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▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ ❊❧❡❝tr♦♥✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s ❋✉❧❧❡r P❤❡♥♦♠❡♥♦♥ ✐♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ▲❛r✐s❛ ▼❛♥✐t❛ ❧♠❛♥✐t❛❅❤s❡✳r✉ ✶✹ ◆♦✈❡♠❜❡r ✷✵✶✽ ▼♦s❝♦✇✱ ❘✉ss✐❛

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ T ˆ ( ϕ ✵ ( x ) + u ϕ ✶ ( x )) dt → ♠✐♥ , x = f ✵ ( x ) + uf ✶ ( x ) ˙ ✵ x ( ✵ ) ∈ B ✵ ⊂ R n , x ( T ) ∈ B T ⊂ R n | u | ≤ ✶ ❍❡r❡ x ✐s ❛ st❛t❡ ✈❛r✐❛❜❧❡✱ u ✐s ❛ s❝❛❧❛r ❝♦♥tr♦❧✱ ϕ i : R n → R ✱ f i : R n → R n ✱ i = ✵ , ✶ ✱ t❤❡ ❢✉♥❝t✐♦♥s ϕ i , f i ❛r❡ s♠♦♦t❤ ❡♥♦✉❣❤✱ B ✵ , B T ❛r❡ s♠♦♦t❤ ♠❛♥✐❢♦❧❞s✳ ❚❤❡ ❛❞♠✐ss✐❜❧❡ ❝♦♥tr♦❧s u ( t ) ♥❡❡❞ t♦ ❜❡ ♠❡❛s✉r❛❜❧❡✱ t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s x ( t ) ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s✳

P♦♥tr②❛❣✐♥✬s ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❉❡✜♥❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H = H ✵ ( x , ψ ) + uH ✶ ( x , ψ ) , ✇❤❡r❡ H ✵ ( x , ψ ) = f ✵ ( x ) ψ − ✶ ✷ ϕ ✵ ( x ) ✱ H ✶ ( x , ψ ) = f ✶ ( x ) ψ − ✶ ✷ ϕ ✶ ( x ) ✳ ❲❡ ❤❛✈❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ x = ∂ H ψ = − ∂ H ˙ ˙ ∂ψ , ✭✶✮ ∂ x ❛♥❞ H ( x ( t ) , ψ ( t ) , u opt ( t )) = ♠❛① ✵ ≤ u ≤ ✶ H ( x ( t ) , ψ ( t ) , u ) ✭✷✮

❙✐♥❣✉❧❛r ❡①tr❡♠❛❧ ❙✐♥❝❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H ✐s ❧✐♥❡❛r ✐♥ ✉✱ ❤❡♥❝❡ t♦ ♠❛①✐♠✐③❡ ✐t ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ u ∈ [ − ✶ , ✶ ] ✇❡ ♥❡❡❞ t♦ ✉s❡ ❜♦✉♥❞❛r② ✈❛❧✉❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s✐❣♥ ♦❢ H ✶ = ψ ✳ ❚❤❡ ♠❛①✐♠✉♠ ❝♦♥❞✐t✐♦♥ ②✐❡❧❞s✿ u = + ✶ ❢♦r H ✶ > ✵ ✱ u = − ✶ ❢♦r H ✶ < ✵ ✳ ❆♥ ❡①tr❡♠❛❧ ( x ( t ) , ψ ( t )) , t ∈ ( t ✵ , t ✶ ) ✱ ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r ✐❢ H ✶ ( x ( t ) , ψ ( t )) = ✵ ❢♦r t ∈ ( t ✵ , t ✶ ) ✳ ❚♦ ✜♥❞ t❤❡ ❝♦♥tr♦❧ ♦♥ s✐♥❣✉❧❛r ❡①tr❡♠❛❧ ( x ( t ) , ψ ( t )) ♦♥❡ ♥❡❡❞s t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ✐❞❡♥t✐t② H ✶ ( x ( t ) , ψ ( t )) = ✵ ✳

❖r❞❡r ♦❢ ❛ s✐♥❣✉❧❛r ❡①tr❡♠❛❧ ❲❡ s❛② t❤❛t ❛ ♥✉♠❜❡r q ✐s ❛♥ ♦r❞❡r ♦❢ ❛ s✐♥❣✉❧❛r tr❛❥❡❝t♦r② ✐✛ � � d k ∂ � H ✶ ( x , ψ ) = ✵ , k = ✵ , . . . , ✷ q − ✶ , � dt k ∂ u ( ✶ ) � � d ✷ q ∂ � H ✶ ( x , ψ ) � = ✵ � dt ✷ q ∂ u ( ✶ ) ✐♥ s♦♠❡ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ s✐♥❣✉❧❛r tr❛❥❡❝t♦r② ( x ( t ) , ψ ( t )) ✳ ■t ✐s ❦♥♦✇♥ t❤❛t ❢♦r ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s ❛ s✐♥❣✉❧❛r ❛r❝ ♦❢ ❡✈❡♥ ♦r❞❡r ✐s ❥♦✐♥❡❞ ✇✐t❤ ❛ ❝❤❛tt❡r✐♥❣ tr❛❥❡❝t♦r②✳ ❆ ❝❤❛tt❡r✐♥❣ tr❛❥❡❝t♦r② ✐s ❛ tr❛❥❡❝t♦r② ✇✐t❤ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧ s✇✐t❝❤✐♥❣s ✐♥ ❛ ✜♥✐t❡ t✐♠❡ ✐♥t❡r✈❛❧✳

❋✉❧❧❡r ♣r♦❜❧❡♠ ▼✐♥✐♠✐③❡ ˆ ∞ s ✷ ( t ) dt ✭✸✮ ✵ s✉❜❥❡❝t t♦ ¨ s ( t ) = u ( t ) , − ✶ ≤ u ( t ) ≤ ✶ ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s s ( ✵ ) = a , s ( ✵ ) = b ˙ ✭✹✮

❖♣t✐♠❛❧ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧

❖♣t✐♠❛❧ ❙♦❧✉t✐♦♥s ✐♥ ❋✉❧❧❡r Pr♦❜❧❡♠ ❉❡♥♦t❡ ˙ s = v ◮ ❚❤❡ ❝✉r✈❡ s = − Cv ✷ sgnv ✐s t❤❡ ♦♣t✐♠❛❧ s✇✐t❝❤✐♥❣ s❡t ♦❢ t❤❡ ❋✉❧❧❡r Pr♦❜❧❡♠✳ ❍❡r❡ C ≈ ✵ , ✹✹✹✻✷✸ . . . ✳ ◮ ❚✇✐st✐♥❣ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥ t❤❡ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s ❛tt❛✐♥ t❤❡ ♦r✐❣✐♥ ✐♥ ❛ ✜♥✐t❡ t✐♠❡ ❛♥❞ ✐♥t❡rs❡❝t t❤❡ s✇✐t❝❤✐♥❣ ❝✉r✈❡ ❛t ❛ ❝♦✉♥t❛❜❧❡ s❡t ♦❢ ♣♦✐♥ts✳ ◮ ❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❡q✉❛❧s ✶ ❢r♦♠ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ s✇✐t❝❤✐♥❣ ❝✉r✈❡ ❛♥❞ ❡q✉❛❧s − ✶ ❢r♦♠ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✐t✳

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❋✉❧❧❡r ♣r♦❜❧❡♠ H ( s , v , φ, ψ ) = − ✶ ✷ s ✷ + v φ + u ψ = H ✵ + uH ✶ ▲❡t ( � s ( t ) , � v ( t ) , � u ( t )) ❜❡ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✐♥ t❤❡ ♣r♦❜❧❡♠✳ ❚❤❡♥ t❤❡r❡ ❡①✐st ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s φ ( t ) , ψ ( t ) s✉❝❤ t❤❛t ˙ − ∂ H φ = ∂ s = s , ˙ − ∂ H ψ = ∂ v = − φ, u = + ✶ ❢♦r ψ > ✵ ❛♥❞ u = − ✶ ❢♦r ψ < ✵ ✳ � � ■❢ ψ = ✵ ❢♦r t ∈ ( t ✵ , t ✶ ) t❤❡♥ ❛♥ ❡①tr❡♠❛❧ ( s ( t ) , v ( t ) , φ ( t ) , ψ ( t )) , t ∈ ( t ✵ , t ✶ ) , ✐s ❛ s✐♥❣✉❧❛r ♦♥❡✳

❙✐♥❣✉❧❛r ❈♦♥tr♦❧ ❉❡♥♦t❡ z = ( s , v , φ, ψ ) ✳ ❲❡ ❤❛✈❡✿ d H ✶ ( z ( t )) = ψ ( t ) ≡ ✵ , dt H ✶ ( z ( t )) = ✵ ⇒ − φ ( t ) = ✵ d ✷ dt ✷ H ✶ ( z ( t )) = ✵ ⇒ − s ( t ) = ✵ , d ✸ dt ✸ H ✶ ( z ( t )) = ✵ ⇒ − v ( t ) = ✵ , d ✹ dt ✹ H ✶ ( z ( t )) = ✵ ⇒ − u ( t ) = ✵ . ✭✺✮ ❚❤❡ s✐♥❣✉❧❛r ❡①tr❡♠❛❧ ✐♥ t❤❡ ❋✉❧❧❡r ♣r♦❜❧❡♠ s = ✵ , v = ✵ ✳

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠ M ✐s t❤❡ ❝❛rt ♠❛ss✱ s ✐s t❤❡ ❝❛rt ♣♦s✐t✐♦♥✱ g ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❣r❛✈✐t②✱ u ✐s t❤❡ ❢♦r❝❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❝❛rt✱ γ i ✐s t❤❡ ❛♥❣❧❡ ♦❢ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ i t❤ ❧✐♥❦ ❢r♦♠ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡✱ m i ✐s t❤❡ ♠❛ss ♦❢ t❤❡ i t❤ ❧✐♥❦✱ r i ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦✇❡r ❡♥❞ ♦❢ t❤❡ i t❤ ❧✐♥❦ t♦ ✐ts ❝❡♥t❡r ♦❢ ♠❛ss✱ I i ✐s t❤❡ ♠♦♠❡♥t ♦❢ ✐♥❡rt✐❛ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ i t❤ ❧✐♥❦✱ ❛♥❞ l i ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ i t❤ ❧✐♥❦ ( i = ✶ , . . . , n ) ✳

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠✳ ▼♦t✐♦♥ ❡q✉❛t✐♦♥s ❚❤❡ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ❛r❡ n n � � ✷ s✐♥ γ i = u a ✶✶ ¨ s + a ✶ , i + ✶ ¨ γ i ❝♦s γ i − a ✶ , i + ✶ ˙ γ i i = ✶ i = ✶ n � a ✶ , i + ✶ ¨ s ❝♦s γ i + a i + ✶ , i + ✶ ¨ γ i + a i + ✶ , j + ✶ ¨ γ j ❝♦s ( γ i + γ j ) − ✭✻✮ j = ✶ � n γ ✷ − a i + ✶ , j + ✶ ˙ j s✐♥ ( γ i + γ j ) − b i s✐♥ γ i = ✵ , i = ✶ , . . . , n . j = ✶

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ s②st❡♠ ✐s ✐♥ ❛ s✉❝✐❡♥t❧② s♠❛❧❧ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ✉♣♣❡r ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥ γ ✶ = ˙ γ ✶ = · · · = γ n = ˙ γ n ≡ ✵ . ✭✼✮ ❲❡ st✉❞② t❤❡ ♣r♦❜❧❡♠ ♦❢ st❛❜✐❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡♥❞✉❧✉♠ ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ♣♦s✐t✐♦♥ ✭✼✮ ✐♥ t❤❡ s❡♥s❡ ♦❢ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥❛❧ ∞ ˆ � γ, γ � dt → ♠✐♥ , ✭✽✮ ✵

▲✐♥❡❛r✐③❡❞ ♠♦❞❡❧✳ ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ∞ ˆ � Kx ( t ) , x ( t ) � dt → ♠✐♥ ✭✾✮ ✵ ♦♥ t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ t❤❡ s②st❡♠ x ( t ) − Λ x ( t ) = Iu ( t ) ¨ ✭✶✵✮ ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s x ( ✵ ) = x ✵ , x ( ✵ ) = y ✵ . ˙ ✭✶✶✮ ❍❡r❡✱ t❤❡ ❝♦♥tr♦❧ u ( t ) ✐s ❛ ❜♦✉♥❞❡❞ s❝❛❧❛r ❢✉♥❝t✐♦♥✿ | u ( t ) | ≤ ✶ , ✭✶✷✮