Controllability and stability of difference equations and - PowerPoint PPT Presentation

Controllability and stability of difference equations and applications Guilherme Mazanti Nonlinear Partial Differential Equations and Applications A conference in the honor of Jean-Michel Coron for his 60 th birthday Paris – June 20 th , 2016 CMAP, École Polytechnique Team GECO, Inria Saclay France

Introduction Stability analysis and applications Relative controllability Outline Introduction 1 Linear difference equations Motivation: hyperbolic PDEs Motivation: previous results Stability analysis and applications 2 Stability analysis Technique of the proof Applications Relative controllability 3 Definition Explicit formula Relative controllability criterion Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Linear difference equations 1 Stability analysis of the difference equation N � Σ stab : x ( t ) = A j ( t ) x ( t − Λ j ) , t ≥ 0 . j =1 2 Relative controllability of the difference equation N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Linear difference equations 1 Stability analysis of the difference equation N � Σ stab : x ( t ) = A j ( t ) x ( t − Λ j ) , t ≥ 0 . j =1 2 Relative controllability of the difference equation N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Λ 1 , . . . , Λ N : ( rationally independent ) positive delays (Λ min = min j Λ j , Λ max = max j Λ j ). x ( t ) ∈ C d , u ( t ) ∈ C m . Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Linear difference equations 1 Stability analysis of the difference equation N � Σ stab : x ( t ) = A j ( t ) x ( t − Λ j ) , t ≥ 0 . j =1 2 Relative controllability of the difference equation N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Λ 1 , . . . , Λ N : ( rationally independent ) positive delays (Λ min = min j Λ j , Λ max = max j Λ j ). x ( t ) ∈ C d , u ( t ) ∈ C m . Motivation: Applications to some hyperbolic PDEs. Generalization of previous results. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Hyperbolic PDEs → difference equations: [Cooke, Krumme, 1968], [Slemrod, 1971], [Greenberg, Li, 1984], [Coron, Bastin, d’Andréa Novel, 2008], [Fridman, Mondié, Saldivar, 2010], [Gugat, Sigalotti, 2010]... Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Hyperbolic PDEs → difference equations: [Cooke, Krumme, 1968], [Slemrod, 1971], [Greenberg, Li, 1984], [Coron, Bastin, d’Andréa Novel, 2008], [Fridman, Mondié, Saldivar, 2010], [Gugat, Sigalotti, 2010]...  ∂ t u i ( t , ξ ) + ∂ ξ u i ( t , ξ ) + α i ( t , ξ ) u i ( t , ξ ) = 0 ,      t ∈ R + , ξ ∈ [0 , Λ i ] , i ∈ � 1 , N � ,  N  �   u i ( t , 0) = m ij ( t ) u j ( t , Λ j ) , t ∈ R + , i ∈ � 1 , N � .    j =1 Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Hyperbolic PDEs → difference equations: [Cooke, Krumme, 1968], [Slemrod, 1971], [Greenberg, Li, 1984], [Coron, Bastin, d’Andréa Novel, 2008], [Fridman, Mondié, Saldivar, 2010], [Gugat, Sigalotti, 2010]...  ∂ t u i ( t , ξ ) + ∂ ξ u i ( t , ξ ) + α i ( t , ξ ) u i ( t , ξ ) = 0 ,      t ∈ R + , ξ ∈ [0 , Λ i ] , i ∈ � 1 , N � ,  N  �   u i ( t , 0) = m ij ( t ) u j ( t , Λ j ) , t ∈ R + , i ∈ � 1 , N � .    j =1 Method of characteristics: for t ≥ Λ max , N N � Λ j � � α j ( t − s , Λ j − s ) ds u j ( t − Λ j , 0) . m ij ( t ) e − u i ( t , 0) = m ij ( t ) u j ( t , Λ j ) = 0 j =1 j =1 Set x ( t ) = ( u i ( t , 0)) i ∈ � 1 , N � . Then x satisfies a difference equation. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Λ 1 Edges: E Vertices: V Λ 3 Λ N Λ 2 ∂ 2 tt u i ( t , ξ ) = ∂ 2 ξξ u i ( t , ξ ) u i ( t , q ) = u j ( t , q ) , ∀ q ∈ V , ∀ i , j ∈ E q + conditions on vertices. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs D’Alembert decomposition on travelling waves: Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: hyperbolic PDEs D’Alembert decomposition on travelling waves: System of 2 N transport equations. Can be reduced to a system of difference equations. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: previous stability results (cf. [Cruz, Hale, 1970], [Henry, 1974], [Michiels et al., 2009]) N � Σ aut stab : x ( t ) = A j x ( t − Λ j ) , t ≥ 0 . j =1 Stability for rationally independent Λ 1 , . . . , Λ N characterized by �� N � j =1 A j e i θ j ρ HS ( A ) = ( θ 1 ,...,θ N ) ∈ [0 , 2 π ] N ρ max . Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: previous stability results (cf. [Cruz, Hale, 1970], [Henry, 1974], [Michiels et al., 2009]) N � Σ aut stab : x ( t ) = A j x ( t − Λ j ) , t ≥ 0 . j =1 Stability for rationally independent Λ 1 , . . . , Λ N characterized by �� N � j =1 A j e i θ j ρ HS ( A ) = ( θ 1 ,...,θ N ) ∈ [0 , 2 π ] N ρ max . Theorem (Hale, 1975; Silkowski, 1976) The following are equivalent: ρ HS ( A ) < 1 ; stab is exponentially stable for some Λ ∈ (0 , + ∞ ) N with Σ aut rationally independent components; Σ aut stab is exponentially stable for every Λ ∈ (0 , + ∞ ) N . Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: previous controllability results N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Stabilization by linear feedbacks u ( t ) = � N j =1 K j x ( t − Λ j ): [Hale, Verduyn Lunel, 2002 and 2003]. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: previous controllability results N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Stabilization by linear feedbacks u ( t ) = � N j =1 K j x ( t − Λ j ): [Hale, Verduyn Lunel, 2002 and 2003]. Spectral and approximate controllability in L p ([ − Λ max , 0] , C d ): [Salamon, 1984]. Controllability and stability of difference equations and applications Guilherme Mazanti

Introduction Stability analysis and applications Relative controllability Introduction Motivation: previous controllability results N � Σ contr : x ( t ) = A j x ( t − Λ j ) + Bu ( t ) , t ≥ 0 . j =1 Stabilization by linear feedbacks u ( t ) = � N j =1 K j x ( t − Λ j ): [Hale, Verduyn Lunel, 2002 and 2003]. Spectral and approximate controllability in L p ([ − Λ max , 0] , C d ): [Salamon, 1984]. Relative controllability in time T > 0: for any initial condition x 0 : [ − Λ max , 0] → C d and final target state x 1 ∈ C d , find u : [0 , T ] → C m such that the solution x with initial condition x 0 and control u satisfies x ( T ) = x 1 . Controllability and stability of difference equations and applications Guilherme Mazanti