Stability analysis for a dissipative feedback control law

STABILITY ANALYSIS FOR A

DISSIPATIVE FEEDBACK CONTROL LAW

SUNGKWON KANG

1. Introduction

Piezo devices such as piezoceramic patches knwon as collocated rate sensor and actuators are commonly used in control of flexible struc- ture (see, e.g., [1]) and noise reduction. Recently, Ito and Kang ([4]) developed a nonlinear feedback control synthesis for regulating fluid flow using these devices. The control law is designed for driving a given fluid flow to a prescribed equilibrium state and enhancing en- ergy dissipation effects. The controlled system becomes stable while control is activated. In this paper, the two-dimensional Navier-Stokes (N-S) equations with periodic boundary conditions are employed. A sufficient condition on control distribution vectors for changing flow state is obtained. Also, energy dissipation effects (exponential stabil- ity property) of the controlled dynamics is analyzed. These results extend those of [4]. Specifically, the condition on control distribution vectors are weakened so that they can be chosen with more flexibilities.

For a convergent numerical scheme for solving the N-5 equations and computational experiments, see [3,4,5].

A control problem and a weak variational form for the N-S equations are explained briefly in Section 2. In Section 3, the well-posedness and the exponential stability properties of the controlled system are given.

Throughout this paper notations are very standard. We will use the notation I. I without any subindex for vector or operator norm. In all such cases the appropriate index for I. I will be understood from the

Received November 9, 1994.

1991 AMS Subject Classification: 49J20, 76D05, 93B52.

Key words: Control system, Navier-Stokes equations, piezo devices, exponential stability.

This research was supported by the 1994 Chosun University Research Funds

and in part by the BSRI-94-1426.

context. For a given Banach space X, X* and (', ·)x*,x denote the strong dual space of X and the dual product, respectively. If X is a Hilbert space, (-,.) is the scalar inner product.

2. Controlled dynamics

Consider a control problem for the two-dimensional N avier-Stokes equations with periodic boundary conditions. For simplicity, the period in space is chosen to be 1. Let n ⁼ (0,1) x (0,1), and let el = (1,0) and

= (0,1) be the canonical basis elements of R

The governing equations are given by

a ;: - v~u + (U' V)u + Vp = -,(t) Lbi

1 bi · (u - ue)dx,

i=1

n

(2.1) V· u-= 0,

u(t,x+ei) = U(t, x), xER

t>O, u(O,x) = uo(x),

where x = (Xl,X2) E R

U = u(t, x) = (Ul(t,X),U2(t, x)) is the ve- locity vector, v > ° is the nondimensional viscosity, p = pet, x) is the pressure, b

= (bil(X), bi2(X)) with V· bi = ^{0, 1 ::; i ::; rn,} are control input vectors,

= u

x) is a desired equilibrium velocity field, and ,(t) is the control law given by

whel."e I~(u ^::- U~)12 = E;=IJn IY(Ui -: uei)1 2

dx,

= JUl' _~2)~ Ue =

(Ueb U e2),

U

R

is the control space, (2.3) gl(U,Ue) ⁼ v

lV(u - ue)l\

g2(U, U

= IV(u - ue )1 ² + 0:2IB*(u - ue)I&,

0:

with 1 ::;

2

is the weight on control action, IB*(U - ue)lb- =

E::l Un bi(x) . (u(t, x) - ue(x) )dx)

^and B* is the adjoint operator of

B which is to be defined later by equation (2.13). This ,et) drives the

given initial state u(O, .) = uo(') to the desired equilibrium state u

and enhances the energy dissipation effects of the system (2.1) (see [4]). It is a "suboptimaf' control synthesis obtained from the following nonlinear optimal control problem:

Find the optimal control,(t) E L2(0, 00; R+) that minimizes the cost functional

subject to the control system (2.1).

Here, by the suboptimal control law we mean that it contains an es- sential part of a linear optimal control law, and one of its special forms satisfies the Hamilton-Jacobi-Bellman equation arising in a nonlinear programming problem which corresponds to the above optimal control problem.

For the well-posedness and the stability analyses of the controlled system (2.1), consider the following function spaces (see [7]).

H;n(n) = {u E H,'::c(R

u(x + ed = u(x), i = 1,2}, (2.4) V = {u E H~(n) x H~(n) V'. u = O},

H = {u E H~(n) x H~(n) : V'. u = O},

where the subscript p stands for "periodic." The Stokes operator A is defined by

(2.5)

for u, v E V and it is given by (2.6) Au =

due to the periodic boundary conditions. Define a bilinear form a on V x Vby

(2.7) a(u, v) = v(Au, v} h(t) t,(L ^b

^udx) (L ^b

^vdx).

where let) is the control law defined by equation (2.2). For any u, v, w E V, define a trilinear form

(2.8)

where Di = a~i' i = 1,2, and a bilinear continuous operator B from V x V into V* by

(2.9) (B(u, v), w)v.,v = b(u; v, w),

where V* is the dual space of V. It is easy to see that, by integration by parts,

(2.10) b(u;v, w) + b(u;w, v) = 0

for all u, v, wE V due to the periodic boundary conditions.

With the trilinear form b, the variational form of the control sys- tem (2.1) becomes

(

~

^{v) + v(Au, v) + btu; u, v)}

(2.11) + ,(t) t,(L ^b

(u - ue)dx) (L ^b

^{vdx) = 0, v E V,}

u(O) = Uo,

where u(t) = u(t, .). Here, the equilibrium state

e ^satisfies -v!:i..ue +

(u

V)u

= ^{0 in V*, Le.,} (vAu e, v) + b(ue;ue,v) = ^{0 for all v E V.}

Thus, from equation (2.11),

(2.12) (~ (u - ue), v) + a(

e, v) + b(

e;

e, v)

+ btu -

e;

e, v) + b(ue;u - ue,v) = 0 for all v E V.

It is easy to observe that the pressure term Vp in equation (2.1) is

dropped in the variational forms (2.11) and (2.12) due to the divergence

free condition.

We now define a nonlinear operator F E £(V, V*) and a control input operator 8 E £(U, H) by

(2.13) F( u) = P B( u, u) and

m

8f

L ^bi(x)fi'

i=1

for all u E V and f

(!I,h,'" ,fm) E LT, where P is the projection operator from _H~(n) x _H~(n) onto the state space H, B is the bilinear operator defined by equation (2.9), and b

E H, 1 :s; i :s; m. Then the adjoint operator 8* E £(H, U) of 8 is given by

8*u = (In ^b

^udx, In ^b

udx," . , In ^b

^{UdX) for all u}

^H.

Thus, the variational form (2.11) is equivalent, in V*, to

(2.14)

dt u(t) d + ^vAu(t) + ^{F( u( t))}

t)88*( u(t) - u

u(O) =

The term -"'((t)88*(u(t) - Ue) is one of passive feedback forms arising in piezo device control mechanisms. These piezo devices have a special sensing and actuating structure. That is, control actions are given at the same locations where sensors are located.

3. Well-posedness and stability

The following well-posedness property of the controlled system (2.1) is an application of arguments in [2.7,8].

THEOREM 3.1. The control system ^(2.14) (equivalently, the sys- tem (2.1)) has a unique global weak solution u(·) E L

(0, x; V) n

C/oc(O, x: H) n ^H

^(0, cx:: V*) provided that u(O)

E H.

Proof. The bilinear form a defined equation (2.7) is continuous and V-coersive, i.e.,

ja(u,v)/:S; ^.M

Iu/v /vjv

and la(u,u)1 ~

lul~ - v lul~

for some constant M

> 0 and for all u, v E V. From equations (2.8) and (2.10), the trilinear form b satisfies that

b(u;v,w) + ^{b(u;w,v) =} 0 and Ib(u;w,u)1 S M

1ulH lulv Iwlv for some constant M

> 0 and for alLu, v, w E V. Also,

Ib(u - Ue;U - ue,·) + b(u - ue;ue,·) + b(ue;u - Ue, ·)!v*

S (M

1u - UelH + M

luelv)lu - uelv·

for some constants M

> 0 and M

> o. Recall that 'Y(t) E £2(0,00; R+) and b

EH, 1 ::; i ::; m. Hence, by the standard arguments such as Lemma 8.4 in [2] or [7, p. 282], the system (2.12) (equivalently, the system (2.14)) has a unique solution U-U

E £2(0, T; V)nC(O, T;H)n HI(O, T;V*) for arbitrary T >. o. The continuity property u -

E C(O, Tj H) follows from [6]. 0

The exponential stability property of the controlled system (2.1) (equivalently, the system (2.14)) follows from the next theorem.

THEOREM

3.2. Assume that the nondimensional viscosity v> 0 is sufficiently small, say, 0 < v < < 1, and that the control vectors b

EH, 1 ::; i ::; m, and a > 0 in equations (2.1)-(2.2) are chosen so that (3.1) v IV( u - ue) 1

+ a 18*( U -:- ue)lh _~ ,Blu - Ue IJI

for some ,B > 0 and for all

E V. Then

(3.2) Iu(t,·) - ueOIH ::; e-Ptl:U(O,·) - Ue(·)\H.

Proof. By substituting v = u -

in equation (2.12), we have the following estimate.

(~ (u - ue), u - ue) + a(u - ue,U - ue) + b(U -

e;

Ue, U- ue)

+ b(U - Ue;Ue,U- u e) + b(Ue;U - Ue, U- ue) ⁼ O.

From equations (2.8) and (2.10), we have b(U - Ue;Ue,U- ue) ⁼ b(U-

Ue;U- Ue,U- ue) = b(Ue;U- Ue,U- ue) = O. Hence, by the definitions

(2.7) and (2.13) of the bilinear form a and the control input operator B, the above equation becomes

(3.3)

Since for any w > 0,

(3.4) __ (elMtlu _ u ^{1 d}

= _ewt_lu _ u 1 d

+ w _ewtlu _ u

2

2 dt

2 dt

2

by integrating equation (3.4) from 0 to t and from equation (3.3),

~elMt ^{lu(t) - ue /2 -} ~ ^{lu(O) - ue}

²

= i t (~1Iu -u

⁺ ~ ^{Iu -}

e ^{W8 ds}

= i t (-vl\7(u - UeW -,(s)IB*(u - u

)1

+ ~ ^{Iu - u e l 2 ) e lM8 ds.}

By the definition (2.2) of ,et), the above equation becomes (3.5)

1 wtl

1

'2 ^e u(t) - uel - '2lu(O) - uel

= - i t (J91(U, ue) + IB*(u - Ue)l292(U, ue) - ~ lu - Ue12) e lM8 _ds, where 91(U, u e) ^and 92(U, ue) are given by equation (2.3). If the condi- tion (3.1) holds,

J91(U,U e) + IB*(u - Ue)l292(U,Ue)

~ J(,8lu -

+ (1 - 2va)/\7(u - ue)l2IB*(u - ^u e )1 2.

Since 1 :::; a :::;

and 0 < v << 1, 1 - 2va ~ O. Hence, from equation (3.5),

~ewtlu(t) ^{- ue}

^{2 + i t (,8 -} ~)Iu ^{- uel}

^{e W8 ds:::;} ~lu(O) ^{- ue}

and, by setting w = 2,8, we have e

u (t) - u

l

lu(O) - u

l

0

REMARK

3.3. In Theorem 3.2, it is easy to see, from the Poincare inequality (see, e.g., [8, p. 117]), that such f3 > 0 exists. The condition on b

E H in equation (3.1) is a constraint for the control distribution vectors. Also, note that the velocity field u(t,·) approaches the desired equilibrium state u

in H as t

and the approaching rate is exponential.

REMARK

3.4. It is easy to observe, from the proof of Theorem 3.2, that the exponential stability property (3.2) holds when the nondimen- sional viscosity v satisfies 0 < v ::; t.

ACKNOWLEDGMENT. The author thanks the refrees for their valu- able comments and suggestions.

References

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4. ,

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5. ,

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Department of Mathematics College of Natural Sciences Chosun University

Kwangju 501-759, Korea