Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Criterions on periodic feedback stabilization for some evolution equations Gengsheng Wang School of Mathematics and Statistics, Wuhan University, P. R. China (Joint work with Yashan Xu, Fudan University) Toulouse, June, 2014 Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Problem and main results in ODE case 1 The sketch proof of Theorem (I) and Theorem (II) 2 Extension to infinitely dimensional cases 3 Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases A ∈ R n × n corresponds to y ′ = Ay . We say that a matrix A is stable if any solution to the verifies � y ( t ) � ≤ Ce − δt � y (0) � , t ≥ 0 , (1.1) for some positive δ and C . When A is not stable, we design a control machine B ∈ R n × m s.t. ( A, B ) is feedback stabilizable (FS, for short), i.e., ∃ a feedback law K ∈ R n × m s.t. any solution y to the equation: y ′ = Ay + BKy , t ≥ 0 , verifies (1.1). T -periodic A ( · ) ∈ L ∞ ( R + ; R n × n ) ( A ( t + T ) = A ( t ) for a.e. t ≥ 0 ) corresponds to the equation y ′ = A ( t ) y . We say that A ( · ) is stable if any solution y verifies (1.1). When A ( · ) is not stable, we design a T -periodic B ( · ) s.t. (( A ( · ) , B ( · )) is T -periodically feedback stabilizable (T-PFS, for short), i.e., ∃ a T -periodic K ( · ) ∈ L ∞ ( R + ; R m × n ) s.t. any solution y to equation: y ′ = A ( t ) y + B ( t ) K ( t ) y verifies (1.1). Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases It is well known that ( i ) A is stable iff σ ( A ) ⊂ C − 1 � { λ ∈ C , Re ( λ ) < 0 } ; ( ii ) ( A, B ) is FS iff rank ( λI − A, B ) = n for all λ ∈ C \ C − ; ( iii ) T -periodic A ( · ) is stable iff σ ( P A ( · ) ) ∈ B . Here B is the unit open ball in C , P A ( · ) � Φ A ( · ) ( T ) , with Φ A ( · ) ( · ) the fundamental as- sociated with A ( · ) . P A ( · ) is called the periodic map (or the Poincar´ e map) associated with A ( · ) . It is natural to ask for a criterion on a T -periodic pair ( A ( · ) , B ( · )) s.t. it is T-PFS. Our aim is ( i ) to build up two criterions (on a T -periodic pair ( A ( · ) , B ( · )) s.t. it is T-PFS); ( ii ) to construct two periodic feedback stabilization laws. (One is T -periodic; while another is nT -periodic.) Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Some preliminaries about a T -periodic pair ( A ( · ) , B ( · )) are given in order: ( i ) Define the null controllable space � x ∈ R n � � � ∃ u ∈ U , t > 0 , s.t. y ( t ; 0 , x, u ) = 0 V ( A ( · ) ,B ( · )) � . (1.2) Here y ( · ; 0 , x, u ) solves y ′ = A ( t ) y + B ( t ) u , y (0) = x , and u ∈ U � L 2 loc ( R + ; R m ) . ( ii ) Write � R n = R n R n 1 ( P A ( · ) ) 2 ( P A ( · ) ) , (1.3) where R n 1 ( P A ( · ) ) and R n 2 ( P A ( · ) ) are invariant under P A ( · ) s.t. 2 ( P A ( · ) ) ) ⊂ B c . σ ( P A ( · ) | R n 1 ( P A ( · ) ) ) ⊂ B , σ ( P A ( · ) | R n Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases ( iii ) Introduce two linear ODEs: n ( t ) − A ( t ) S n ( t ) − S n ( t ) A ( t ) ∗ + 1 εB ( t ) B ( t ) ∗ = 0 , t ∈ [0 , nT ] S ′ S n ( nT ) = I ; (1.4) S ′ ( t ) − A ( t ) S ( t ) − S ( t ) A ( t ) ∗ + 1 εB ( t ) B ( t ) ∗ = 0 , t ∈ [0 , T ] S ( T ) = P A ( · ) XX ∗ P ∗ A ( · ) , (1.5) where X is an invertible matrix in R n × n and ε > 0 . Write S ε n ( · ) and S ε ( · ) for the solutions of (1.4) and (1.5) respec- tively. We proved that S ε n ( · ) and S ε ( · ) are positive matrix-valued functions on [0 , nT ] and [0 , T ] ; and � � − 1 ≥ 0 . ¯ Q ( A ( · ) ,B ( · )) � lim S ε n (0) (1.6) ε → 0 + Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Theorem ( I ) Let ( A ( · ) , B ( · )) be a T -periodic pair. Then the following statements are equivalent: ( a ) ( A ( · ) , B ( · )) is nT -periodically stabilizable. ( b ) ( A ( · ) , B ( · )) is T -periodically stabilizable. ( c ) � ¯ � ( A ( · ) ,B ( · )) ¯ Q ∼ 1 σ Q ( A ( · ) ,B ( · )) P A ( · ) ⊂ B , where ¯ A ( · ) ,B ( · ) is the Moore-Penrose inverse of ¯ Q ∼ 1 Q ( A ( · ) ,B ( · )) . ( d ) R n 2 ( P A ( · ) ) ⊂ V ( A ( · ) ,B ( · )) . Say ( A ( · ) , B ( · )) is kT -PFS if ∃ a kT -periodic K ( · ) in L ∞ ( R + ; R n × m ) s.t. any solution y ( · ) to equation: y ′ = A ( t ) y + B ( t ) K ( t ) y verifies (1.1), i.e., � y ( t ) � ≤ Ce − δt � y (0) � for all t ≥ 0 . Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Our PFS laws are constructed as follows. We define an nT -periodic K ε n ( · ) ∈ L ∞ ( R + ; R m × n ) by � − 1 n ( t ) = − 1 εB ( t ) ∗ ( S ε K ε n ( t ) for a.e. t ∈ [0 , nT ] , (1.7) for a.e. t ∈ R + , K ε n ( t ) = K ε n ( t + nT ) , and a T -periodic K ε ( · ) ∈ L ∞ ( R + ; R m × n ) by � − 1 K ε ( t ) = − 1 εB ( t ) ∗ ( S ε ( t ) for a.e. t ∈ [0 , T ] , (1.8) K ε ( t ) = K ε ( t + T ) , for a.e. t ∈ R + . Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Theorem ( II ) Let ( A ( · ) , B ( · )) be a T -PFS pair. ( i ) K ε n ( · ) defined by (1.7) with � � − 1 − ¯ S ε � n (0) Q ( A ( · ) ,B ( · )) � < 1 , is an nT -PFS law. ( ii ) There are an invertible matrix X ∈ R n × n and ε 0 > 0 s.t. K ε ( · ) given by (1.8) with ε ≤ ε 0 is a T -PFS law for this pair. The matrix X in the above can be constructed. Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases Several remarks are given in order. (1) The condition ( d ) in Theorem ( I ) is a geometric condition which says, in plain language, that the bad invariant subspace of P A ( · ) is contained in the null controllable subspace. The condition ( c ) is an algebraic condition which is comparable to the T -periodic stable condition on A ( · ) . The first one is as � ¯ � Q ∼ 1 ( A ( · ) ,B ( · )) ¯ σ Q ( A ( · ) ,B ( · )) P A ( · ) ⊂ B , while the second one is as σ ( P A ( · ) ) ∈ B . Besides, from Condition ( c ) , we can easily derive the Kalman rank condition for the case when ( A ( · ) , B ( · )) = ( A, B ) . Thus, Condition ( c ) is a natural extension of Kalman’s rank condition from time- invariant pairs to time-period pairs. Gengsheng Wang Periodic feedback stabilization
Problem and main results in ODE case The sketch proof of Theorem (I) and Theorem (II) Extension to infinitely dimensional cases (2) R. Brockett formulated the following problem in 1999: What are the condition on a triple ( A, B, C ) ( n × n , n × m and p × n matrices) ensuring the existence of a periodic K ( · ) s.t. the system y ′ = Ay + BK ( t ) Cy is asymptotically stable? After this, G. A. Leonov (2001) reformulated the Brockett problem as: Can the time periodic matrices K ( · ) aid in the stabilization? He further provided some examples which give the positive answer for the reformulated Brockeet problem. Based on Theorem I , we found that when ( A ( · ) , B ( · )) = ( A, B ) , it is feedback stabilizable by a constant matrix iff it is T -PFS for some T iff it is T -PFS for any T . Hence, time periodic matrices K ( · ) will not aid in the stabilization for any ( A, B, C ) with rank C = n , i.e., the reformulated Brockett problem has possibly positive answer only if rank C < n . Thus, we conclude that time periodic matrices may aid in the observation feedback stabilization, but not state feedback stabilization. Gengsheng Wang Periodic feedback stabilization
Recommend
More recommend
