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

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✐♠❡ ❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡ � �� � f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim 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✐❝❧❡❀

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✐♠❡ ❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡ � �� � f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim 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 )

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❱❛r✐❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ ❛✈❡r❛❣❡ ❝❤❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♦❢ t✐♠❡ � �� � f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim 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 � dx � v ( t ) := d � x dt ( t ) , dy dt ( t ) , dz � dt ( t ) := 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

❘❛t❡ ◗✉❛♥t✐t② ❘❛t❡ ◗✉❛♥t✐t② ❘❛t❡ � � ❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❤❡ ✐❞❡❛ ■♥ ♦♥❡ ❝❡❧❧ ✱ ❢♦r ♦♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♦♥ � ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ❙♣❧✐❝✐♥❣ ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♦♥

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

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

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

◗✉❛♥t✐t② � � ❙♦♠❡ 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 ) ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣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 )  du  α ( t ) − β ( t ) u ( t ) dt ( 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 )  du  α ( t ) − β ( t ) u ( t ) dt ( t ) =   ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ds   dt ( t ) = β ( t ) u ( t ) − γ ( t ) s ( t )  ∅ ❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s ( t ) ❉❡❣r❛❞❛t✐♦♥ ❘❛t❡ γ ( 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 ✮

❖✉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❡❝❛❧❧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   β ( t ) u ( t ) − γ ( t ) s ( t ) dt ( t ) = 

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

❋✐♥❛❧ ❡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 ✮✳

✏✭▲✐♥❡❛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 ) − γ 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 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 ) − γ s ( t )  ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✑

✵ ✵ ■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳ ❙♦❧✉t✐♦♥ ■❢ ✵ ✱ ✵ ✵ ❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst ❡q✉❛t✐♦♥ du dt ( t ) = α − u ( 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 u ✵ > α 1.0 1.0 0.8 0.8 0.6 0.6 α α u(t) u(t) 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 t t

❙♦♠❡ 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 u ✵ > α 1.0 1.0 0.8 0.8 0.6 0.6 α α u(t) u(t) 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 t t ■♥ ❛❧❧ ❝❛s❡s✱ lim t →∞ u ( t ) = α ✱ ✐✳❡✳ u ( t ) ≈ α ✐❢ 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 ) − γ s ( 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 ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � � γ + u ✵ − α s ✵ + α − u ✵ s ( t ) = α γ − ✶ − α γ − ✶ e − t + e − γ t . γ

▼❛♥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 ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � � γ + u ✵ − α s ✵ + α − u ✵ s ( t ) = α γ − ✶ − α γ − ✶ e − t + e − γ t . γ

▼❛♥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 ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � �  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ � = ✶  γ s ( t ) =   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 ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � �  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ � = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) .

❇✉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 ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � �  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ � = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) . ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❙♦❧✉t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♦❧✉t✐♦♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ � �  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ � = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) . ▼❛♥② ❣r❛♣❤✐❝❛❧ ♣♦ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❤❛✈❡ t →∞ s ( t ) = α lim 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 = ✵ . ✼✺ u ✵ = ✸ , s ✵ = ✶ , α = ✵ . ✷✺ , γ = ✵ . ✼✺ α/γ 0.30 1.5 0.25 0.20 s(t) s(t) 0.15 1.0 0.10 0.05 0.5 α/γ 0.00 0 2 4 6 8 10 0 2 4 6 8 10 t t u ✵ = ✸ , s ✵ = ✹ , α = ✸✵ , γ = ✺ u ✵ = ✹ , s ✵ = ✻ , α = ✶ , γ = ✶ 6 α/γ 6 5 5 4 s(t) s(t) 4 3 2 3 1 α/γ 0 2 0 2 4 6 8 10 0 2 4 6 8 10 t t

■♥ ♣❛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❤ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim 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❤ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim γ . ■♥ ♣❛rt✐❝✉❧❛r✱ u ( t ) lim s ( t ) = γ. 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❤ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim γ . ■♥ ♣❛rt✐❝✉❧❛r✱ u ( t ) lim s ( t ) = γ. t →∞ ❙t❡❛❞② st❛t❡ ❲❤❡♥ t ≫ ✵✱ t❤❡ s②st❡♠ r❡❛❝❤❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❤ s ( t ) ≈ α u ( t ) ≈ α, u ( t ) ≈ γ s ( t ) . γ , ❛♥❞

❙♦♠❡ 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 = ✵ . ✼✺ u ✵ = ✸ , s ✵ = ✹ , α = ✵ , γ = ✵ . ✼✺ 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 u u 1.0 1.0 0.5 0.5 0.0 0.0 0 1 2 3 4 0 1 2 3 4 s s u ≥ γ s u ≤ γ s ❚❤❡ s②st❡♠ r❡❛❝❤❡s t❤❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❤❡ str❛✐❣❤t ❧✐♥❡ u = γ s ✳

❙♦♠❡ 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 = ✵ . ✼✺ u ✵ = ✸ , s ✵ = ✹ , α = ✵ , γ = ✵ . ✼✺ 3.0 3.0 2.5 2.5 2.0 2.0 1.5 = ⇒ 1.5 u u 1.0 1.0 0.5 0.5 0.0 0.0 0 1 2 3 4 0 1 2 3 4 s s u ≥ γ s u ≤ γ s ❚❤❡ s②st❡♠ r❡❛❝❤❡s t❤❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❤❡ str❛✐❣❤t ❧✐♥❡ u = γ s ✳ 3.0 2.5 2.0 u 1.5 1.0 0.5 0.0 0 1 2 3 4 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

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

❲❛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 s j ( t ) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ j t❤ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❼ ❊❛❝❤ s j ( t ) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳

❉❡✜♥✐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 s j ( t ) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ j t❤ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❼ ❊❛❝❤ s j ( t ) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳ ❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦ β j = ✶ ❢♦r ❛❧❧ j ✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡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 s j ( t ) ❜❡ t❤❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡✮ q✉❛♥t✐t② ♦❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ j t❤ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❼ ❊❛❝❤ s j ( t ) ✈❡r✐✜❡s t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ ✇✐t❤ ✐ts ♦✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳ ❲❛r♥✐♥❣✦ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥✦ β j = ✶ ❢♦r ❛❧❧ j ✿ t❤❡ r❛t❡s ♦❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ ❣❡♥❡s✦ ❉❡✜♥✐t✐♦♥ ❚❤❡ ❘◆❆ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❡❧❧ ✭❛t t✐♠❡ t ✮ ✐s � ds ✶ � d � s dt ( t ) , ..., ds p dt ( t ) := dt ( t ) .

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 ✷ = ✶

❙♦♠❡ 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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ • ❆rr♦✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♦❝✐t② 0.5 • ❘❡❞ ♣♦✐♥t✿ st❡❛❞② st❛t❡ 0.0 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❼ ❆rr♦✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♦❝✐t② 0.5 • ❘❡❞ ♣♦✐♥t✿ st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❼ ❆rr♦✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♦❝✐t② 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❼ ❆rr♦✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♦r② ♦❢ 2.5 ● t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏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 ✷ = ✶ 3.0 ❼ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♦r② ♦❢ 2.5 ● t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● 0.5 • ❘❡❞ ♣♦✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

✏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 ✷ = ✶ 3.0 ● ❼ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♦r② ♦❢ 2.5 ● t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● ❼ ❘❡❞ ♣♦✐♥t✿ 0.5 ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1

❙♦♠❡ 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 ✷ = ✶ 3.0 ● ❼ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♦r② ♦❢ 2.5 ● t❤❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❼ ❆rr♦✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♦❝✐t② ● ❼ ❘❡❞ ♣♦✐♥t✿ 0.5 ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❤②s✐❝❛❧ ✈❡❧♦❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✦

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 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✳❈✳❀

❙♦♠❡ 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✳✳✳

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♦♠ ❬✶❪✮

❙♦♠❡ 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

❆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✳

❊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 ✳

❙♦♠❡ 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 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❼ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● 0.0 ● 0 1 2 3 4 s 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 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❼ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s 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❡✱ ❡st✐♠❛t✐♦♥ ♦❢ ✭s❧♦♣❡✮✿ ✵✳✼✸✹✾✸ ❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❼ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s Pr♦❝❡ss ✶✳ ❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❤❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ✷✳ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ❡①tr❡♠❡ ❝❡❧❧s

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❚❤❡♦r✐t✐❝❛❧ ❡st✐♠❛t✐♦♥ ♦❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❼ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s 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❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❼ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● 0.0 ● 0 1 2 3 4 s ❊st✐♠❛t✐♦♥ ♦❢ γ ✿ ✵✳✾✼✹✸✸✳✳✳ ❈♦rr❡❝t✐♦♥s❄ ❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ❝❧✉st❡rs ♦❢ ✈❡r② ❝♦rr❡❧❛t❡❞ ❝❡❧❧s✳✳✳

❊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❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❼ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s

❈♦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❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❼ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s ❊st✐♠❛t✐♦♥ ♦❢ γ ✿ ✵✳✾✼✹✸✸✳✳✳

❙♦♠❡ 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❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❼ α = ✸ , γ = ✵ . ✼✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❼ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❼ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♦r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s ❊st✐♠❛t✐♦♥ ♦❢ γ ✿ ✵✳✾✼✹✸✸✳✳✳ ❈♦rr❡❝t✐♦♥s❄ ❊st✐♠❛t✐♦♥ ♦♥ ✈❡r② ❝♦rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♦♠❡ ❝❡❧❧s✳✳✳

❜✉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 ✳✳✳

❙♦♠❡ 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✳✳✳

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 ▼♦❞❡❧ ■■ ✿ ✐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 ✵ − γ s ✵ ✳

❚❤❡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 ✵ − γ s ✵ ✳ ❼ ▼♦❞❡❧ ■■ ✿ u ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱ � � s ( t ) = u ✵ s ✵ − u ✵ e − γ t . γ + γ

❙♦♠❡ 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 ✵ − γ s ✵ ✳ ❼ ▼♦❞❡❧ ■■ ✿ u ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t❀ t❤❡♥✱ � � s ( t ) = u ✵ s ✵ − u ✵ e − γ t . γ + γ ❚❤❡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

❘◆❆ ❡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✳ 2.0 1.5 1 − cos(t) 1.0 0.5 0.0 0 5 10 15 20 25 t

❙♦♠❡ 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✳ 2.0 1.5 1 − cos(t) 1.0 0.5 0.0 0 5 10 15 20 25 t ❘◆❆ ❡q✉❛t✐♦♥s  du  dt ( t ) = [ ✶ − cos( t )] − u ( t )   ds   u ( t ) − γ s ( t ) dt ( t ) = 

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❙♦❧✉t✐♦♥s ❯♥s♣❧✐❝❡❞ ❘◆❆ � � u ( t ) = ✶ − ✶ u ✵ − ✶ e − t . ✷ (cos( t ) + sin( t )) + ✷ u ✵ = ✵ u ✵ = ✸ 3.0 1.5 2.5 2.0 1.0 u(t) u(t) 1.5 0.5 1.0 0.5 0.0 0 5 10 15 20 25 0 5 10 15 20 25 t 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 ✵ − ✶ / ✷ s ✵ − ✶ γ + ✶ / ✷ − u ✵ γ − ✶ γ − ✶ e − t + e − γ t + ✷ ( ✶ + γ ✷ ) γ − ✶ ❛♥❞✱ ✐❢ γ = ✶✱ �� � � s ( t ) = ✶ − ✶ u ✵ − ✶ e − t . ✷ sin( t ) + t + s ✵ − ✶ ✷

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① ❙♦❧✉t✐♦♥s u ✵ = s ✵ = ✵ , γ = ✵ . ✷ u ✵ = ✼ , s ✵ = ✺ , γ = ✵ . ✷ 5 8 4 7 3 s(t) s(t) 6 2 1 5 0 0 5 10 15 20 25 0 5 10 15 20 25 t t u ✵ = ✶ , s ✵ = ✷ , γ = ✷ . ✺ u ✵ = ✷✵ , s ✵ = ✺ , γ = ✶ 2.0 8 1.5 6 s(t) s(t) 1.0 4 0.5 2 0 5 10 15 20 25 0 5 10 15 20 25 t t

❙♦♠❡ r❡❝❛❧❧s ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❘◆❆ ✈❡❧♦❝✐t② ❊st✐♠❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❆♣♣❡♥❞✐① P❤❛s❡ ♣♦rtr❛✐ts u ✵ = s ✵ = ✵ , γ = ✵ . ✷ u ✵ = ✼ , s ✵ = ✺ , γ = ✵ . ✷ 7 1.5 6 5 1.0 4 u(t) u(t) 3 0.5 2 1 0.0 5 6 7 8 0 1 2 3 4 5 s(t) s(t) u ✵ = ✶ , s ✵ = ✷ , γ = ✷ . ✺ u ✵ = ✷✵ , s ✵ = ✺ , γ = ✶ 20 1.6 1.4 15 1.2 u(t) u(t) 1.0 10 0.8 5 0.6 0.4 0 0.5 1.0 1.5 2.0 2 4 6 8 s(t) s(t)