 5 10 15 20 s(t) u(t)

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐①

❘❡❢❡r❡♥❝❡s ■

• ✳ ▲❛ ▼❛♥♥♦ ❡t ❛❧✳

❘◆❆ ✈❡❧♦❝✐t② ♦❢ s✐♥❣❧❡ ❝❡❧❧s✳ ◆❛t✉r❡✱ ✺✻✵✿✹✾✹✕✺✶✻✱ ✷✵✶✽✳