Input-to-state stability of systems of partial differential - PowerPoint PPT Presentation

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Input-to-state stability of systems of partial differential equations Andrii Mironchenko Centre for Industrial Mathematics, University of Bremen, Germany February 14, 2011, Elgersburg Workshop 2011, Elgersburg, Germany joint work with Sergey Dashkovskiy Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 1 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Outline 1 Basic notions 2 Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions Linearisation 3 Monotone control systems 4 Interconnections of control systems 5 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 2 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Semigroups and their Generators Let X be a Banach space, and L ( X ) be the space of bounded operators, defined on X . Definition (Strongly continuous semigroup) A family of operators { T ( t ) , t ≥ 0 } ⊂ L ( X ) , is called a strongly continuous semigroup (for short C 0 -semigroup), if it holds 1 T ( 0 ) = I 2 T ( t + s ) = T ( t ) T ( s ) , ∀ t , s ≥ 0. 3 For all x ∈ X function t �→ T ( t ) x belongs to C ([ 0 , ∞ ) , X ) Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 3 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Definition (Analytic semigroup) The C 0 -semigroup is called analytic, if in addition it holds: T ( t ) x → x , when t → + 0. t �→ T ( t ) x is real analytic on 0 < t < ∞ for every x ∈ X . Definition (Generator of a C 0 -semigroup) Linear operator L , defined by 1 Lx = lim t ( T ( t ) x − x ) t → + 0 with domain 1 D ( L ) = { x ∈ X : lim t ( T ( t ) x − x ) exists } t → + 0 is called an infinitesimal generator of a C 0 -semigroup T ( t ) . Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 4 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Definition of control system Let ( X , � · � X ) be a state space, ( U , � · � U ) be an input space and U c be the set of admissible input functions: R + → U . Definition (Control system) The triple Σ = ( X , U c , φ ) is a control system, if: φ ( t , t , x , · ) = x for all t ≥ 0. ∀ t ≥ r ≥ s ≥ 0, ∀ x ∈ X , ∀ u 1 ∈ U [ s , r ] , u 2 ∈ U [ r , t ] it holds c c φ ( t , r , φ ( r , s , x , u 1 ) , u 2 ) = φ ( t , s , x , u ) , where � u 1 ( τ ) , τ ∈ [ s , r ] , u ( τ ) := u 2 ( τ ) , τ ∈ [ r , t ] . ∀ x ∈ X , u ∈ U c the map t → φ ( t , 0 , x , u ) is in C ([ 0 , ∞ ) , X ) φ is continuous in two last arguments. Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 5 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Stability notions Let Σ = ( X , U c , φ ) be time-invariant and φ ( t , 0 , 0 , 0 ) ≡ 0. Definition (Global asymptotic stability at zero) Σ is globally asymptotically stable at zero (0-GAS), if ∃ β ∈ KL : ∀ φ 0 ∈ X , ∀ t ≥ 0 it holds � φ ( t , 0 , φ 0 , 0 ) � X ≤ β ( � φ 0 � X , t ) . Definition (Local input-to-state stability) Σ is locally input-to-state stable (LISS), if ∃ ρ x , ρ u > 0 and ∃ β ∈ KL , γ ∈ K , such that ∀ t ≥ 0, ∀ φ 0 : � φ 0 � X ≤ ρ x and ∀ u ∈ U c : � u � U c ≤ ρ u it holds � φ ( t , t 0 , φ 0 , u ) � X ≤ max { β ( � φ 0 � X , t ) , γ ( � u � U c ) } . Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 6 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Definition (exponential LISS and ISS) If β ( r , t ) = Me ω t r , for some ω < 0, then ( X , U c , φ ) is locally exponentially ISS If one can choose ρ x = ρ u = ∞ , then ( X , U c , φ ) is ISS β ( � φ 0 � , t ) � x ( t ) � γ ( � u � ) t Figure: Input-to-state stability in max-formulation Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 7 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Outline 1 Basic notions 2 Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions Linearisation 3 Monotone control systems 4 Interconnections of control systems 5 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 8 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko LISS-Lyapunov functions Definition (Local ISS-Lyapunov function (LISS-LF)) A smooth function V : D → R + , D ⊂ X , 0 ∈ int ( D ) is LISS-LF for system ( X , U c , φ ) , if there exist ρ x , ρ u > 0, functions ψ 1 , ψ 2 ∈ K ∞ , χ ∈ K and positive definite function α , such that: ψ 1 ( � x � X ) ≤ V ( x ) ≤ ψ 2 ( � x � X ) , ∀ x ∈ D and ∀ x ∈ D : � x � X ≤ ρ x , ∀ u ∈ U : � u � U ≤ ρ u it holds: ˙ � x � X ≥ χ ( � u � U ) ⇒ V ( x ) ≤ − α ( � x � X ) , (1) where 1 ˙ V ( x ) = lim t ( V ( φ ( t , 0 , x , u )) − V ( x )) . t → + 0 Function χ is called Lyapunov gain. Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 9 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Lyapunov characterisation of LISS Theorem Let Σ = ( X , U c , φ ) be a time-invariant control system. If Σ possesses a LISS-Lyapunov function, then Σ is LISS. Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 10 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Lyapunov characterisation of LISS Theorem Let Σ = ( X , U c , φ ) be a time-invariant control system. If Σ possesses a LISS-Lyapunov function, then Σ is LISS. Example: semilinear heat equation � ∂ t = ∂ 2 s ∂ s ∂ x 2 − f ( s ) + u ( x , t ) , x ∈ ( 0 , π ) , t > 0 , (2) s ( 0 , t ) = s ( π, t ) = 0 . We assume, that f is locally Lipschitz, monotonically increasing up to infinity, f ( − r ) = − f ( r ) for all r ∈ R and u ( · , t ) ∈ L 2 ( 0 , π ) . Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 10 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Example: Formulation and Lyapunov function We define: As = d 2 s D ( A ) = H 1 0 ( 0 , π ) ∩ H 2 ( 0 , π ) . with dx 2 Operator A generates an analytic semigroup on L 2 ( 0 , π ) . System (2) takes form ds dt = As − f ( s ) + u , t > 0 . (3) Equation (3) defines the control system with state space X = H 1 0 ( 0 , π ) and input space U = L 2 ( 0 , π ) . �� π � 1 2 . The norm on H 1 0 s 2 0 ( 0 , π ) we define as � s � H 1 0 ( 0 ,π ) = x ( x ) dx � π � s ( x ) � � 1 2 s 2 V ( s ) = x ( x ) + f ( y ) dy dx . (4) 0 0 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 11 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Verification of the first property: � s ( x ) � π 1 x ( x ) dx = 1 2 s 2 2 � s � 2 f ( y ) dy ≥ 0 ⇒ V ( s ) ≥ H 1 0 ( 0 ,π ) 0 0 The derivative of V along the trajectories is: � π � π ˙ ( s xx ( x ) − f ( s ( x ))) 2 dx + V ( s ) = − ( s xx ( x ) − f ( s ( x )))( − u ) dx . 0 0 Define � π ( s xx ( x ) − f ( s ( x ))) 2 dx . I ( s ) := 0 Using Cauchy-Schwarz inequality for the second term, we have: ˙ � V ( s ) ≤ − I ( s ) + I ( s ) � u � L 2 ( 0 ,π ) . (5) Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 12 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Lyapunov function � π ( s xx ( x ) − f ( s ( x ))) 2 dx . I ( s ) := 0 One can prove directly: � π s 2 I ( s ) ≥ xx ( x ) dx . 0 For s ∈ H 1 0 ( 0 , π ) ∩ H 2 ( 0 , π ) it holds (a corollary of Friedrich’s inequality), that: � π � π s 2 s 2 xx ( x ) dx ≥ x ( x ) dx . 0 0 Overall, we have: I ( s ) ≥ � s � 2 0 ( 0 ,π ) . (6) H 1 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 13 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Gains Now we choose the gain as χ ( r ) = ar , a > 1 . If � s � H 1 0 ( 0 ,π ) ≥ χ ( � u � L 2 ( 0 ,π ) ) , we obtain V ( s ) ≤ − I ( s )+ 1 0 ( 0 ,π ) ≤ − ( 1 − 1 a ) I ( s ) ≤ − ( 1 − 1 ˙ � a ) � s � 2 I ( s ) � s � H 1 0 ( 0 ,π ) . H 1 a This proves, that V is ISS-Lyapunov function, and consequently, our control system (with X = H 1 0 ( 0 , π ) , U = L 2 ( 0 , π ) ) is ISS. Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 14 / 28

Centre for Elgersburg Workshop 2011 Industrial Mathematics Mironchenko Outline 1 Basic notions 2 Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions Linearisation 3 Monotone control systems 4 Interconnections of control systems 5 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections 15 / 28