Lecture 2: Existence, uniqueness, and regularity in the Lipschitz - PowerPoint PPT Presentation

Lecture 2: Existence, uniqueness, and regularity in the Lipschitz case Habib Ammari Department of Mathematics, ETH Z¨ urich Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Banach fixed point theorem • DEFINITION: Contraction Let • ( X , d ): metric space. • F : X → X : contraction if there exists 0 < λ < 1 s.t. for all x , y ∈ X d ( F ( x ) , F ( y )) ≤ λ d ( x , y ) . • THEOREM: Banach fixed point theorem • ( X , d ): complete metric space (i.e., every Cauchy sequence of elements of X : convergent); • F : X → X : contraction . • There exists a unique x ∈ X s.t. F ( x ) = x . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Gronwall’s lemma LEMMA: Gronwall’s lemma • I = [0 , T ]; φ ∈ C 0 ( I ). • There exist two constants α, β ∈ R , β ≥ 0, s.t. � t ( ∗ ) φ ( t ) ≤ α + β φ ( s ) d s for all t ∈ I . 0 • ⇒ φ ( t ) ≤ α e β t for all t ∈ I . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • PROOF: • ϕ : I → R � t ϕ ( t ) := α + β φ ( s ) d s . 0 • φ ∈ C 0 ⇒ ϕ ∈ C 1 , d ϕ d t = βφ ( t ) for all t ∈ I . • ( ∗ ) ⇒ d ϕ d t ≤ βϕ. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • ψ ( t ) := exp( − β t ) ϕ ( t ) for t ∈ I , d ψ d t = − β e − β t ϕ ( t ) + e − β t d ϕ d t � � − βϕ ( t ) + d ϕ = e − β t ≤ 0 . d t • ψ (0) = ϕ (0) = α ⇒ ψ ( t ) ≤ α for t ∈ I , ϕ ( t ) ≤ α e β t ; • ⇒ φ ( t ) ≤ ϕ ( t ) ≤ α e β t for all t ∈ I . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Cauchy-Lipschitz theorem • I = [0 , T ]; d : positive integer; f : I × R d → R d . • Suppose that f ∈ C 0 ( I × R d ). • DEFINITION: Lipschitz condition • There exists a constant C f ≥ 0 s.t., for any x 1 , x 2 ∈ R d and any t ∈ I , ( ∗∗ ) | f ( t , x 1 ) − f ( t , x 2 ) | ≤ C f | x 1 − x 2 | . • f satisfies a Lipschitz condition on I . • C f : Lipschitz constant for f . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • THEORM: Cauchy-Lipschitz theorem • Consider  d x d t = f ( t , x ) , t ∈ [0 , T ] ,  x 0 ∈ R d . x (0) = x 0 ,  • If f ∈ C 0 ( I × R d ) satisfies the Lipschitz condition ( ∗∗ ) on [0 , T ], then there exists a unique solution x ∈ C 1 ( I ) on [0 , T ]. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • PROOF: • � t x ( t ) = x 0 + f ( s , x ( s )) d s , ∀ t ∈ [0 , T ] . 0 • Define F : C 0 ([0 , T ]; R d ) → C 0 ([0 , T ]; R d ) by � t F ( y ) := x 0 + f ( s , y ( s )) d s . 0 • For y ∈ C 0 ([0 , T ]; R d ), norm of y : {| y ( t ) | e − C f t } ; � y � := sup t ∈ [0 , T ] • C f : Lipschitz constant for f . | y ( t ) | ⇒ C 0 ([0 , T ]; R d ) • Equivalent to the usual norm sup t ∈ [0 , T ] equipped with the new norm: complete. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Compute | F [ y 1 ]( t ) − F [ y 2 ]( t ) | e − C f t � F [ y 1 ] − F [ y 2 ] � = sup t ∈ [0 , T ] � t e − C f t ≤ sup | f ( s , y 1 ( s )) − f ( s , y 2 ( s )) | d s t ∈ [0 , T ] 0 � t e − C f t C f ≤ sup | y 1 ( s ) − y 2 ( s ) | d s t ∈ [0 , T ] 0 � t e − C f t C f e C f s e − C f s | y 1 ( s ) − y 2 ( s ) | d s ≤ sup t ∈ [0 , T ] 0 � t { e − C f t C f e C f s d s }� y 1 − y 2 � ≤ sup t ∈ [0 , T ] 0 ≤ (1 − e − C f T ) � y 1 − y 2 � . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Banach fixed point theorem in a complete metric space ⇒ there exists a unique y ∈ C 0 ([0 , T ]; R d ) s.t. F ( y ) = y . • ⇒ Existence and uniqueness of a solution. • Picard iteration y ( n +1) = F [ y ( n ) ] : Cauchy sequence and converges to the unique fixed point y . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • REMARK: • Existence and uniqueness theorem: holds true if R d : replaced with a Banach space (a complete normed vector space). • Same proof. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • REMARK: • If f : continuous, there is no guarantee that the initial value problem possesses a unique solution. • EXAMPLE: • Consider d x 2 3 , d t = x x (0) = 0 . • There are two solutions given by x 1 ( t ) = t 3 27 and x 2 ( t ) = 0. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • THEOREM: Cauchy-Peano existence theorem • f : continuous. • There exists a solution x ( t ): at least defined for small t . • PROOF: Use Arzela-Ascoli theorem. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • DEFINITION: Equicontinuity • A family of functions F : equicontinuous on [ a , b ] if for any given ǫ > 0, there exists δ > 0 s.t. | f ( t ) − f ( s ) | < ǫ whenever | t − s | < δ for every function f ∈ F and t , s ∈ [ a , b ]. • DEFINITION: Uniform boundedness • A family of continuous functions F on [ a , b ]: uniformly bounded if there exists a positive number M s.t. | f ( t ) | ≤ M for every function f ∈ F and t ∈ [ a , b ]. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • THEOREM: Arzela-Ascoli • Suppose that the sequence of functions { f n ( t ) } n ∈ N on [ a , b ]: uniformly bounded and equicontinuous. • There exists a subsequence { f n k ( t ) } k ∈ N : uniformly convergent on [ a , b ]. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • EXAMPLE: • Consider d x d t = x 2 , x (0) = x 0 � = 0 . • Separation of variables ⇒ d x x 2 = d t . • ⇒ � d x − 1 x = x 2 = t + C , • ⇒ 1 x = − t + C . • x (0) = x 0 ⇒ x 0 x ( t ) = 1 − x 0 t . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • If x 0 > 0, x ( t ) blows up when t → 1 x 0 from below. • If x 0 < 0, the singularity: in the past ( t < 0). • Only solution defined for all positive and negative t : constant solution x ( t ) = 0, corresponding to x 0 = 0. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Continuity of the solution. • THEOREM: • f satisfies the Lipschitz condition. • x 1 ( t ) and x 2 ( t ): two solutions of corresponding to the initial data x 1 (0) and x 2 (0), respectively. • Continuity with respect to the initial data: | x 1 ( t ) − x 2 ( t ) | ≤ e C f t | x 1 (0) − x 2 (0) | for all t ∈ [0 , T ] . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • PROOF: • d d t | x 1 ( t ) − x 2 ( t ) | 2 = 2( f ( t , x 1 ( t )) − f ( t , x 2 ( t )))( x 1 ( t ) − x 2 ( t )) ≤ 2 C f | x 1 ( t ) − x 2 ( t ) | 2 , t ∈ [0 , T ] , • ⇒ � � d | x 1 ( t ) − x 2 ( t ) | 2 e − 2 C f t ≤ 0 . d t • Integration from 0 to t : | x 1 ( t ) − x 2 ( t ) | 2 e − 2 C f t ≤ | x 1 (0) − x 2 (0) | 2 . • ⇒ | x 1 ( t ) − x 2 ( t ) | ≤ | x 1 (0) − x 2 (0) | e C f t . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Differentiability with respect to the initial data. • Formal: differentiate the solution x with respect to the initial data ⇒  ∂ x ( t ) ∂ x ( t , x ( t )) ∂ x ( t ) d = ∂ f ∂ x 0 ,   d t ∂ x 0  ( ∗ ∗ ∗ ) ∂ x ( t )  = 1 .   ∂ x 0 • THEOREM: • f ∈ C 1 . • x 0 �→ x ( t ): differentiable and ∂ x ( t ) /∂ x 0 : unique solution of the linear equation ( ∗ ∗ ∗ ). Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • PROOF: • ∆ x ( t , x 0 , h ) := x ( t , x 0 + h ) − x ( t , x 0 ): difference quotient. • Mean-value theorem ⇒ � t ∆ x ( t , x 0 , h ) = h + ( f ( s , x ( s , x 0 + h )) − f ( s , x ( s , x 0 ))) d s 0 � t = h + ( f ( s , x ( t , x 0 ) + ∆ x ( s , x 0 , h )) − f ( s , x ( s , x 0 ))) d s 0 � t ∂ f = h + ∂ x ( s , x ( s , x 0 ) + τ ∆ x )∆ x d s . 0 • τ = τ ( s , x 0 , h ) ∈ [0 , 1]. Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • There exists a positive constant M s.t. | ∂ f ∂ x | < M ⇒ � t | ∆ x | ≤ | h | + M | ∆ x ( s , x 0 , h ) | d s . 0 • Gronwall’s lemma ⇒ | ∆ x ( t , x 0 , h ) | ≤ | h | e Mt . Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • v ( t ): unique solution of ( ∗ ∗ ∗ ). • Compute ∆ x ( t , x 0 , h ) − v ( t ) h � t � f ( s , x ( s , x 0 + h )) − f ( s , x ( s , x 0 )) � − ∂ f = ∂ x ( s , x ( s , x 0 )) v ( s ) d s h 0 � t ∆ x ( s , x 0 , h ) � ∂ f � ∂ x ( s , x ( s , x 0 ) + τ ∆ x ( s , x 0 , h )) − ∂ f = ∂ x ( s , x ( s , x 0 )) d s h 0 � t ∂ f � ∆ x ( s , x 0 , h ) � + ∂ x ( s , x ( s , x 0 )) − v ( s ) d s . h 0 Numerical methods for ODEs Habib Ammari

Existence, uniqueness, and regularity • Uniform continuity of ∂ f ∂ x ⇒ For any ǫ > 0 there exists h 0 > 0 s.t., for any | h | ≤ h 0 , the first term on the right-hand side: of order O ( ǫ ). • Gronwall’s lemma ⇒ for | h | small enough, | ∆ x ( t , x 0 , h ) − v | ≤ ǫ MTe MT . h • ⇒ x 0 �→ x ( t ): differentiable and its derivative given by ∂ x ∂ x 0 = v . Numerical methods for ODEs Habib Ammari