Criterions on periodic feedback stabilization for some evolution equations

Transcription

1 Criterions on periodic feedback stabilization for some evolution equations School of Mathematics and Statistics, Wuhan University, P. R. China (Joint work with Yashan Xu, Fudan University) Toulouse, June, 2014

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3 A R n n corresponds to y = Ay. stable if any solution to the verifies We say that a matrix A is 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).

4 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 associated with A( ). P A( ) is called the periodic map (or the Poincaré 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.)

5 Some preliminaries about a T -periodic pair (A( ), B( )) are given in order: (i) Define the null controllable space V (A( ),B( )) { x R n u U, t > 0, s.t. y(t; 0, x, u) = 0 }. (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 1 (P A( ) ) R n 2 (P A( ) ), (1.3) where R n 1 (P A( )) and R n 2 (P A( )) are invariant under P A( ) s.t. σ(p A( ) R n 1 (P A( ) )) B, σ(p A( ) R n 2 (P A( ) )) B c.

6 (iii) Introduce two linear ODEs: S n(t) A(t)S n (t) S n (t)a(t) + 1 ε B(t)B(t) = 0, t [0, nt ] S n (nt ) = I; (1.4) S (t) A(t)S(t) S(t)A(t) + 1 ε B(t)B(t) = 0, t [0, T ] S(T ) = P A( ) XX PA( ), (1.5) where X is an invertible matrix in R n n and ε > 0. Write Sn( ) ε and S ε ( ) for the solutions of (1.4) and (1.5) respectively. We proved that Sn( ) ε and S ε ( ) are positive matrix-valued functions on [0, nt ] and [0, T ]; and Q (A( ),B( )) lim ε 0 + ( S ε n (0) ) 1 0. (1.6)

7 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) σ ( Q 1 (A( ),B( )) Q (A( ),B( )) P A( ) ) B, where Q 1 A( ),B( ) is the Moore-Penrose inverse of 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.

8 Our PFS laws are constructed as follows. We define an nt -periodic Kn( ) ε L (R + ; R m n ) by Kn(t) ε = 1 ε B(t) (Sn(t) ε ) 1 for a.e. t [0, nt ], (1.7) Kn(t) ε = Kn(t ε + nt ), for a.e. t R +, and a T -periodic K ε ( ) L (R + ; R m n ) by K ε (t) = 1 ε B(t) (S ε (t) ) 1 for a.e. t [0, T ], K ε (t) = K ε (t + T ), for a.e. t R +. (1.8)

9 Theorem (II) Let (A( ), B( )) be a T -PFS pair. (i) Kn( ) ε defined by (1.7) with ( Sn(0) ε ) 1 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.

10 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 σ ( Q 1 (A( ),B( )) 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 timeinvariant pairs to time-period pairs.

11 (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 rankc = n, i.e., the reformulated Brockett problem has possibly positive answer only if rankc < n. Thus, we conclude that time periodic matrices may aid in the observation feedback stabilization, but not state feedback stabilization.

12 (3) By our understanding, the produce to stabilize periodically a system: y (t) = A(t)y(t), (where A( ) is T -periodic) is as follows: one first builds up a T -periodic B( ) L (R + ; R n m ) s.t. [A( ), B( )] is T -periodically stabilizable, and then design a periodic K( ) L (R + ; R m n ) s.t. A( )+B( )K( ) is exponentially stable. We call B( ) as a control machine and K( ) as a feedback law. Control machines could be treated as control equipments which belongs to the category of hardware, while feedback law could be treated as control programs which belongs to the category of software. Thus, it is interesting to ask the question: How to design a simple T -periodic B( ) for a given T -periodic A( ) s.t. [A( ), B( )] is T -periodically stabilizable? With the aid of the geometric condition (d) in Theorem (I), we provided the answer for the aforementioned question in certain sense.

13 Given a T -periodic A( ), define CB A { B R n m (A( ), B) is T -PFS }. For each B CB A, denote by M( B) the number of columns of B. Set M(CB A ) min { M( B) } B CBA. Each matrix B, with M(CB A ) columns, in CB A could be one of the simplest ones.(notice that we can construct for given A( ) a 1-dim B( ) s.t. (A( ), B( )) is T -PFS.) We found a way to determine the number M(CB A ) and to design a matrix B CB A with M(CB A ) columns. In particular, when A( ) A, our way leads to M(CB A ) = max m(λ), λ σ(a)\c where m(λ) denotes the geometric multiplicity of the eigenvalue λ. This coincides with one of results obtained by M. Badra and T. Takahashi (2011).

14 (4) There have been studies on PFS criteria. H. Kano and T. Nishmara (1985) established the following equivalence: T-PFS H-stabilization. By H-stabilization of (A( ), B( )), it means that for each λ σ(p A( ) ) with λ > 1, P A( ) η = λη and B (t)(φ(t) ) 1 η = 0, t (0, T ) η = 0. This is the unique continuation of the adjoint equation where initial data are eigenfunctions of the periodic map corresponding to bad eigenfunctions.

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16 We start with proving (a) (c) of Theorem (I). Given a T -periodic (A( ), B( )). Write V V A( ),B( ) ; P P A( ),B( ) ; Φ Φ A( ),B( ) ; Q = Q A( ),B( ). Write y( ; t, x, u) for the solution to Let y = A(s)y + B(s)u, s t; y(t) = x. V n = {x R n u U s.t. y(nt ; 0, x, u) = 0} V. Define an LQ problem (LQ) ε t,x: W ε (t, x) inf J ε (u), u L 2 (t,nt ;R m ) where nt J ε (u) = ε u(s) 2 ds + y(nt ; t, x, u) 2. t

17 Observation 1: V = V n (One of the keys); PV = V = P 1 V ; P V = V = (P ) 1 V. Observation 2: W ε (t, x) =< x, Q ε n(t)x > for all t 0, x R n, where Q ε n solves the Riccati equation: Q + A Q + QA 1 ε QBB Q = 0, t [0, nt ]; Q(nT ) = I. Observation 3: S ε n( ) = Q ε n( ) 1 where S ε n solves (1.4). Observation 4: Q (given by (1.6)) is well defined and Q 0. Observation 5: V = N( Q), V = R( Q). (One of the keys )

18 Observation 6: T -periodic A 11, A 12, A 22 in L (R + ; R n n ) and B 1 L (R + ; R n m ) s.t. any solution to y = Ay + Bu can be expressed as y = y 1 + y 2, where y 1 = A 11 y 1 + A 12 y 2 + B 1 u; y 2 = A 22 y 2 ; (2.1) y 1 (t) Φ(t)V, y 2 (t) (Φ(t) 1 ) )V, t 0; (2.2) y 2 (nt ) = Q 1 QP n y 2 (0) V. (2.3) From (2.1), we see that to ensure (A( ), B( )) is PFS, A 22 ( ) must be stable. So the aim is to make the first equation in (2.1) is PFS. The term A 12 y 2 may cause some trouble. Fortunately, we can handle it.

19 More details about the proof of (a) (c). Recall that in Theorem (I), (a) says that (A( ), B( )) is nt -PFS; while (c) says that σ( Q 1 QP) B. By the linear algebra theory, we proved σ( Q 1 QP) B σ( Q 1 QP n ) B. Thus, to show (a) (c) in Theorem (I), we only need to prove σ( Q 1 QP n ) B (a). (2.4)

20 First of all, from (2.2), (2.3), as well as Observation 1, we have y 1 (nt ) V = V n ; y 2 (nt ) = Q 1 QP n y 2 (0) V. (2.5) We now construct K ε n(t) = 1 ε B (t)(s ε n(t)) 1 for a.e. t (0, nt ]. Extend it nt -periodically over R +. Let y n,ε ( ; x) be the solution to y = Ay + BK ε ny, t [0, nt ], y(0) = x. (2.6)

21 The proof of σ( Q 1 QP n ) B (a): When σ( Q 1 QP n ) B, with the aid of (2.5), as well as Observation 6, we can show that ε 0, δ 0 (0, 1), k N s.t. for each ε ε 0, y n,ε (knt ; x) δ 0 x for all x R n. (2.7) Then we extend nt -periodically K ε n( ) over R +. After that, with the help of (2.7), we get that K ε n( ) is an nt -PFS law. This leads to (a) in Theorem (I).

22 Conversely, let (a) hold. Write K for an nt -PFS law. Consider y = Ay + BKy, t R + ; y(0) = x. (2.8) Write y( ; x, K) for its solution. decomposition: By Observation 6, we have the y( ; x, K) = y 1 ( ; x) + y 2 ( ; x), as well as the the property y 1 (knt ; x) y 2 (knt ; x).these help us to prove lim y 2(knT ; x) = 0 for each x R n. (2.9) k Meanwhile, we verified that (2.9) σ( Q 1 QP n ) B. This completes the proof of (a) (c).

23 The next step in the proof of Theorem (I) is to show (a) (b), i.e., (A( ), B( )) is nt -PFS (A( ), B( )) is T -PFS. Two keys in its proof are as: (i) to structure an invertible matrix X by setting X = (ξ 1,..., ξ k1, η 1,..., η n k1 ) where {η 1,..., η n k1 } is any basis of V, but {η 1,..., η n k1 } is a special basis of V.(ii) To study the LQ problem T inf ε u(t) 2 dt + X 1 P 1 y(t ; 0, x, u) 2. u L 2 (0,T ;R m ) 0 Finally, we proved (b) (d) ( i.e., T -PFS R n 2 (P A( )) V (A( ),B( )) ), with the help of Observation 5 (i.e., V = N( Q), V = R( Q)).

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25 Consider linear periodic evolution equation: y (t) + Ay(t) + D(t)y(t) = B(t)u(t) in R +. (3.1) We assume that (H 1 ) The operator A, with its domain D( A), generates a compact semigroup {S(t)} t 0 in a Hilbert space H, with its innerproduct, and norm. (H 2 ) The operator-value function D( ) L 1 loc (R+ ; L(H)) is T - periodic, i.e., D(t + T ) = D(t) for a.e. t > 0, where T > 0 and L(H) denotes to the space of all linear bounded operators on H.

26 (H 3 ) The control machine B( ) L (R + ; L(U, H)) is T -periodic, where U is another Hilbert space (with its inner product, U and U ), and L(U, H) stands for the space of all linear bounded operator from U to H. For each h H, s 0 and u( ) L 2 (R + ; U), Equ. (3.1), with the initial condition y(s) = h, has a unique mild solution in C([s, ); H). We denote this solution by y( ; s, h, u).

27 Definition 1. Equ. (3.1) is said to be periodic feedback stabilizable (PFS) if T -periodic K( ) L (R + ; L(H, U)) s.t. the feedback equation y + Ay + D(t)y = B(t)K(t)y, in R + (3.2) is exponential stable. Any such a K( ) is called an LPFS law for (3.1). Definition 2. Let Z be a subspace of U. Equ. (3.1) is said to be PFS w.r.t. Z if T -periodic K( ) L (R + ; L(H, Z)) s.t. Equ. (3.2) is exponential stable. Any such a K( ) is called an PFS law for Equ. (3.1) w.r.t. Z.

28 Let U F S { Z U subspace (3.1) is PFS w.r.t. Z }. (3.3) The main purpose of our study on Equ. (3.1) are (i) to provide three criteria for judging whether a subspace Z belongs to U F S ; (ii) to show that if U U F S, then there is a finite dimensional subspace Z in U F S. These three criteria are related with the following subjects: the attainable subspace of Equ. (3.1); the unstable subspace (of (3.1) with the null control) provided by the Kato projection; the Poincaré map associated to A( ); and two unique continuation property of the dual equation of (3.1) with the null control.

29 Before stating main results, some preliminaries are given in order: (a) The Poincaré map. Let {Φ(t, s)} 0 s t< be the evolution system generated by ( A D( )). By the T -periodicity of D( ), one can easily check that Φ(t + T, s + T ) = Φ(t, s) for all 0 s t <. (3.4) Introduce the Poincaré map: P(t) Φ(t + T, t), t R +. (3.5)

30 Let L L(H). Write and H c H C H + ih the complexification of H; It is proved that L c (α + iβ) = Lα + ilβ, α, β H. σ(p(t) c ) \ {0} = {λ j } j=1, t 0, (3.6) where λ j, j 1, are all distinct non-zero eigenvalues of the compact operator P(t) c s.t. lim j λ j = 0. Hence, unique n N + s.t. λ j 1, j {1, 2,..., n} and λ j < 1, j {n + 1, n + 2,... }. (3.7) Let l j be the algebraic multiplicity of λ j. Write n 0 l l n. (3.8)

31 (b) The Kato projection. Let δ = max { λ j j > n } < 1. (3.9) Arbitrarily fix a δ ( δ, 1). Let Γ be the circle B(0, δ) with the clockwise direction in C. Define the Kato projection: P (t) = 1 ( λi P(t) c ) 1 dλ, t 0. (3.10) 2πi Γ

32 We have for each t 0, the operator P (t), defined by P (t) P (t) H (the restriction of P (t) on H), (3.11) is a projection on H, and P ( ) is T -periodic. H = H 1 (t) H 2 (t), where H 1 (t) = P (t)h and H 2 (t) = (I P (t))h, both H 1 (t) and H 2 (t) are invariant subspace of P(t), dim H 1 (t) = n 0 ; σ(p(t) c H1 (t) c) = {λ j} n j=1, σ(p(t)c H2 (t) c)\{0} = {λ j} j=n+1.

33 For simplicity, we write H 1 H 1 (0), H 2 H 2 (0), P P (0), P P(0). (3.12) The subspaces H 1 and H 2 are respectively called the unstable subspace (with the finite dimension n 0 ) and the stable subspace of Equ. (3.1) with the null control. Each eigenvalue in {λ j } n j=1 (or in {λ j} j=n+1 ) is called an unstable (or stable) eigenvalue of P c. Each eigenfunction of P c corresponding to an unstable (or stable) eigenvalue is called an unstable (or stable) eigenfunction of P c.

34 (c) Attainable subspace. k N, V Z k { kt 0 Given a subspace Z U, write for each Φ(kT, s)b(s)u(s) ds } u( ) L 2 (R + ; Z). (3.13) It is called the attainable subspace of Equ. (3.1) (over (0, kt )) w.r.t. Z. Let V k Z = P V k Z, k N+. (3.14) V k Z is the projection of the attainable subspace V Z k into the unstable subspace H 1, under the Kato project P.

35 Theorem (III) Let P, P and H i, i = 1, 2 be given by (3.12). let n 0 be given by (3.8). Then for each subspace Z U, the following statements are equivalent: (a) Equ. (3.1) is PFS w.r.t. Z, i.e., Z U F S ; (b) The subspace Z satisfies V Z n 0 = H 1 ; (c) The subspace Z satisfies ξ P H 1, (B( ) Z ) Φ(n 0 T, ) ξ = 0 over (0, n 0 T ) ξ = 0. (d) The subspace Z satisfies µ / B, ξ H c, (µi P c )ξ = 0, (B( ) Z ) c Φ(T, ) c ξ = 0 over (0, T ) ξ = 0.

36 Theorem (IV) Equ. (3.1) is LPFS iff it is LPFS w.r.t. a finite dimensional subspace Z (of U) with dimz n 0. The key to show Theorem (III) is to built up the equivalence (a) (b).the condition (b) is a geometric condition which says that the projection of the attainable subspace Vn Z 0 is the unstable subspace. The functions Φ(n 0 T, ) ξ with ξ H and Φ(T, ) c ξ with ξ H C are respectively the solutions to the dual equations and ψ t A ψ(t) D(t) ψ(t) = 0, t (0, n 0 T ), ψ(n 0 T ) = ξ ψ t A c ψ(t) D(t) c ψ(t) = 0, t (0, T ), ψ(t ) = ξ.

37 Thus, the condition (c) in Theorem (III) presents a unique continuation property for solutions of the first dual equation with initial data in P H 1 ; while the condition (d) in Theorem (III) presents a unique continuation property for solutions of the second dual e- quation where the initial data are unstable eigenfunctions of P c. There have been studies on the equivalent conditions of periodic feedback stabilization for linear evolution systems. A. Lunardi (1991) and G. Da Prato and A. Ichikawa (1990) established an e- quivalent condition on stabilizability for linear time-periodic parabolic equations with open looped controls. Their equivalence can be stated, under our framework, as follows: the condition (d) (where Z = U) is equivalent to the statement that for any h H, a control u h ( ) C(R + ; U) with sup e δt t R + u h (t) U bounded, s.t. the solution y( ; 0, h, u h ) is stable.

38 Meanwhile, it was pointed out in the paper of A. Lunardi s that when open-loop stabilization controls exists, one can construct a periodic feedback stabilization law. From this point of view, the equivalence (a) (d) in Theorem (III) is not new, though our way to approach the equivalence differs from theirs. A byproduct of our study shows that when both D( ) and B( ) are time invariant, linear time-periodic function K( ) will not aid the linear stabilization of Equ. (3.1), i.e., Equ. (3.1) is linear T -periodic feedback stabilizable for some T > 0 iff Equ. (3.1) is linear timeinvariant feedback stabilizable. On the other hand, when Equ. (3.1) is periodic time varying, we provide an example to explain that linear time-periodic K( ) do aid in the linear stabilization of this equation.

39 We next give two applications of Theorem (III). (i) Let Ω R d (d 1) be a bounded domain with a C 2 -boundary Ω. Let Q Ω R + and Σ Ω R +. Let ω Ω be a non-empty open subset with the characteristic function χ ω. Let T > 0 and a L (Q) be T -periodic w.r.t. the time variable, i.e., a(, t + T ) = a(, t) over Ω for a.e. t. Consider { y y + ay = χ ω u in Q, (3.15) y = 0 on Σ, where u L 2 (R + ; L 2 (Ω)).

40 We can set up (3.15) in our framework: H = U = L 2 (Ω); A = with D(A) = H 1 0 (Ω) H2 (Ω); D(t) : L 2 (Ω) = H L 2 (Ω) = H by D(t)z(x) = a(x, t)z(x), x Ω; B(t) : L 2 (Ω) = U L 2 (Ω) = H by B(t)v(x) = χ ω v(x), x Ω. One can check that A, D( ), B( ), defined above, verify assumptions (H 1 ), (H 2 ) and (H 3 ). By the equivalence (a) (d) in Theorem (III), and by the unique continuation property for linear parabolic equations, we have Corollary (I) Equation (3.15) is PFS w.r.t. P H.

41 (ii) Write λ 1 and λ 2 for the first and second eigenvalues of. Let ξ j be an eigenfunction corresponding to λ j. Consider { y y λ 2 y = u, ξ 1 ξ 1 in Q, (3.16) y = 0 on Σ, where u L 2 (R + ; L 2 (Ω)). By a direct calculation, one has V n0 = Span {ξ 1 } and H 1 Span {ξ 1, ξ 2 }. Hence, by the geometric condition in Theorem (III), Equ. (3.16) is not PFS.

42 Criterions on periodic feedback stabilization for some evolution equations School of Mathematics and Statistics, Wuhan University, P. R. China (Joint work with Yashan Xu, Fudan University) Toulouse, June, 2014

43 Thank you!