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■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s ✐♥ ❈♦♥tr♦❧ Pr♦❜❧❡♠s ✇✐t❤ ❙t❛t❡ ❈♦♥str❛✐♥ts ❆✳❱✳ ❆r✉t②✉♥♦✈ P❛❞♦✈❛✱ ❙❡♣t❡♠❜❡r✱ ✷✵✶✼ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✇✐t❤ ✐♥❡q✉❛❧✐t② st❛t❡ ❝♦♥str❛✐♥ts✳ ▼✐♥✐♠✐③❡ ϕ ( p ) , s✉❜❥❡❝t t♦ x = f ( x , u , t ) , ˙ p = ( x ✵ , x ✶ ) ∈ S , ✭✶✮ u ( t ) ∈ U ❛✳❛✳ t ∈ [ t ✵ , t ✶ ] , g ( x ( t ) , t ) ≤ ✵ ∀ t ∈ [ t ✵ , t ✶ ] . x = dx ❍❡r❡✱ ˙ dt ✱ t ∈ [ ✵ , ✶ ] ❞❡s✐❣♥❛t❡s t✐♠❡✱ x ✐s t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ ✇❤✐❝❤ t❛❦❡s ✈❛❧✉❡s ✐♥ R n ❀ x ✵ = x ( t ✵ ) ✱ x ✶ = x ( t ✶ ) ❀ ✈❡❝t♦r u ∈ U ⊂ R m ✐s t❤❡ ❝♦♥tr♦❧ ♣❛r❛♠❡t❡r✳ ❚❤❡ s❡ts S ✱ U ❛r❡ ❝❧♦s❡❞✳ ❱❡❝t♦r p = ( x ✵ , x ✶ ) ✐s t❡r♠❡❞ ❡♥❞♣♦✐♥t✳ ❆ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ u : [ t ✵ , t ✶ ] → U ✐s t❡r♠❡❞ ❝♦♥tr♦❧ ✭♦r✱ ❝♦♥tr♦❧ ❢✉♥❝t✐♦♥✮✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❉❡✜♥✐t✐♦♥ ❚❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss ( x ∗ ( t ) , u ∗ ( t )) ♦❢ ✭✶✮ s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ♣r♦✈✐❞❡❞ t❤❛t t❤❡r❡ ❡①✐st ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs✿ ❛ ♥✉♠❜❡r λ ∈ [ ✵ , ✶ ] ✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ψ : [ t ✵ , t ✶ ] → R n ✱ ❛♥❞ ❛ ❇♦r❡❧ ♠❡❛s✉r❡ η ∈ C ∗ ([ t ✵ , t ✶ ]) ✱ η ≥ ✵✱ s✉❝❤ t❤❛t λ + sup | ψ ( t ) | = ✶ , t ∈ [ t ✵ , t ✶ ] d ψ ( t ) = − H ′ x ( x ∗ ( t ) , u ∗ ( t ) , ψ ( t ) , t ) dt + g ′ x ( x ∗ ( t ) , t ) d η, t ∈ [ t ✵ , t ✶ ] , ( ψ ( t ✵ ) , − ψ ( t ✶ )) ∈ λϕ ′ ( p ∗ ) + N S ( p ∗ ) , max u ∈ U H ( x ∗ ( t ) , u , ψ ( t ) , t ) = H ( x ∗ ( t ) , u ∗ ( t ) , ψ ( t ) , t ) ❛✳❛✳ t ∈ [ t ✵ , t ✶ ] , � � � g ( x ∗ ( t ) , t ) , d η = ✵ . [ t ✵ , t ✶ ] ❍❡r❡✱ H ( x , u , ψ, t ) := � ψ, f ( x , u , t ) � ✱ p ∗ = ( x ∗ ( t ✵ ) , x ∗ ( t ✶ )) ✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❆✳❨❛ ❉✉❜♦✈✐ts❦✐✐ ❛♥❞ ❆✳❆✳ ▼✐❧②✉t✐♥ ♣r♦♣♦s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss ( x ∗ ( t ) , u ∗ ( t )) ✐s ♦♣t✐♠❛❧ t♦ ♣r♦❜❧❡♠ ✭✶✮✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss ( x ∗ ( t ) , u ∗ ( t )) s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❉❡❣❡♥❡r❛❝② ♦❢ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❙✉♣♣♦s❡ t❤❛t ♦♥❡ ♦❢ t❤❡ ❡♥❞♣♦✐♥ts✱ ❡✐t❤❡r x ✵ ✱ ♦r x ✶ ✱ ✐s ✜①❡❞✳ ■♥ t❤✐s ❝❛s❡✱ ❛s ✐s ❡❛s② t♦ ✈❡r✐❢②✱ t❤❡ ❉✉❜♦✈✐ts❦✐✐✲▼✐❧②✉t✐♥ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❝❛♥ ❜❡ s❛t✐s✜❡❞ ❜② ❛♥② ❢❡❛s✐❜❧❡ ♣❛✐r ❝♦♥tr♦❧✴tr❛❥❡❝t♦r②✳ ■♥❞❡❡❞✱ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ t❤✐s ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ s❡t ♦❢ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ✭✐♥ ❝❛s❡ t❤❡ ❧❡❢t ❡♥❞♣♦✐♥t ✐s ✜①❡❞✮✿ λ = ✵ , η = δ ( t ✵ ) , ψ ( · ) : ψ ( t ) = ✵ ∀ t ∈ ( t ✵ , t ✶ ] , ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❡rs λ = ✵ , η = δ ( t ✶ ) , ψ ( · ) : ψ ( t ) = ✵ ∀ t ∈ [ t ✵ , t ✶ ) , ✐❢ t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ✐s ✜①❡❞✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❉❡❣❡♥❡r❛❝② ♦❢ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❆✳❨❛✳ ❉✉❜♦✈✐ts❦✐✐ ❛♥❞ ❱✳❆✳ ❉✉❜♦✈✐ts❦✐✐ ❢♦✉♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡ ✭✶✾✽✺✮✳ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r n = m = k = ✶✱ t ✵ = ✵✱ t ✶ = ✶✱ � ✶  u ( t ) dt → min ,     ✵   x = tu , ˙  x ✵ = ✵ ,   u ( t ) ∈ [ − ✶ , ✶ ] ,     x ( t ) ≥ ✵ .  ❚❤❡ ♦♣t✐♠❛❧ ♣r♦❝❡ss ✐s x = u = ✵✱ ❜✉t t❤❡r❡ ❛r❡ ♦♥❧② ❞❡❣❡♥❡r❛t❡ ♠✉❧t✐♣❧✐❡rs s❛t✐s❢②✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ✐t✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝② ❉❡✜♥✐t✐♦♥ ❚❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ s❛✐❞ t♦ ❜❡ r❡❣✉❧❛r ♣r♦✈✐❞❡❞ t❤❛t ❢♦r ❛❧❧ x , t : g ( x , t ) ≤ ✵ t❤❡r❡ ❡①✐sts z = z ( x , t ) s✉❝❤ t❤❛t � ∂ g j � ∂ x ( x , t ) , z > ✵ ∀ j ∈ J ( x , t ) . ❍❡r❡✱ J ( x , t ) := { j : g j ( x , t ) = ✵ } ✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝② ❉❡✜♥✐t✐♦♥ ❚❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ s❛✐❞ t♦ ❜❡ ❝♦♠♣❛t✐❜❧❡ ❛t p ∗ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥t ❝♦♥str❛✐♥ts ♣r♦✈✐❞❡❞ t❤❛t ∃ ε > ✵ : { p ∈ R ✷ n : | p ∗ − p | ≤ ε, p ∈ S } ⊆ { p ∈ R ✷ n : g ( x ✵ , t ✵ ) ≤ ✵ , g ( x ✶ , t ✶ ) ≤ ✵ } . ❚❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ♦❢ ❝♦♥str❛✐♥ts ✐s ♥♦t ❛♥ ❡①tr❛ r❡q✉✐r❡♠❡♥t✳ ■t ❝❛♥ ❛❧✇❛②s ❜❡ ❛❝❤✐❡✈❡❞ ❜② r❡♣❧❛❝✐♥❣ t❤❡ s❡t S ✇✐t❤ t❤❡ s❡t S ∩ { p ∈ R ✷ n : g ( x ✵ , t ✵ ) ≤ ✵ , g ( x ✶ , t ✶ ) ≤ ✵ } . ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝② ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ Γ( x , u , t ) = g ′ x ( x , t ) f ( x , u , t ) + g ′ t ( x , t ) . ❉❡✜♥✐t✐♦♥ ❆ ❢❡❛s✐❜❧❡ ❛r❝ x ∗ ( t ) ✐s s❛✐❞ t♦ ❜❡ ❝♦♥tr♦❧❧❛❜❧❡ ❛t t❤❡ ❡♥❞♣♦✐♥ts ✇✳r✳t✳ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts✱ ♣r♦✈✐❞❡❞ t❤❛t t❤❡r❡ ❡①✐st ✈❡❝t♦rs γ r ∈ ❝♦♥✈ Γ( x r , U , r ) s✉❝❤ t❤❛t✿ ( − ✶ ) r γ j r < ✵ ∀ j ∈ J ( x r , r ) , r = ✵ , ✶ . ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

◆♦♥✲❞❡❣❡♥❡r❛t❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss ( x ∗ ( t ) , u ∗ ( t )) ✐s ♦♣t✐♠❛❧ ✐♥ ♣r♦❜❧❡♠ ✭✶✮✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ r❡❣✉❧❛r ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥ts ❝♦♥str❛✐♥ts ❛t p ∗ ✱ ❛♥❞ t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss ( x ∗ ( t ) , u ∗ ( t )) s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✱ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ str❡♥❣t❤❡♥❡❞ ♥♦♥✲tr✐✈✐❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ✈❛❧✐❞ λ + ℓ ( t ∈ [ t ✵ , t ✶ ] : | ψ ( t ) | � = ✵ ) > ✵ , ✇❤❡r❡ ℓ st❛♥❞s ❢♦r t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳