Criterions on periodic feedback stabilization for some evolution - - PowerPoint PPT Presentation

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

1

Problem and main results in ODE case

2

The sketch proof of Theorem (I) and Theorem (II)

3

Extension to infinitely dimensional cases

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 ∈ Rn×n corresponds to y′ = Ay. We say that a matrix A is stable if any solution to the verifies y(t) ≤ Ce−δty(0), t ≥ 0, (1.1) for some positive δ and C. When A is not stable, we design a control machine B ∈ Rn×m s.t. (A, B) is feedback stabilizable (FS, for short), i.e., ∃ a feedback law K ∈ Rn×m s.t. any solution y to the equation: y′ = Ay + BKy, t ≥ 0, verifies (1.1). T-periodic A(·) ∈ L∞(R+; Rn×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+; Rm×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(PA(·)) ∈ B. Here B is the unit

• pen ball in C, PA(·) ΦA(·)(T), with ΦA(·)(·) the fundamental as-

sociated with A(·). PA(·) 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

• rder:

(i) Define the null controllable space V(A(·),B(·))

• x ∈ Rn

∃ 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 L2

loc(R+; Rm).

(ii) Write Rn = Rn

1(PA(·))

• Rn

2(PA(·)),

(1.3) where Rn

1(PA(·)) and Rn 2(PA(·)) are invariant under PA(·) s.t.

σ(PA(·)|Rn

1 (PA(·))) ⊂ B,

σ(PA(·)|Rn

2 (PA(·))) ⊂ Bc. 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:    S′

n(t) − A(t)Sn(t) − Sn(t)A(t)∗ + 1

εB(t)B(t)∗ = 0, t ∈ [0, nT] Sn(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) = PA(·)XX∗P∗

A(·),

(1.5) where X is an invertible matrix in Rn×n and ε > 0. Write Sε

n(·) and Sε(·) for the solutions of (1.4) and (1.5) respec-

• tively. We proved that Sε

n(·) and Sε(·) are positive matrix-valued

functions on [0, nT] and [0, T]; and ¯ Q(A(·),B(·)) lim

ε→0+

• Sε

n(0)

−1 ≥ 0. (1.6)

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) σ ¯ Q∼1

(A(·),B(·)) ¯

Q(A(·),B(·))PA(·)

• ⊂ B,

where ¯ Q∼1

A(·),B(·) is the Moore-Penrose inverse of ¯

Q(A(·),B(·)). (d) Rn

2(PA(·)) ⊂ V(A(·),B(·)).

Say (A(·), B(·)) is kT-PFS if ∃ a kT-periodic K(·) in L∞(R+; Rn×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−δty(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 Kε

n(·) ∈ L∞(R+; Rm×n) by

   Kε

n(t) = −1

εB(t)∗(Sε

n(t)

−1 for a.e. t ∈ [0, nT], Kε

n(t) = Kε n(t + nT),

for a.e. t ∈ R+, (1.7) and a T-periodic Kε(·) ∈ L∞(R+; Rm×n) by    Kε(t) = −1 εB(t)∗(Sε(t) −1 for a.e. t ∈ [0, T], Kε(t) = Kε(t + T), for a.e. t ∈ R+. (1.8)

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) Kε

n(·) defined by (1.7) with

• Sε

n(0)

−1 − ¯ Q(A(·),B(·)) < 1, is an nT-PFS law. (ii) There are an invertible matrix X ∈ Rn×n and ε0 > 0 s.t. Kε(·) given by (1.8) with ε ≤ ε0 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 PA(·) 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(·))PA(·)

• ⊂ B,

while the second one is as σ(PA(·)) ∈ 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 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.

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

(3) By our understanding, the produce to stabilize periodically a system: y′(t) = A(t)y(t), (where A(·) is T-periodic) is as fol- lows: one first builds up a T-periodic B(·) ∈ L∞(R+; Rn×m) s.t. [A(·), B(·)] is T-periodically stabilizable, and then design a periodic K(·) ∈ L∞(R+; Rm×n) s.t. A(·)+B(·)K(·) is exponentially stable. We call B(·) as a control machine and K(·) as a feedback law. Con- trol 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.

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

Given a T-periodic A(·), define CBA B ∈ Rn×m (A(·), B) is T-PFS

• .

For each B ∈ CBA, denote by M( B) the number of columns of

• B. Set M(CBA) min
• M(

B)

• B ∈ CBA
• . Each matrix

B, with M(CBA) columns, in CBA 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(CBA) and to design a matrix B ∈ CBA with M(CBA) columns. In particular, when A(·) ≡ A, our way leads to M(CBA) = max

λ∈σ(A)\C− m(λ),

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).

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

(4) There have been studies on PFS criteria. H. Kano and T. Nish- mara (1985) established the following equivalence: T-PFS ⇔ H-stabilization. By H-stabilization of (A(·), B(·)), it means that for each λ ∈ σ(PA(·)) with |λ| > 1, P∗

A(·)η = λη and B∗(t)(Φ(t)∗)−1η = 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.

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

1

Problem and main results in ODE case

2

The sketch proof of Theorem (I) and Theorem (II)

3

Extension to infinitely dimensional cases

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

We start with proving (a) ⇔ (c) of Theorem (I). Given a T-periodic (A(·), B(·)). Write V VA(·),B(·); P PA(·),B(·); Φ ΦA(·),B(·); ¯ Q = ¯ QA(·),B(·). Write y(·; t, x, u) for the solution to y′ = A(s)y + B(s)u, s ≥ t; y(t) = x. Let Vn = {x ∈ Rn | ∃u ∈ U s.t. y(nT; 0, x, u) = 0} ⊂ V. Define an LQ problem (LQ)ε

t,x:

W ε(t, x) inf

u∈L2(t,nT;Rm) Jε(u),

where Jε(u) = ε nT

t

u(s)2ds + y(nT; t, x, u)2.

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

Observation 1: V = Vn (One of the keys); PV = V = P−1V ; P∗V ⊥ = V ⊥ = (P∗)−1V ⊥. Observation 2: W ε(t, x) =< x, Qε

n(t)x > for all t ≥ 0, x ∈ Rn,

where Qε

n solves the Riccati equation:

Q′ + A∗Q + QA − 1 εQBB∗Q∗ = 0, t ∈ [0, nT]; Q(nT) = I. Observation 3: Sε

n(·) = Qε n(·)−1 where Sε 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 )

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

Observation 6: ∃ T-periodic A11, A12, A22 in L∞(R+; Rn×n) and B1 ∈ L∞(R+; Rn×m) s.t. any solution to y′ = Ay + Bu can be expressed as y = y1 + y2, where y′

1 = A11y1 + A12y2 + B1u; y′ 2 = A22y2;

(2.1) y1(t) ∈ Φ(t)V, y2(t) ∈ (Φ(t)−1)∗)V ⊥, t ≥ 0; (2.2) y2(nT) = ¯ Q∼1 ¯ QPny2(0) ∈ V ⊥. (2.3) From (2.1), we see that to ensure (A(·), B(·)) is PFS, A22(·) must be stable. So the aim is to make the first equation in (2.1) is PFS. The term A12y2 may cause some trouble. Fortunately, we can handle it.

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

More details about the proof of (a) ⇔ (c). Recall that in Theo- rem (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 ¯ QPn) ∈ B. Thus, to show (a) ⇔ (c) in Theorem (I), we only need to prove σ( ¯ Q∼1 ¯ QPn) ∈ B ⇔ (a). (2.4)

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

First of all, from (2.2), (2.3), as well as Observation 1, we have y1(nT) ∈ V = Vn; y2(nT) = ¯ Q∼1 ¯ QPny2(0) ∈ V ⊥. (2.5) We now construct Kε

n(t) = −1

εB∗(t)(Sε

n(t))−1 for a.e. t ∈ (0, nT].

Extend it nT-periodically over R+. Let yn,ε(·; x) be the solution to y′ = Ay + BKε

ny, t ∈ [0, nT], y(0) = x.

(2.6)

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

The proof of σ( ¯ Q∼1 ¯ QPn) ∈ B ⇒ (a): When σ( ¯ Q∼1 ¯ QPn) ∈ B, with the aid of (2.5), as well as Observation 6, we can show that ∃ ε0, δ0 ∈ (0, 1), ∃ k ∈ N s.t. for each ε ≤ ε0, yn,ε(knT; x) ≤ δ0x for all x ∈ Rn. (2.7) Then we extend nT-periodically Kε

n(·) over R+. After that, with

the help of (2.7), we get that Kε

n(·) is an nT-PFS law. This leads

to (a) in Theorem (I).

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

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. By Observation 6, we have the decomposition: y(·; x, K) = y1(·; x) + y2(·; x), as well as the the property y1(knT; x)⊥y2(knT; x).These help us to prove lim

k→∞ y2(knT; x) = 0 for each x ∈ Rn.

(2.9) Meanwhile, we verified that (2.9) ⇔ σ( ¯ Q∼1 ¯ QPn) ∈ B. This completes the proof of (a) ⇔ (c).

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

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 = (ξ1, . . . , ξk1, η1, . . . , ηn−k1) where {η1, . . . , ηn−k1} is any basis of V ⊥, but {η1, . . . , ηn−k1} is a special basis of V .(ii) To study the LQ problem inf

u∈L2(0,T;Rm)

T εu(t)2dt + X−1P−1y(T; 0, x, u)2. Finally, we proved (b) ⇔ (d) ( i.e., T-PFS ⇔ Rn

2(PA(·)) ⊂ V(A(·),B(·))),

with the help of Observation 5 (i.e., V = N( ¯ Q), V ⊥ = R( ¯ Q)).

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

1

Problem and main results in ODE case

2

The sketch proof of Theorem (I) and Theorem (II)

3

Extension to infinitely dimensional cases

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

Consider linear periodic evolution equation: y′(t) + Ay(t) + D(t)y(t) = B(t)u(t) in R+. (3.1) We assume that (H1) The operator −A, with its domain D(−A), generates a com- pact semigroup {S(t)}t≥0 in a Hilbert space H, with its inner- product ·, · and norm · . (H2) The operator-value function D(·) ∈ L1

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.

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

(H3) 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

• perator from U to H.

For each h ∈ H, s ≥ 0 and u(·) ∈ L2(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).

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

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.

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

Let UFS

• 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 be- longs to UFS; (ii) to show that if U ∈ UFS, then there is a finite dimensional subspace Z in UFS. These three criteria are related with the following subjects: the at- tainable subspace of Equ. (3.1); the unstable subspace (of (3.1) with the null control) provided by the Kato projection; the Poincar´ e map associated to A(·); and two unique continuation property of the dual equation of (3.1) with the null control.

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

Before stating main results, some preliminaries are given in order: (a) The Poincar´ e map. Let {Φ(t, s)}0≤s≤t<∞ be the evolution system generated by (−A − D(·)). By the T-periodicity of D(·),

• ne can easily check that

Φ(t + T, s + T) = Φ(t, s) for all 0 ≤ s ≤ t < ∞. (3.4) Introduce the Poincar´ e map: P(t) Φ(t + T, t), t ∈ R+. (3.5)

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

Let L ∈ L(H). Write Hc H

• C H + iH the complexification of H;

and Lc(α + iβ) = Lα + iLβ, α, β ∈ H. It is proved that σ(P(t)c) \ {0} = {λj}∞

j=1, ∀t ≥ 0,

(3.6) where λj, j ≥ 1, are all distinct non-zero eigenvalues of the compact

• perator P(t)c s.t. limj→∞ |λj| = 0. Hence, ∃ unique n ∈ N+ s.t.

|λj| ≥ 1, j ∈ {1, 2, . . . , n} and |λj| < 1, j ∈ {n + 1, n + 2, . . . }. (3.7) Let ℓj be the algebraic multiplicity of λj. Write n0 ℓ1 + · · · + ℓn. (3.8)

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

(b) The Kato projection. Let ¯ δ = max

• |λj|
• 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 2πi

• Γ
• λI − P(t)c−1 dλ, t ≥ 0.

(3.10)

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

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 = H1(t) H2(t), where H1(t) = P(t)H and H2(t) = (I − P(t))H, both H1(t) and H2(t) are invariant subspace of P(t), dim H1(t) = n0; σ(P(t)c|H1(t)c) = {λj}n

j=1, σ(P(t)c|H2(t)c)\{0} = {λj}∞ j=n+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

For simplicity, we write H1 H1(0), H2 H2(0), P P(0), P P(0). (3.12) The subspaces H1 and H2 are respectively called the unstable sub- space (with the finite dimension n0) and the stable subspace of

• Equ. (3.1) with the null control.

Each eigenvalue in {λj}n

j=1 (or in {λj}∞ j=n+1) is called an unstable

(or stable) eigenvalue of Pc. Each eigenfunction of Pc corresponding to an unstable (or stable) eigenvalue is called an unstable (or stable) eigenfunction of Pc.

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

(c) Attainable subspace. Given a subspace Z ⊂ U, write for each k ∈ N, V Z

k

kT Φ(kT, s)B(s)u(s) ds

• u(·) ∈ L2(R+; Z)
• .

(3.13) It is called the attainable subspace of Equ. (3.1) (over (0, kT)) w.r.t.

• Z. Let
• V Z

k = PV Z k , k ∈ N+.

(3.14)

• V Z

k is the projection of the attainable subspace V Z k into the unstable

subspace H1, under the Kato project P.

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 (III) Let P, P and Hi, i = 1, 2 be given by (3.12). let n0 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 ∈ UFS; (b) The subspace Z satisfies V Z

n0 = H1;

(c) The subspace Z satisfies ξ ∈ P ∗H1, (B(·)|Z)∗Φ(n0T, ·)∗ξ = 0 over (0, n0T) ⇒ ξ = 0. (d) The subspace Z satisfies µ / ∈ B, ξ ∈ Hc, (µI − P∗c)ξ = 0, (B(·)|Z)∗cΦ(T, ·)∗cξ = 0 over (0, T) ⇒ ξ = 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 (IV)

• Equ. (3.1) is LPFS iff it is LPFS w.r.t. a finite dimensional subspace

Z (of U) with dimZ ≤ n0. 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 V Z

n0 is the unstable subspace.

The functions Φ(n0T, ·)∗ξ with ξ ∈ H and Φ(T, ·)∗cξ with ξ ∈ HC are respectively the solutions to the dual equations ψt − A∗ψ(t) − D(t)∗ψ(t) = 0, t ∈ (0, n0T), ψ(n0T) = ξ and ψt − A∗cψ(t) − D(t)∗cψ(t) = 0, t ∈ (0, T), ψ(T) = ξ.

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

Thus, the condition (c) in Theorem (III) presents a unique contin- uation property for solutions of the first dual equation with initial data in P ∗H1; 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 period- ic 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 parabol- ic 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 uh(·) ∈ C(R+; U) with supt∈R+ e¯

δtuh(t)U bounded, s.t.

the solution y(· ; 0, h, uh) is stable.

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

Meanwhile, it was pointed out in the paper of A. Lunardi’s that when

• pen-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 time- invariant 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.

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

We next give two applications of Theorem (III). (i) Let Ω ⊂ Rd (d ≥ 1) be a bounded domain with a C2-boundary ∂Ω. Let Q Ω × R+ and Σ ∂Ω × R+. Let ω ⊂ Ω be a non-empty open subset with the characteristic function χω. 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 = χωu in Q,

y = 0 on Σ, (3.15) where u ∈ L2(R+; L2(Ω)).

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

We can set up (3.15) in our framework: H = U = L2(Ω); A = −∆ with D(A) = H1

0(Ω)∩H2(Ω); D(t) : L2(Ω) = H → L2(Ω) = H by

D(t)z(x) = a(x, t)z(x), x ∈ Ω; B(t) : L2(Ω) = U → L2(Ω) = H by B(t)v(x) = χωv(x), x ∈ Ω. One can check that A, D(·), B(·), defined above, verify assump- tions (H1), (H2) and (H3). By the equivalence (a) ⇔ (d) in Theo- rem (III), and by the unique continuation property for linear parabol- ic equations, we have Corollary (I) Equation (3.15) is PFS w.r.t. P ∗H.

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

(ii) Write λ1 and λ2 for the first and second eigenvalues of −∆. Let ξj be an eigenfunction corresponding to λj. Consider

• y′ − ∆y − λ2y = u, ξ1ξ1 in Q,

y = 0 on Σ, (3.16) where u ∈ L2(R+; L2(Ω)). By a direct calculation, one has Vn0 = Span {ξ1} and H1 ⊇ Span {ξ1, ξ2}. Hence, by the geometric condition in Theorem (III), Equ. (3.16) is not PFS.

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

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

Thank you !

Gengsheng Wang Periodic feedback stabilization