A nonsmooth Chow-Rashevski’s theorem Ermal Feleqi University of Vlora, Albania Optimization, State Constraints and Geometric Control Padova, May 25, 2018 Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 1 / 18
References F. Rampazzo & H. Sussmann, Set-valued di ff erentials and a 1. nonsmooth version of Chow-Rashevski’s theorem , Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, December 2001. F. Rampazzo and H. Sussmann, Commutators of flow maps of 2. nonsmooth vector fields , J. Di ff erential Equations 2007 Sketch of results in E. Feleqi & F. Rampazzo, Integral representations for 1. bracket-generating multi-flows , Discrete Contin. Dyn. Syst. Ser. A., 2015. E. Feleqi & F. Rampazzo, Iterated Lie brackets for nonsmooth 2. vector fields , NoDEA - Nonlinear Di ff erential Equations Appl., 2017. E. Feleqi & F. Rampazzo, An L ∞ -Chow-Rashevski’s Theorem , work 3. in progress. Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 2 / 18
Controllability Given X = ( X 1 , . . . , X p ) vector fields . on some open set Ω ⊂ R n . X − trajectory : = concatenation of a finite no. of integral curves of X 1 , . . . , X p , − X 1 , . . . , − X p . Definition X controllable in Ω if ∀ x , y ∈ Ω ∃X − trajectory ξ : [ t 1 , t 2 ] → Ω s.t. ξ ( t 1 ) = x , ξ ( t 2 ) = y . Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 3 / 18
Lie bracket Definition Given two vector fields X , Y [ X , Y ] = XY − YX ≡ DY · X − DX · Y . Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 4 / 18
Lie bracket Definition Given two vector fields X , Y [ X , Y ] = XY − YX ≡ DY · X − DX · Y . Main fact needed here e − tY ◦ e − tX ◦ e tY ◦ e tX ( x ) = x + t 2 [ X , Y ]( x ∗ ) + o ( t 2 ) as ( t , x ) → (0 , x ∗ ). Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 4 / 18
Iterated Lie brackets Iterated brackets of a family X 1 , . . . , X p of vector fields: degree 1 X 1 , . . . , X p Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 5 / 18
Iterated Lie brackets Iterated brackets of a family X 1 , . . . , X p of vector fields: degree 1 X 1 , . . . , X p degree 2 ( L ie Bracket or Commutator ) [ X i , X j ] : = X i X j − X j X i ≡ ∇ X j X i − ∇ X i X j Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 5 / 18
Iterated Lie brackets Iterated brackets of a family X 1 , . . . , X p of vector fields: degree 1 X 1 , . . . , X p degree 2 ( L ie Bracket or Commutator ) [ X i , X j ] : = X i X j − X j X i ≡ ∇ X j X i − ∇ X i X j degree 3 � � [ X i , X j ] , X k Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 5 / 18
Iterated Lie brackets Iterated brackets of a family X 1 , . . . , X p of vector fields: degree 1 X 1 , . . . , X p degree 2 ( L ie Bracket or Commutator ) [ X i , X j ] : = X i X j − X j X i ≡ ∇ X j X i − ∇ X i X j degree 3 � � [ X i , X j ] , X k degree 4 [[[ X i , X j ] , X k ] , X ℓ ] . . . [[ X i , X j ] , [ X k , X ℓ ]] Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 5 / 18
Iterated Lie brackets Iterated brackets of a family X 1 , . . . , X p of vector fields: degree 1 X 1 , . . . , X p degree 2 ( L ie Bracket or Commutator ) [ X i , X j ] : = X i X j − X j X i ≡ ∇ X j X i − ∇ X i X j degree 3 � � [ X i , X j ] , X k degree 4 [[[ X i , X j ] , X k ] , X ℓ ] . . . [[ X i , X j ] , [ X k , X ℓ ]] et cetera.... Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 5 / 18
LARC, Chow-Rashevski’s theorem and other deep results Theorem Assume X 1 , . . . , X p satisfy L ie A lgebra R ank C ondition or H örmander’s C ondition or, that is, � � = R n . span iterated Lie brackets at x (LARC) Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 6 / 18
LARC, Chow-Rashevski’s theorem and other deep results Theorem Assume X 1 , . . . , X p satisfy L ie A lgebra R ank C ondition or H örmander’s C ondition or, that is, � � = R n . span iterated Lie brackets at x (LARC) Then (Chow-Rashevski) Any two points can be connected by an 1 . X -trajectory: Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 6 / 18
LARC, Chow-Rashevski’s theorem and other deep results Theorem Assume X 1 , . . . , X p satisfy L ie A lgebra R ank C ondition or H örmander’s C ondition or, that is, � � = R n . span iterated Lie brackets at x (LARC) Then (Chow-Rashevski) Any two points can be connected by an 1 . X -trajectory: T ( y , x ) ≤ C | y − x | 1 / k Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 6 / 18
LARC, Chow-Rashevski’s theorem and other deep results Theorem Assume X 1 , . . . , X p satisfy L ie A lgebra R ank C ondition or H örmander’s C ondition or, that is, � � = R n . span iterated Lie brackets at x (LARC) Then (Chow-Rashevski) Any two points can be connected by an 1 . X -trajectory: T ( y , x ) ≤ C | y − x | 1 / k (Hörmander) 2 . p � X 2 L = is hypoelliptic j j = 1 (Bony) L satisfies the strong maximum principle. 3 . Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 6 / 18
A set-valued-bracket (Franco and Hector) If X 1 , X 2 are C 0 , 1 ,we set � � [ X 1 , X 2 ] set ( x ) : = co v = lim j →∞ [ X 1 , X 2 ]( x j ) , where 1. x j ∈ D i ff ( X 1 ) ∩ D i ff ( X 2 ) for all j , 2. lim j →∞ x j = x Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 7 / 18
A set-valued-bracket (Franco and Hector) If X 1 , X 2 are C 0 , 1 ,we set � � [ X 1 , X 2 ] set ( x ) : = co v = lim j →∞ [ X 1 , X 2 ]( x j ) , where 1. x j ∈ D i ff ( X 1 ) ∩ D i ff ( X 2 ) for all j , 2. lim j →∞ x j = x Properties: x �→ [ X 1 , X 2 ] set ( x ) u. s.c., comp. convex valued; robust Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 7 / 18
A set-valued-bracket (Franco and Hector) If X 1 , X 2 are C 0 , 1 ,we set � � [ X 1 , X 2 ] set ( x ) : = co v = lim j →∞ [ X 1 , X 2 ]( x j ) , where 1. x j ∈ D i ff ( X 1 ) ∩ D i ff ( X 2 ) for all j , 2. lim j →∞ x j = x Properties: x �→ [ X 1 , X 2 ] set ( x ) u. s.c., comp. convex valued; robust Applications commutativity, simultaneous rectification, asymptotic formulas, Chow-Rashevski type theorem. (H. Sussmann, F. Rampazzo, 2001, 2007). Frobenius type thm (F. Rampazzo 2007). Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 7 / 18
A set-valued-bracket (Franco and Hector) If X 1 , X 2 are C 0 , 1 ,we set � � [ X 1 , X 2 ] set ( x ) : = co v = lim j →∞ [ X 1 , X 2 ]( x j ) , where 1. x j ∈ D i ff ( X 1 ) ∩ D i ff ( X 2 ) for all j , 2. lim j →∞ x j = x Properties: x �→ [ X 1 , X 2 ] set ( x ) u. s.c., comp. convex valued; robust Applications commutativity, simultaneous rectification, asymptotic formulas, Chow-Rashevski type theorem. (H. Sussmann, F. Rampazzo, 2001, 2007). Frobenius type thm (F. Rampazzo 2007). Asymptotic formula: As | t | + | x − x ∗ | → 0, e − tX 2 ◦ e − tX 1 ◦ e tX 2 ◦ e tX 1 ( x ) − x ∈ t 2 [ X 1 , X 2 ]( x ∗ ) + t 2 o (1) Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 7 / 18
Higher-order set-valued brackets If X 1 , X 2 are C 1 , 1 and X 3 is C 0 , 1 , we set [[ X 1 , X 2 ] , X 3 ] set ( x ) � � : = co v = lim j →∞ DX 3 ( y j ) · [ X 1 , X 2 ]( x j ) − D [ X 1 , X 2 ]( x j ) · X 3 ( y j ) , where 1. x j ∈ D i ff ( DX 1 ) ∩ D i ff ( DX 2 ) ∀ j , y j ∈ D i ff ( X 3 ) ∀ j , 2. lim j →∞ ( x j , y j ) = ( x , x ). Properties: Chart-invariant, robust, u.s.c. with comp, conv values Asymptotic formula: As | t | + | x − x ∗ | → 0 e − tX 3 ◦ e − tX 1 ◦ e − tX 2 ◦ e tX 1 ◦ e tX 2 ◦ e tX 3 ◦ e − tX 2 ◦ e − tX 1 ◦ e tX 2 ◦ e tX 1 ( x ) − x � ������������������������� �� ������������������������� � � ������������������������� �� ������������������������� � Ψ − 1 Ψ ∈ t 3 � X 1 , [ X 2 , X 3 ] � ( x ∗ ) + t 3 o (1) . Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 8 / 18
Theorem (A generalization of Chow-Rashevski’s theorem) Assume ∃ iterated brackets B 1 , . . . , B r , possibly set-valued, of the vector fields X 1 , . . . , X p s.t. at x ∗ � � = R n span v 1 , . . . , v r ∀ v 1 ∈ B 1 , . . . v r ∈ B r . (GHC) Ermal Feleqi (University of Vlora, Albania) A Chow-Rashvski type theorem Padova, May 25, 2018 9 / 18
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