Optimal control of systems with a unitary ... - Semantic Scholar
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Systems & Control Letters 48 (2003) 329 – 340 www.elsevier.com/locate/sysconle
Optimal control of systems with a unitary semigroup and with colocated control and observation George Weiss∗ Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London UK, SW7 2BT Received 2 September 2001; received in revised form 3 February 2002; accepted 1 March 2002 This paper is dedicated to the memory of J.-L. Lions
Abstract We solve the quadratic optimal control problem on an in1nite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The di3culty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Semigroup of unitary operators; Colocated control and observation; Conservative linear system; Quadratic optimal control; Unbounded control and observation operators; Output feedback; Riccati equation
1. Problem formulation and main result In this paper we investigate the standard quadratic optimal control problem for a class of systems described by a second-order di@erential equation in a Hilbert space. Such a di@erential equation is rather common in describing undamped oscillatory systems, such as waves, beams or plates, where the control
This work was supported in part by EPSRC grant GR/R05048/01. ∗ Tel.: +44-171-594-6196; fax: +44-171-823-8125. E-mail address: [email protected] (G. Weiss). URL: http://www.ee.ic.ac.uk/CAP
and the sensing are colocated, for example, acting through the same part of the boundary. The original second-order equation can of course be rewritten as a 1rst-order di@erential equation in a product Hilbert space X called the state space. The operator semigroup associated with the 1rst-order equation is unitary, so that the system is not strongly stable. The system may have unbounded control and observation operators, and these are adjoint to each other (this is the formal meaning of “colocated”). The main di3culty is that this oscillatory system may be ill-posed. Actually, it may violate up to three out of the four conditions for well-posedness listed in Curtain and Weiss [4] (the one it does not violate is semigroup generation). Thus, it does not 1t
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/03/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 2 7 6 - 1
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
into any framework established for the treatment of the quadratic optimal control problem. For example, in Lasiecka and Triggiani [7,8], either the semigroup is assumed to be analytic, or the control operator is assumed to be admissible, which is not necessarily the case here. In Callier and Winkin [2], Curtain [3], Sta@ans [10,11] as well as in Weiss and Weiss [17], the system to be controlled is assumed to be well-posed. (All these references make also various other assumptions.) In spite of the di3culty explained above, we provide a surprisingly simple solution to our optimal control problem, by a transformation which leads to a conservative linear system of a special kind, studied in Tucsnak and Weiss [12,16]. Let H be a Hilbert space, and let A0 : D(A0 ) → H be a self-adjoint, positive and boundedly invertible operator. We introduce the scale of Hilbert spaces H ; ∈ R, as follows: for every ¿ 0; H = D(A0 ), with the norm z = A0 zH . The space H− is de1ned by duality with respect to the pivot space H as follows: H− = H∗ for ¿ 0. Equivalently, H− is the completion of H with respect to the norm z− = A− 0 zH . The operator A0 can be extended (or restricted) to each H , such that it becomes a bounded operator A0 : H → H−1
∀ ∈ R:
Let C1 be a bounded linear operator from H1=2 to U, where U is another Hilbert space. We identify U with its dual, so that U = U∗ . We denote B1 = C1∗ , so that B1 ∈ L(U; H−1=2 ). The system u studied here is described by d2 z(t) + A0 z(t) = B1 u(t); dt 2
(1.1)
z(0) = z0 ;
(1.2)
y(t) =
z(0) ˙ = w0 ;
d C1 z(t); dt
(1.3)
where t ∈ [0; ∞) is the time. Eq. (1.1) is understood as an equation in H−1=2 , i.e., all the terms are in H−1=2 . The signal u is the input function, with values in U, and the signal y is the output function, with values in U as well. The state x(t) of this system, its initial state x0 and its state space X are de1ned
by
x(t) =
z(t)
z(t) ˙
x0 =
;
z0
w0
;
X = H1=2 × H:
(1.4)
As mentioned earlier, systems in the class just described may be not well-posed, so we have to be careful about the meaning of state trajectories and output functions. To discuss this, we rewrite (1.1) as a 1rst-order di@erential equation: x(t) ˙ = Au x(t) + Bu u(t); where Au =
0
I
−A0
0
(1.5)
;
Bu =
0 B1
;
(1.6)
D(Au ) = H1 × H1=2 : It is easy to check that Au is skew-adjoint on X and hence, it generates a strongly continuous group of unitary operators on X, denoted by Tu . Such a group may describe, for example, oscillations of an undamped Pexible structure. The superscript “u” used above and in u may stand for “unitary” or for “unstable”. It will be useful to note that for every s ∈ C with −s2 ∈ (A0 ), sI I (sI − Au )−1 = (s2 I + A0 )−1 : (1.7) −A0 sI The operators Tut (with t ∈ R) have a natural bounded u extension to the Hilbert space X−1 de1ned by u X−1 = H × H−1=2 ;
the generator of this extended semigroup is an extension of Au whose domain is X, and we have u ). The resolvent (sI − Au )−1 (whenBu ∈ L(U; X−1 u ever it exists) has a bounded extension to X−1 . We use the same notation for the original operators and the extended ones. Now it is clear that, if x0 ∈ X and u ∈ L2 ([0; ∞); U), then the state trajectory x is a u continuous function with values in X−1 , given by t x(t) = Tut x0 + Tut− Bu u() d: (1.8) 0
u
If B were an admissible control operator for Tu , in the sense of [13] or [14], then x from (1.8) would be a continuous X-valued function of t. However, under
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
the given assumptions, this may be not true. Note that we have x(s) ˆ = (sI − Au )−1 x0 + (sI − Au )−1 Bu u(s); ˆ where a hat denotes the Laplace transformation and s ∈ C0 , where C0 is the open right half-plane. In particular, looking at z, the 1rst component of x, we obtain, using formula (1.7), that for all s ∈ C0 , 2
z(s) ˆ = s(s I + A0 ) 2
−1
+ (s I + A0 )
2
z0 + (s I + A0 )
−1
−1
w0
B1 u(s); ˆ
(1.9)
where z0 and w0 are the components of x0 , as in (1.4). Now we discuss the interpretation of the output Eq. (1.3). If x is only known to be a continuous funcu tion with values in X−1 , then (1.3) makes no sense, because z(t) (the 1rst component of x(t)) is not in the domain of C1 (which is H1=2 ). Even if it happens that z(t) is in the domain of C1 , it is still unclear if we can di@erentiate C1 z(t) with respect to t. We shall overcome these di3culties in de1ning y by using the Laplace transformation. Indeed, if C1 were bounded, i.e., if C1 ∈ L(H; U), then (1.9) and (1.3) would imply that for every u ∈ L2 ([0; ∞); U), y(s) ˆ = −C1 A0 (s2 I + A0 )−1 z0 + C1 s(s2 I + A0 )−1 w0 + C1 s(s2 I + A0 )−1 B1 u(s) ˆ
∀s ∈ C0 :
(1.10)
This expression for y(s) ˆ is well de1ned for every x0 ∈ X (i.e., for every z0 ∈ H1=2 and w0 ∈ H ) and for every u ∈ L2 ([0; ∞); U), even if we remove the boundedness assumption on C1 . In this paper, we are only interested in the situation when the output signal y is in L2 ([0; ∞); U). Recall that, according to a well-known theorem of Paley and Wiener, the Laplace transformation is an isomorphism from L2 ([0; ∞); U) to the Hardy space H2 (U) of U-valued analytic functions on the right half-plane C0 . These facts may serve as an intuitive justi1cation for the following de1nition. Denition 1.1. We use the standing assumptions on A0 ; B1 ; C1 , as stated before (1.1). For every x0 ∈ X, we de1ne the set Dx0 by Dx0 = {u ∈ L2 ([0; ∞); U) | yˆ given by (1:10) is in H2 (U)}:
(1.11)
If x0 ∈ X and u ∈ Dx0 , then we de1ne the corresponding output function y of the system u as the inverse Laplace transform of yˆ de1ned in (1.10).
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It is easy to see that D0 is a vector space and, if u ∈ Dx0 then Dx0 = u + D0 . Hence, Dx0 is either empty or it is a linear manifold whose supporting vector space is D0 . If u; x0 and y are as in the de1nition, then (1.10) can be rewritten (using (1.7)) in the form y(s) ˆ = Bu∗ (sI − Au )−1 x0 + Gu (s)u(s); ˆ
(1.12)
where Bu∗ = [0 C1 ] and, for all s ∈ C0 , Gu (s) = C1 s(s2 I + A0 )−1 B1 : We call Gu the transfer function of u , because (1.12) looks like the formula for the Laplace transform of the output function of a well-posed linear system with transfer function Gu (see [10,14]), even though u may be not well-posed. The above de1nition immediately raises the following questions: (1) Is the set Dx0 rich enough (in particular, not empty)? (2) If u; x0 and y are as in the de1nition and z is the 1rst component of the state trajectory x from (1.8) (equivalently, z is given by (1.9)), does y satisfy (1.3) in some reasonable sense? Both answers are positive, and they are contained in the following proposition. Proposition 1.2. With the above notation, for every x0 ∈ X; Dx0 is an in4nite-dimensional linear manifold. For u ∈ Dx0 , the state trajectory x from (1.8) is a continuous function with values in X (so that its 4rst component z is continuous with values in H1=2 ). Moreover, the function C1 z is in the Sobolev space H1 (0; ∞; U), and its distributional derivative is the output function y ∈ L2 ([0; ∞); U). The proof of Proposition 1.2 will be given in Section 3. Remark 1.3. The output signal y of u could be de1ned via (1.10) for every x0 ∈ X and for every u ∈ L2 ([0; ∞); U). To see this, 1rst we factor C1 = Cb A01=2 ;
B1 = A01=2 Bb ;
where Cb ∈ L(H; U) and Bb ∈ L(U; H ) (here the subscript “b” stands for “bounded”). Then we have from (1.10), via a short computation, y(s) ˆ = Cb [I − s2 (s2 I + A0 )−1 ] ×[ − A01=2 z0 + sA0−1=2 w0 + sBb u(s)]: ˆ
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
Since (s2 I + A0 )−1 and u(s) ˆ are uniformly bounded on the right half-plane where Re s ¿ 1, it follows that |y(s)| ˆ 6 K|s|3 on this half-plane. Hence, the function 1 q(s) ˆ = y(s ˆ + 1) (s + 1)4 is in the Hardy space H2 (U), so that it is the Laplace transform of q ∈ L2 ([0; ∞); U). From here, after extending q to be zero for t ¡ 0, we can de1ne d4 y(t) = 4 (et q(t)); dt in the sense of distributions in D (R). However, such a de1nition of the output signal is not needed in this paper, because for the quadratic optimal control problem we only consider those inputs which produce an output in L2 ([0; ∞); U). We associate to the system u from (1.1) to (1.3) the following cost function: ∞ J (x0 ; u) = [y(t)2 + r 2 u(t)2 ] dt; (1.13) 0
where r ¿ 0. Clearly, J (x0 ; u) is 1nite for every u ∈ Dx0 . The optimal control problem is to 1nd, for each x0 ∈ X, the function u ∈ Dx0 which minimizes J (x0 ; u). Moreover, it is desirable to express this optimal input function in feedback form, i.e., to express u(t) as a function of x(t). We introduce the operator A : D(A) → X by 0 I ; A= 1 −A0 − B1 C1 r z 1 D(A) = ∈ H1=2 × H1=2 | A0 z + B1 C1 w∈H : r w It is easy to verify (see, for example, [16, Section 5]) that A is dissipative and onto, and hence it generates a contraction semigroup on X , denoted by T. Theorem 1.4. (1) With the above notation, for every x0 ∈ X, the function ’ de4ned for t ¿ 0 by ’(t) = [C1 0]Tt x0 is in H1 (0; ∞; U). We de4ne the function u0 : [0; ∞) → U by u0 = −(1=r)’(t), ˙ i.e., for almost every t ¿ 0, 1 d [C1 0]Tt x0 : u0 (t) = − (1.14) r dt Then u0 ∈ Dx0 ⊂ L2 ([0; ∞); U ) and J (x0 ; u0 ) 6 rx0 2 .
(2) For every x0 ∈ X, there exists a unique uopt ∈ Dx0 , called the optimal input function, which minimizes J (x0 ; u) over all u ∈ Dx0 . (3) If (A0 ) has measure zero in R, then the optimal input function uopt is the function u0 de4ned in (1.14). (4) If x0 ∈ X is such that limt→∞ Tt x0 = 0, then (again) the optimal input function uopt is u0 de4ned in (1.14), and moreover J (x0 ; u0 ) = rx0 2 . (5) The input function u0 can be obtained by closing the output-feedback loop 1 (1.15) u(t) = − y(t) r around the original system u described by (1.1)– (1.3). The closed-loop semigroup corresponding to this feedback is T. Note that if A−1 0 is compact (which is usually the case in applications), then (A0 ) is countable, so that (A0 ) has measure zero. Then, according to (3) above, we have solved the optimal control problem for u . Moreover, according to (5) above, we have expressed the optimal input in feedback form. If the condition in (3) is not satis1ed, then the solution of the optimal control problem (which exists according to (2)) may be much more di3cult to express. The proof is provided in Section 3. Note that for any x0 ∈ X, the 1rst component of x(t) = Tt x0 is continuously di@erentiable as an H -valued function of t and its derivative is the second component of x(t). (The semigroup Tu also has this property.) Similarly, if x0 ∈ D(A) then the 1rst component of x(t) is continuously di@erentiable as an H1=2 -valued function of t and its derivative is the second component of x(t). From this property of T it follows that if x0 ∈ D(A), then u0 from (1.14) can also be expressed as 1 1 (1.16) u0 (t) = − Bu∗ Tt x0 = − [0 C1 ]Tt x0 : r r In this form, the formula for u0 looks like what we would expect, based on 1nite-dimensional optimal control theory (i.e., considering A0 ; B1 and C1 to be matrices). Remark 1.5. The semigroup T is called strongly stable if limt→∞ Tt x0 = 0 for all x0 ∈ X. Several equivalent conditions for the strong stability of semigroups with this structure are given in [12]. Suppose that T is strongly stable, so that (by point (4) above) for
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
every x0 ∈ X we have uopt = u0 . Then the formula J (x0 ; u0 ) = rx0 2 means that the optimal cost operator corresponding to our optimal control problem is P = rI , so that J (x0 ; uopt ) = Px0 ; x0 . It is easy to see that this P satis1es 1 Au∗ P + PAu + Bu Bu∗ = 2 PBu Bu∗ P Au∗ = −Au : r This is the algebraic Riccati equation that we would expect to hold based on the theory with bounded control and observation operators in Curtain and Zwart [5], or on the theory that allows unbounded operators in Lasiecka and Triggiani [7,8], even though the assumptions in [7,8] are not satis1ed. However, from this fact we cannot conclude directly that the feedback 1 1 u(t) = − 2 Bu∗ Px(t) = − y(t) r r leads to an optimal input function, because (as already mentioned) there is no Riccati equation theory that covers our ill-posed system u . Besides, the above Riccati equation with P = rI holds regardless if T is strongly stable (and regardless if (A0 ) has measure zero), so that it holds also for systems in our class where uopt = u0 and/or where P is not the optimal cost operator. It is trivial to 1nd examples where P = rI is not the optimal cost operator: take C1 = 0. 2. Reduction to another optimal control problem In this section we introduce a conservative linear system using the operators A0 ; B1 and C1 from description (1.1)–(1.3) of the original unstable system u . The input, state and output spaces remain U; X and U. We show that the optimal control problem for u is equivalent to an optimal control problem for . First we rewrite the cost J (x0 ; u) from (1.13) using the parallelogram identity: y(t)2 + r 2 u(t)2 = 12 [ru(t) − y(t)2 + ru(t) + y(t)2 ]: Thus, if we denote y1 (t) = ru(t) − y(t);
u1 (t) = ru(t) + y(t);
(2.1)
and if we regard y1 as the new output function and u1 as the new input function, then J (x0 ; u) = J1 (x0 ; u1 ) 1 ∞ = [y1 (t)2 + u1 (t)2 ] dt: 2 0
333
In terms of the new signals y1 and u1 , Eqs. (1.1) and (1.3) become d2 d 1 1 z(t) + A0 z(t) + B1 C1 z(t) = B1 u1 (t); dt 2 r dt r (2.2) d C1 z(t) + u1 (t): (2.3) dt Now, we introduce the scaled versions of C1 and B1 de1ned by
2 2 C1 ; B0 = B1 ; C0 = r r so that B0 = C0∗ . Then (2.2) and (2.3) can be rewritten as d2 d 1 1 z(t) + A0 z(t) + B0 C0 z(t) = B0 √ u1 (t); dt 2 2 dt 2r 1 d 1 √ y1 (t) = − C0 z(t) + √ u1 (t): dt 2r 2r Finally, we introduce the scaled versions of y1 and u1 by 1 1 ˜ = √ u1 (t): (2.4) y(t) ˜ = √ y1 (t); u(t) 2r 2r Then the last two equations become d2 d 1 z(t) + A0 z(t) + B0 C0 z(t) = B0 u(t); ˜ (2.5) dt 2 2 dt d y(t) ˜ = − C0 z(t) + u(t); ˜ (2.6) dt and the cost function becomes J (x0 ; u) = J˜ (x0 ; u) ˜ ∞ 2 2 [y(t) ˜ + u(t) ˜ ] dt: (2.7) =r y1 (t) = −2
0
Transformations (2.1) and (2.4) are shown as a block diagram in Fig. 1. If we regard u˜ as the new input signal and y˜ as the new output signal, then this is a new system , with the same state and the same state space as for u . However, is much “nicer” because it is well-posed, as we shall see. It is important to note that the transformations (2.1) and (2.4) are reversible, as expressed in matrix form: u˜ r 1 u 1 =√ ; 2r r −1 y˜ y u 1 1 u˜ 1 =√ : (2.8) 2r r −r y y˜
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
Fig. 1. The conservative system with input u˜ and output y, ˜ as obtained from the possibly ill-posed system u (with input u and output y).
Thus, u; ˜ z and y˜ satisfy (2.5) and (2.6) (in the general sense of the equality of the Laplace transforms of the sides) if and only if the corresponding u; z and y satisfy (1.1) and (1.3) (again in the sense of Laplace transforms). Some properties of the system . The system described by (2.5) and (2.6) 1ts into the framework of the papers [16] and [12] by Tucsnak and the author. We know from [16, Theorem 1.1] that (2.5) and (2.6), together with (1.2) de1ne a conservative linear system with input and output space U and state space X. For the concept of a conservative linear system we refer to Arov and Nudelman [1], Weiss et al. [15,16]. The fact that is conservative implies, in particular, that is a well-posed linear system and for every ! ¿ 0 we have the balance equation ! 2 x(!)2 + y(t) ˜ dt 0
=x(0)2 +
0
!
2 u(t) ˜ dt:
(2.9)
Moreover, a similar balance equation holds for the dual system of . The semigroup generator of is A as de1ned before Theorem 1.4, see [16, Theorem 1.3], so that its semigroup is the contraction semigroup T appearing in Theorem 1.4. For every s ∈ C0 , the operator s2 I + A0 + (s=2)B0 C0 ∈ L(H1=2 ; H−1=2 ) is invertible, see [16, Proposition 5.3], and we denote −1 s V (s) = s2 I + A0 + B0 C0 2 ∈ L(H−1=2 ; H1=2 ):
(2.10)
We denote by X−1 the completion of X with respect to the norm x0 −1 = (I − A)−1 x0 : The semigroup T has a continuous extension to X−1 , whose generator is an extension of A, with domain
X. Hence, for every s ∈ (A); (sI − A)−1 can be extended to a bounded operator from X−1 to X. We use the same notation for the original operators and the extended ones. It has been proved in [5, Section 5] that H1=2 × H−1=2 (which obviously contains X) is a subspace of X−1 and on this subspace we have 1 [I − V (s)A ] V (s) 0 ; (sI − A)−1 = s (2.11) −V (s)A0 sV (s) for every s ∈ C0 . The control and observation operators of are given by 0 ; C = [0 − C0 ]: B= B0 Note that B maps into H1=2 × H−1=2 and hence, into X−1 . The domain of C is D(A). The input maps of are de1ned, as usual, by ! T!− Bu() ˜ d; (2.12) #! u˜ = 0
for all ! ¿ 0. We have #! ∈ L(L2 ([0; !]; U); X) with #! 6 1 for all ! ¿ 0, see [16, Proposition 6.1]. The state trajectory of corresponding to the initial state x0 ∈ X and the input function u˜ ∈ L2 ([0; ∞); U) is given by x(t) = Tt x0 + #t u˜
∀t ¿ 0:
This is a continuous and bounded X-valued function of t. If we denote its 1rst component by z(t), then its second component is z(t) ˙ and z satis1es (2.5) for almost every t ¿ 0 (as an equation in H−1=2 ), see [16, Theorem 1.1] for details. The extended output map of is de1ned, as usual, by ($x0 )(t) = CTt x0
∀x0 ∈ D(A); t ¿ 0;
and this operator has a unique continuous extension to X, denoted by the same symbol, so that $ ∈ L(X; L2 ([0; ∞); U)). Moreover, we have
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
$ 6 1, see [16, Proposition 6.2]. We have y = $x0 if and only if y(s) ˆ = C(sI − A)−1 x0 . We have seen in Section 1 that the (possibly ill-posed) system u from (1.1) to (1.3) has the following transfer function: Gu (s) = C1 s(s2 I + A0 )−1 B1 =
r C0 s(s2 I + A0 )−1 B0 ; 2
(2.13)
which is analytic on C0 , the open right half-plane. This means that if x0 = 0 and u ∈ D0 , then uˆ and y, ˆ the Laplace transforms of u and y, are related by ˆ y(s) ˆ = Gu (s)u(s)
∀s ∈ C0 :
The following proposition lists some properties of the transfer function of , denoted by G, which is related to Gu . Proposition 2.1. The transfer function of is given by G(s) = I − C0 sV (s)B0
∀s ∈ C0 ;
(2.14)
where V (s) is the operator de4ned in (2.10). We have G(s) = (rI − Gu (s))(rI + Gu (s))−1
∀s ∈ C0 : (2.15)
The function G satis4es G(s) 6 1 for all s ∈ C0 . If ! ∈ R is such that !2 ∈ (A0 ), then G has an analytic extension to a neighborhood of i! and G∗ (i!)G(i!) = G(i!)G∗ (i!) = I:
(2.16)
In particular, if (A0 ) has measure zero, then (2.16) holds for almost every ! ∈ R. Proof. Eq. (2.14) and the fact that G(s) 6 1 are contained in [16, Theorem 1.3], along with other properties of G. It is easy to check, using (2.1) and (2.4) (or using the block diagram in Fig. 1), that the transfer function of is also given by (2.15). For any ! ∈ R such that !2 ∈ (A0 ), it follows from (2.13) that Gu has an analytic continuation to a neighborhood of i!. For such !, we can factor Gu (i!) = iT (!), where T (!) is a self-adjoint operator in L(U). We have (rI −G(i!))∗ =rI +iT (!); (rI +G(i!))∗ =rI −iT (!),
335
so that using (2.15), G∗ (i!)G(i!) = (rI − iT (!))−1 (rI + iT (!)) ×(rI − iT (!))(rI + iT (!))−1 : Since the factors on the right-hand side commute, we obtain G∗ (i!)G(i!)=I . The proof of G(i!)G∗ (i!)= I is similar. Remark 2.2. A bounded analytic L(U)-valued function de1ned on C0 which satis1es G∗ (i!)G(i!)=I for almost every ! ∈ R is called inner. The values G(i!) are de1ned for almost every ! ∈ R by non-tangential strong limits, see [9, Theorem 4.5]. If the order of the factors G∗ (i!) and G(i!) is reversed, then G is called co-inner. Suppose that is a conservative linear system with semigroup T and transfer function G. It is not di3cult to prove that if T is strongly stable, then G is inner. Similarly, if T∗ is strongly stable, then G is co-inner. 3. Proof of the main results We continue to use the notation from Sections 1 and 2. The following proposition shows that u0 , our candidate optimal input function from Theorem 1.4, can be expressed using $, the extended output map of . Proposition 3.1. For every x0 ∈ X, the function ’ from Theorem 1.4 is in H1 (0; ∞; U), so that u0 = −(1=r)’˙ ∈ L2 ([0; ∞); U). This function u0 from (1.14) is also given by 1 u0 = √ $x0 : 2r
(3.1)
If the input function of u is u0 and its initial state is x0 , then the corresponding output function of u (see (1.10) or (1.12) is −ru0 . Proof. Denoting z(t) = [I 0]Tt x0 , it follows from [16, Theorem 1.3] that z is a solution of (2.5) corresponding to u˜ = 0. Now it follows from [16, Theorem 1.1] that C0 z ∈ H1 (0; ∞; U), and hence the same is true for ’. We see from (2.6) (with u˜ = 0) that ($x0 )(t) = −(d=dt)C0 z(t). Using that C1 = r=2C0 , we obtain (3.1).
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
If the system has input function u˜ = 0 and initial state x0 , then its output function is y˜ =$x0 . According to (2.8), the corresponding signals u and y are 1 u = √ y˜ = u0 ; 2r
1 y = − √ r y˜ = −ru0 : 2r
This proves the last statement in the proposition. We denote by F the extended input–output operator of . Thus, F is a bounded shift-invariant operator on L2 ([0; ∞); U) and y = Fu if and only if yˆ = Guˆ (see [16, Section 3]). If the input function of is u˜ ∈ L2 ([0; ∞); U) and its initial state is x0 ∈ X, then its output function is (as for any well-posed system) y˜ = $x0 + Fu; ˜
(3.2)
and y˜ ∈ L2 ([0; ∞); U). In the following proposition, we use F to describe Dx0 . Proposition 3.2. For every x0 ∈ X, the set Dx0 de4ned in (1.11) is described by Dx0 = u0 + (I + F)L2 ([0; ∞); U); where u0 is the function de4ned in (1.14). Proof. The last part of Proposition 3.1 together with (1.12) implies that − r uˆ 0 (s) = Bu∗ (sI − Au )−1 x0 + Gu (s)uˆ 0 (s):
(3.3)
Let x0 ∈ X and suppose that u ∈ L2 ([0; ∞); U) is of the form given in the proposition, i.e., u = u0 + (I + F)v, with v ∈ L2 ([0; ∞); U). Then uˆ = uˆ 0 + (I + G)v. ˆ Substituting this into (1.12) and using (3.3), we obtain that y, the corresponding output function of u , is given by yˆ = −r uˆ 0 + Gu (I + G)v: ˆ Note that (2.15) implies Gu (I + G) = r(I − G), so that yˆ = −r uˆ 0 + r(I − G)v; ˆ which shows that y ∈ L2 ([0; ∞); U) (because u0 ∈ L2 ([0; ∞); U)). Hence, u ∈ Dx0 . Conversely, let x0 ∈ X and suppose that u ∈ Dx0 . Then, by the de1nition of Dx0 , the corresponding output function y of u is also in L2 ([0; ∞); U). We see from (2.8) that u, ˜ the corresponding input function of , is also in L2 ([0; ∞); U). The corresponding output
function of ; y˜ ∈ L2 ([0; ∞); U) is given by (3.2). Using (2.8), then (3.2) and 1nally (3.1), we have 1 1 ˜ = √ [$x0 + (I + F)u] u = √ (u˜ + y) ˜ 2r 2r 1 ˜ = u0 + (I + F) √ u: 2r √ ˜ we see that u has the structure Denoting v = 1= 2r u, claimed in the proposition. Proof of Proposition 1.2. The fact that Dx0 is an in1nite-dimensional linear manifold follows from Proposition 3.2. Let x0 ∈ X be the initial state of u and let u ∈ Dx0 be its input function. Thus, the corresponding output function of u ; y is in L2 ([0; ∞); U). From (2.8) we see that u, ˜ the corresponding input function of , is in L2 ([0; ∞); U). The state trajectory x of u corresponding to x0 and u is the same as the ˜ Now state trajectory of corresponding to x0 and u. the properties of x and C1 z claimed in Proposition 1.2 follow from [16, Theorem 1.1]. The following proposition is a general result about conservative linear systems. It is simple and probably well known to specialists in conservative systems, but we do not know a good reference for it. Proposition 3.3. Let be a conservative linear system with input space U, state space X, semigroup T, extended output map $ and extended input–output map F. (a) If u ∈ L2 ([0; ∞); U) is such that Fu=u, then $∗ Fu = 0. (b) If x0 ∈ X is such that limt→∞ Tt x0 = 0, then F∗ $x0 = 0. Proof. We will need the input maps of , denoted (as usual) by #! , see (2.12). Let Y denote the output space of . Let P! denote the truncation operator which maps L2 ([0; ∞); U) onto L2 ([0; !]; U), and similarly for Y in place of U. We introduce $! = P! $;
F! = P! FP!
∀! ¿ 0;
which are the usual operators appearing in the de1nition of a well-posed linear system, see for example [14]. It is clear that we have, for any z0 ∈ X and any
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
v ∈ L2 ([0; ∞); U), lim $! z0 = $z0 ;
!→∞
lim F! v = Fv:
!→∞
(3.4)
The fact that is conservative means that the operators T ! #! ! = $ ! F! are unitary, see [12,16,15] for details. From !∗ ! = I we see that T∗! #! + $!∗ F! = 0
or equivalently
#!∗ T! + F!∗ $! = 0:
(3.5)
Now we prove point (a). If Fu = u, then #! u converges to zero, because of the balance equation (2.9) rewritten for the initial state zero: #! u2 + F! u2 = P! u2 : Now we see from the 1rst equation in (3.5) and from the uniform boundedness of the operators T∗! that lim $!∗ F! u = 0:
t→∞
This implies that for any z0 ∈ X we have limt→∞ F! u; $! z0 = 0. From here, using (3.4) we see that Fu; $z0 = 0, which implies that $∗ Fu = 0. We proceed to the proof of (b). If T! x0 converges to zero as t → ∞, then we see from the second equation in (3.5) and from the uniform boundedness of the operators #!∗ that lim F!∗ $! x0 = 0:
!→∞
This implies that for any v ∈ L2 ([0; ∞); U) we have lim!→∞ $! x0 ; F! v = 0. From here, using (3.4) we see that $x0 ; Fv = 0, so that F∗ $x0 = 0. Proof of Theorem 1.4. (1) We know from Proposition 3.1 that indeed ’ ∈ H1 (0; ∞; U), so that u0 ∈ L2 ([0; ∞); U). From the same proposition we know that the output function of u corresponding to the input function u0 is −ru0 , which implies that u0 ∈ Dx0 . Substituting into (2.8) we see that u˜ = 0, so that according to (2.7) we obtain J (x0 ; u) = ry ˜ 2. Since (according to (3.2)) we have y˜ = $x0 , we get J (x0 ; u) = r$x0 2 . Since $ 6 1, we obtain that J (x0 ; u) 6 rx0 2 . (2) The optimal control problem for u has been reduced, via the transformations (2.8), to the optimal
337
control problem for . This can be addressed using the techniques in Sta@ans [10] or Weiss and Weiss [17]. The paper [17] usually assumes that the system to be controlled is weakly regular, but in [17, Section 7] it is pointed out that for the results in that section, the regularity assumption is not needed. For subjective reasons, we will now use the terminology and a result from that section. The Popov function corresponding to with the cost (2.7) is ((i!) = r[G∗ (i!)G(i!) + I ];
(3.6)
which is positive and bounded from below. Hence, by [17, Proposition 7.2] there is a unique optimal input function corresponding to every initial state of , and this can be translated via (2.8) into a unique optimal input function for u . (3) If (A0 ) has measure zero, then (according to Proposition 2.1) G is inner. The Popov function from (3.6) becomes ((i!) = 2rI , which implies that the Toeplitz operator with symbol ( is also 2rI . Using the formula from [17, Proposition 7.2], we see that the optimal input function for is u˜ opt = − 12 F∗ $x0 : The fact that G is inner implies that we have Fu = u for all u ∈ L2 ([0; ∞); U). According to point (a) of Proposition 3.3 we have $∗ F =0, whence F∗ $ =0. Thus, the above formula for u˜ opt shows that in fact u˜ opt = 0. The corresponding output function of is of course $x0 . Using transformation (2.8) to compute the corresponding input of u , we obtain √ that the optimal input function of u is uopt = (1= 2r)$x0 . According to Proposition 3.1, this is the same as u0 from (1.4). (4) If x0 is such that Tt x0 converges to zero, then according to point (b) of Proposition 3.3 we have F∗ $x0 = 0. Using again the formula from [17, Proposition 7.2], we see that the optimal input function for is u˜ opt = 0. By the same argument as in the proof of (3), we obtain that the optimal input function of u is uopt = u0 . Since Tt x0 converges to zero, from the balance Eq. (2.9) with u=0 ˜ we see that $x0 =x0 . We have seen in the proof of (1) that J (x0 ; u) = r$x0 2 . Combining this with our earlier conclusion, we obtain that J (x0 ; u) = rx0 2 . (5) Closing feedback (1.15) around u (i.e., imposing relation (1.15) on u and y) is equivalent, according to (2.8), to imposing the restriction u˜ = 0 on . It is clear that this leads to a unique input function, state
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
trajectory and output function for u . We know from the last part of Proposition 3.1 that the corresponding input function is u0 . It is clear that the state trajectory of corresponding to u˜ = 0 is x(t) = Tt x0 , and this is the same as the state trajectory of u with feedback (1.15). Thus, the closed-loop semigroup is T.
4. An example based on the wave equation We describe a challenging example of an unstable system u of the type introduced in Section 1. We assume that ) ⊂ Rn is a (possibly unbounded) domain with Lipschitz boundary * (such a boundary admits corners and edges). *0 and *1 are non-empty open subsets of * such that *0 ∩ *1 = ∅ and *0 ∪ *1 = *. V We assume We denote by x the space variable (x ∈ )). that the PoincarWe inequality holds for ) and *0 . This means that there exists a c ¿ 0 such that for every f ∈ H1 ()) with f|*0 = 0, 2 (∇f)(x) d x ¿ c |f(x)|2 d x: )
)
This holds, in particular, if ) is bounded. A function b ∈ L∞ (*1 ) is given, with b(x) = 0 for almost every x ∈ *1 . The equations of the system u are z(x; X t) = Yz(x; t) z(x; t) = 0
on ) × [0; ∞);
y(x; t) = b(x)z(x; ˙ t) z(x; 0) = z0 (x);
H1*0 ()) = {g ∈ H1 ()) | (I − R)0g = 0}; gH1 = ∇gH n : The Neumann trace 01 is an operator originally deV by 1ned on C 1 ()) 01 f =
@ f|*1 = ∇f; / |*1 ; @/
where / is the unit vector in the outward normal direction to *1 , which is de1ned almost everywhere on *1 . Thus, 01 is the outward normal derivative restricted to *1 . Using Green’s formula, it is possible to extend 01 to all those f ∈ H1*0 ()) for which Yf ∈ L2 ()) (Yf is computed in the sense of distributions on )). For the details we refer to [16, Section 7] (without any claim of originality). We put
A0 z = −Yz;
on *1 × [0; ∞); on );
(4.1)
where u is the input function and y is the output function. The functions z0 and w0 are the initial state of the system. We shall often write z(t) to denote a function of x, meaning that z(t)(x) = z(x; t), and similarly for other functions. To put Eqs. (4.1) into the framework (1.1)–(1.3) studied in this paper, we introduce the Hilbert spaces H = L2 ()) and U = L2 (*1 ). The Dirichlet trace opV erator 0 is initially de1ned for any function g ∈ C 1 ()) by 0g = g|* :
We call 00 g the Dirichlet trace of g on *1 . If we regard L2 (*1 ) as a subspace of L2 (*), then I − R is the restriction from L2 (*) onto L2 (*0 ) and we de1ne the Hilbert space
We de1ne the operator A0 : D(A0 ) ⊂ L2 ()) → L2 ()) by
on *1 × [0; ∞);
z(x; ˙ 0) = w0 (x)
00 g = R0g:
Z0 = {f ∈ H1*0 ()) | Yf ∈ L2 ()); 01 f ∈ bL2 (*1 )}:
on *0 × [0; ∞);
@ z(x; t) = b(x)u(x; t) @/
If we regard 0g as an element of L2 (*), then the operator 0 has a continuous extension to H1 ()). We denote by R the usual restriction operator mapping L2 (*) onto L2 (*1 ) and for all g ∈ H1 ()) we put
D(A0 ) = {z ∈ Z0 | 01 z = 0}:
Then A0 is self-adjoint, positive and boundedly invertible (the bounded invertibility of A0 follows from the PoincarWe inequality). The norms z and the spaces H , with ∈ R, are de1ned as in the Section 1. In particular, it can be checked (see [16, Section 7]) that H1=2 = D(A01=2 ) = H1*0 ()) and z21=2
=
A01=2 z2H
=
)
∇z(x)2 d x:
We de1ne the operator C1 ∈ L(H1=2 ; U ) by V 0: C1 = b0
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
Here, bV is the operator of pointwise multiplication with the complex conjugate of the function b introduced earlier. We put B1 = C1∗ , as in Section 1. An explicit description of B1 can be found in [16, Section 7]. We will also need √ √ C0 = 2C1 ; B0 = 2B1 ; to make it easier to follow [16], which is written in terms of C0 and B0 . It can be checked (see again [16, Section 7]) that we have Z0 = H1 + A−1 0 B0 U. We de1ne the operators G0 ; G1 : Z0 → U by G1 = b−1 01 ;
G0 =
√1 2
G1 :
Note that b−1 01 cannot be de1ned on the larger space of those f ∈ H1*0 ()) for which Yf ∈ L2 ()), but on Z0 , its de1nition makes sense because 01 f ∈ bL2 (*1 ). Clearly we have G0 H1 = {0} and it can be checked (see [16, Section 7]) that G0 A−1 0 B0 = I
or equivalently
G1 A−1 0 B1 = I:
(4.2)
We need one more operator: L0 : Z0 → H is de1ned by L0 = A0 − B0 G0 = A0 − B1 G1 . The fact that L0 maps indeed into H follows from (4.2), see [16, Section 6]. It is not di3cult to check that in fact L0 =−4, see [16, Section 7]. Now assuming that z(t) ∈ Z0 ; z(t) ˙ ∈ H1=2 and z(t) X ∈ H , we can rewrite (4.1) in the form z(t) X + L0 z(t) = 0; z(0) = z0 ;
G1 z(t) = u(t);
z(0) ˙ = w0 ;
y(t) = C1 z(t): ˙
(4.3)
Using formulas (4.2) and L0 = A0 − B1 G1 , it is easy to transform these into the equations (1.1)–(1.3). The transformations from (1.1)–(1.3) to (4.1) work also in the opposite way, if we assume again that z(t) ∈ Z0 ; z(t) ˙ ∈ H1=2 and z(t) X ∈ H. The state space X is de1ned, as in Section 1, by X = H1=2 × H , so that X = H1*0 ()) × L2 ()): It is known that for n ¿ 1, this system is ill-posed. In fact, using the notation from (1.6), Bu is not admissible for the unitary group generated by Au , see Lasiecka
339
and Triggiani [6]. We de1ne the cost function ∞ [y(t)2 + u(t)2 ] dt; J (x0 ; u) = 0
which corresponds to (1.13) with r = 1. Now we see that C0 and B0 de1ned above are the same as those de1ned in Section 2. The semigroup T is de1ned as in Section 1. It is proved in [12] that T is always strongly stable. Hence, we can apply point (4) of Theorem 1.4 to conclude that the optimal input function is generated by the feedback u = −y. Moreover, the optimal cost operator corresponding to this system with this cost function is P = I , and the closed-loop semigroup is T.
References [1] D.Z. Arov, M.A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996) 1–45. [2] F. Callier, J. Winkin, LQ-optimal control of in1nitedimensional systems by spectral factorization, Automatica 28 (1992) 757–770. [3] R.F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems, Math. Control, Signals Systems 14 (2001) 299–337. [4] R.F. Curtain, G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), in: F. Kappel, K. Kunisch, W. Schappacher (Eds.), Control and Estimation of Distributed Parameter Systems, BirkhXauser, Basel, 1989, pp. 41–59. [5] R.F. Curtain, H.J. Zwart, An Introduction to In1niteDimensional Linear Systems Theory, Springer, New York, 1995. [6] I. Lasiecka, R. Triggiani, A cosine operator approach to modeling L2 (0; T ; L2 (*))-boundary input hyperbolic equations, Appl. Math. Optim. 7 (1981) 35–93. [7] I. Lasiecka, R. Triggiani, Di@erential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Vol. 164, Springer, Berlin, 1991. [8] I. Lasiecka, R. Triggiani, Control Theory for Partial Di@erential Equations: Continuous and Approximation Theories, Cambridge University Press, Cambridge, UK, 2000. [9] M. Rosenblum, J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, New York, 1985. [10] O.J. Sta@ans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc. 349 (1997) 3679– 3715. [11] O.J. Sta@ans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim. 37 (1998) 131–164.
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[12] M. Tucsnak, G. Weiss, How to get a conservative well-posed linear system out of thin air, Part II: controllability and stability, 2001, submitted for publication. [13] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim. 27 (1989) 527–545. [14] G. Weiss, Transfer functions of regular linear systems. Part I: characterizations of regularity, Trans. Amer. Math. Soc. 342 (1994) 827–854.
[15] G. Weiss, O.J. Sta@ans, M. Tucsnak, Well-posed linear systems—a survey with emphasis on conservative systems, Appl. Math. Comput. Sci. 11 (2001) 101–127. [16] G. Weiss, M. Tucsnak, How to get a conservative well-posed linear system out of thin air, Part I: well-posedness and energy balance, 2001, submitted for publication. [17] M. Weiss, G. Weiss, Optimal control of stable weakly regular linear systems, Math. Control, Signals Systems 10 (1997) 287–330.
Systems & Control Letters 48 (2003) 329 – 340 www.elsevier.com/locate/sysconle
Optimal control of systems with a unitary semigroup and with colocated control and observation George Weiss∗ Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London UK, SW7 2BT Received 2 September 2001; received in revised form 3 February 2002; accepted 1 March 2002 This paper is dedicated to the memory of J.-L. Lions
Abstract We solve the quadratic optimal control problem on an in1nite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The di3culty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Semigroup of unitary operators; Colocated control and observation; Conservative linear system; Quadratic optimal control; Unbounded control and observation operators; Output feedback; Riccati equation
1. Problem formulation and main result In this paper we investigate the standard quadratic optimal control problem for a class of systems described by a second-order di@erential equation in a Hilbert space. Such a di@erential equation is rather common in describing undamped oscillatory systems, such as waves, beams or plates, where the control
This work was supported in part by EPSRC grant GR/R05048/01. ∗ Tel.: +44-171-594-6196; fax: +44-171-823-8125. E-mail address: [email protected] (G. Weiss). URL: http://www.ee.ic.ac.uk/CAP
and the sensing are colocated, for example, acting through the same part of the boundary. The original second-order equation can of course be rewritten as a 1rst-order di@erential equation in a product Hilbert space X called the state space. The operator semigroup associated with the 1rst-order equation is unitary, so that the system is not strongly stable. The system may have unbounded control and observation operators, and these are adjoint to each other (this is the formal meaning of “colocated”). The main di3culty is that this oscillatory system may be ill-posed. Actually, it may violate up to three out of the four conditions for well-posedness listed in Curtain and Weiss [4] (the one it does not violate is semigroup generation). Thus, it does not 1t
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/03/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 2 7 6 - 1
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
into any framework established for the treatment of the quadratic optimal control problem. For example, in Lasiecka and Triggiani [7,8], either the semigroup is assumed to be analytic, or the control operator is assumed to be admissible, which is not necessarily the case here. In Callier and Winkin [2], Curtain [3], Sta@ans [10,11] as well as in Weiss and Weiss [17], the system to be controlled is assumed to be well-posed. (All these references make also various other assumptions.) In spite of the di3culty explained above, we provide a surprisingly simple solution to our optimal control problem, by a transformation which leads to a conservative linear system of a special kind, studied in Tucsnak and Weiss [12,16]. Let H be a Hilbert space, and let A0 : D(A0 ) → H be a self-adjoint, positive and boundedly invertible operator. We introduce the scale of Hilbert spaces H ; ∈ R, as follows: for every ¿ 0; H = D(A0 ), with the norm z = A0 zH . The space H− is de1ned by duality with respect to the pivot space H as follows: H− = H∗ for ¿ 0. Equivalently, H− is the completion of H with respect to the norm z− = A− 0 zH . The operator A0 can be extended (or restricted) to each H , such that it becomes a bounded operator A0 : H → H−1
∀ ∈ R:
Let C1 be a bounded linear operator from H1=2 to U, where U is another Hilbert space. We identify U with its dual, so that U = U∗ . We denote B1 = C1∗ , so that B1 ∈ L(U; H−1=2 ). The system u studied here is described by d2 z(t) + A0 z(t) = B1 u(t); dt 2
(1.1)
z(0) = z0 ;
(1.2)
y(t) =
z(0) ˙ = w0 ;
d C1 z(t); dt
(1.3)
where t ∈ [0; ∞) is the time. Eq. (1.1) is understood as an equation in H−1=2 , i.e., all the terms are in H−1=2 . The signal u is the input function, with values in U, and the signal y is the output function, with values in U as well. The state x(t) of this system, its initial state x0 and its state space X are de1ned
by
x(t) =
z(t)
z(t) ˙
x0 =
;
z0
w0
;
X = H1=2 × H:
(1.4)
As mentioned earlier, systems in the class just described may be not well-posed, so we have to be careful about the meaning of state trajectories and output functions. To discuss this, we rewrite (1.1) as a 1rst-order di@erential equation: x(t) ˙ = Au x(t) + Bu u(t); where Au =
0
I
−A0
0
(1.5)
;
Bu =
0 B1
;
(1.6)
D(Au ) = H1 × H1=2 : It is easy to check that Au is skew-adjoint on X and hence, it generates a strongly continuous group of unitary operators on X, denoted by Tu . Such a group may describe, for example, oscillations of an undamped Pexible structure. The superscript “u” used above and in u may stand for “unitary” or for “unstable”. It will be useful to note that for every s ∈ C with −s2 ∈ (A0 ), sI I (sI − Au )−1 = (s2 I + A0 )−1 : (1.7) −A0 sI The operators Tut (with t ∈ R) have a natural bounded u extension to the Hilbert space X−1 de1ned by u X−1 = H × H−1=2 ;
the generator of this extended semigroup is an extension of Au whose domain is X, and we have u ). The resolvent (sI − Au )−1 (whenBu ∈ L(U; X−1 u ever it exists) has a bounded extension to X−1 . We use the same notation for the original operators and the extended ones. Now it is clear that, if x0 ∈ X and u ∈ L2 ([0; ∞); U), then the state trajectory x is a u continuous function with values in X−1 , given by t x(t) = Tut x0 + Tut− Bu u() d: (1.8) 0
u
If B were an admissible control operator for Tu , in the sense of [13] or [14], then x from (1.8) would be a continuous X-valued function of t. However, under
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
the given assumptions, this may be not true. Note that we have x(s) ˆ = (sI − Au )−1 x0 + (sI − Au )−1 Bu u(s); ˆ where a hat denotes the Laplace transformation and s ∈ C0 , where C0 is the open right half-plane. In particular, looking at z, the 1rst component of x, we obtain, using formula (1.7), that for all s ∈ C0 , 2
z(s) ˆ = s(s I + A0 ) 2
−1
+ (s I + A0 )
2
z0 + (s I + A0 )
−1
−1
w0
B1 u(s); ˆ
(1.9)
where z0 and w0 are the components of x0 , as in (1.4). Now we discuss the interpretation of the output Eq. (1.3). If x is only known to be a continuous funcu tion with values in X−1 , then (1.3) makes no sense, because z(t) (the 1rst component of x(t)) is not in the domain of C1 (which is H1=2 ). Even if it happens that z(t) is in the domain of C1 , it is still unclear if we can di@erentiate C1 z(t) with respect to t. We shall overcome these di3culties in de1ning y by using the Laplace transformation. Indeed, if C1 were bounded, i.e., if C1 ∈ L(H; U), then (1.9) and (1.3) would imply that for every u ∈ L2 ([0; ∞); U), y(s) ˆ = −C1 A0 (s2 I + A0 )−1 z0 + C1 s(s2 I + A0 )−1 w0 + C1 s(s2 I + A0 )−1 B1 u(s) ˆ
∀s ∈ C0 :
(1.10)
This expression for y(s) ˆ is well de1ned for every x0 ∈ X (i.e., for every z0 ∈ H1=2 and w0 ∈ H ) and for every u ∈ L2 ([0; ∞); U), even if we remove the boundedness assumption on C1 . In this paper, we are only interested in the situation when the output signal y is in L2 ([0; ∞); U). Recall that, according to a well-known theorem of Paley and Wiener, the Laplace transformation is an isomorphism from L2 ([0; ∞); U) to the Hardy space H2 (U) of U-valued analytic functions on the right half-plane C0 . These facts may serve as an intuitive justi1cation for the following de1nition. Denition 1.1. We use the standing assumptions on A0 ; B1 ; C1 , as stated before (1.1). For every x0 ∈ X, we de1ne the set Dx0 by Dx0 = {u ∈ L2 ([0; ∞); U) | yˆ given by (1:10) is in H2 (U)}:
(1.11)
If x0 ∈ X and u ∈ Dx0 , then we de1ne the corresponding output function y of the system u as the inverse Laplace transform of yˆ de1ned in (1.10).
331
It is easy to see that D0 is a vector space and, if u ∈ Dx0 then Dx0 = u + D0 . Hence, Dx0 is either empty or it is a linear manifold whose supporting vector space is D0 . If u; x0 and y are as in the de1nition, then (1.10) can be rewritten (using (1.7)) in the form y(s) ˆ = Bu∗ (sI − Au )−1 x0 + Gu (s)u(s); ˆ
(1.12)
where Bu∗ = [0 C1 ] and, for all s ∈ C0 , Gu (s) = C1 s(s2 I + A0 )−1 B1 : We call Gu the transfer function of u , because (1.12) looks like the formula for the Laplace transform of the output function of a well-posed linear system with transfer function Gu (see [10,14]), even though u may be not well-posed. The above de1nition immediately raises the following questions: (1) Is the set Dx0 rich enough (in particular, not empty)? (2) If u; x0 and y are as in the de1nition and z is the 1rst component of the state trajectory x from (1.8) (equivalently, z is given by (1.9)), does y satisfy (1.3) in some reasonable sense? Both answers are positive, and they are contained in the following proposition. Proposition 1.2. With the above notation, for every x0 ∈ X; Dx0 is an in4nite-dimensional linear manifold. For u ∈ Dx0 , the state trajectory x from (1.8) is a continuous function with values in X (so that its 4rst component z is continuous with values in H1=2 ). Moreover, the function C1 z is in the Sobolev space H1 (0; ∞; U), and its distributional derivative is the output function y ∈ L2 ([0; ∞); U). The proof of Proposition 1.2 will be given in Section 3. Remark 1.3. The output signal y of u could be de1ned via (1.10) for every x0 ∈ X and for every u ∈ L2 ([0; ∞); U). To see this, 1rst we factor C1 = Cb A01=2 ;
B1 = A01=2 Bb ;
where Cb ∈ L(H; U) and Bb ∈ L(U; H ) (here the subscript “b” stands for “bounded”). Then we have from (1.10), via a short computation, y(s) ˆ = Cb [I − s2 (s2 I + A0 )−1 ] ×[ − A01=2 z0 + sA0−1=2 w0 + sBb u(s)]: ˆ
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Since (s2 I + A0 )−1 and u(s) ˆ are uniformly bounded on the right half-plane where Re s ¿ 1, it follows that |y(s)| ˆ 6 K|s|3 on this half-plane. Hence, the function 1 q(s) ˆ = y(s ˆ + 1) (s + 1)4 is in the Hardy space H2 (U), so that it is the Laplace transform of q ∈ L2 ([0; ∞); U). From here, after extending q to be zero for t ¡ 0, we can de1ne d4 y(t) = 4 (et q(t)); dt in the sense of distributions in D (R). However, such a de1nition of the output signal is not needed in this paper, because for the quadratic optimal control problem we only consider those inputs which produce an output in L2 ([0; ∞); U). We associate to the system u from (1.1) to (1.3) the following cost function: ∞ J (x0 ; u) = [y(t)2 + r 2 u(t)2 ] dt; (1.13) 0
where r ¿ 0. Clearly, J (x0 ; u) is 1nite for every u ∈ Dx0 . The optimal control problem is to 1nd, for each x0 ∈ X, the function u ∈ Dx0 which minimizes J (x0 ; u). Moreover, it is desirable to express this optimal input function in feedback form, i.e., to express u(t) as a function of x(t). We introduce the operator A : D(A) → X by 0 I ; A= 1 −A0 − B1 C1 r z 1 D(A) = ∈ H1=2 × H1=2 | A0 z + B1 C1 w∈H : r w It is easy to verify (see, for example, [16, Section 5]) that A is dissipative and onto, and hence it generates a contraction semigroup on X , denoted by T. Theorem 1.4. (1) With the above notation, for every x0 ∈ X, the function ’ de4ned for t ¿ 0 by ’(t) = [C1 0]Tt x0 is in H1 (0; ∞; U). We de4ne the function u0 : [0; ∞) → U by u0 = −(1=r)’(t), ˙ i.e., for almost every t ¿ 0, 1 d [C1 0]Tt x0 : u0 (t) = − (1.14) r dt Then u0 ∈ Dx0 ⊂ L2 ([0; ∞); U ) and J (x0 ; u0 ) 6 rx0 2 .
(2) For every x0 ∈ X, there exists a unique uopt ∈ Dx0 , called the optimal input function, which minimizes J (x0 ; u) over all u ∈ Dx0 . (3) If (A0 ) has measure zero in R, then the optimal input function uopt is the function u0 de4ned in (1.14). (4) If x0 ∈ X is such that limt→∞ Tt x0 = 0, then (again) the optimal input function uopt is u0 de4ned in (1.14), and moreover J (x0 ; u0 ) = rx0 2 . (5) The input function u0 can be obtained by closing the output-feedback loop 1 (1.15) u(t) = − y(t) r around the original system u described by (1.1)– (1.3). The closed-loop semigroup corresponding to this feedback is T. Note that if A−1 0 is compact (which is usually the case in applications), then (A0 ) is countable, so that (A0 ) has measure zero. Then, according to (3) above, we have solved the optimal control problem for u . Moreover, according to (5) above, we have expressed the optimal input in feedback form. If the condition in (3) is not satis1ed, then the solution of the optimal control problem (which exists according to (2)) may be much more di3cult to express. The proof is provided in Section 3. Note that for any x0 ∈ X, the 1rst component of x(t) = Tt x0 is continuously di@erentiable as an H -valued function of t and its derivative is the second component of x(t). (The semigroup Tu also has this property.) Similarly, if x0 ∈ D(A) then the 1rst component of x(t) is continuously di@erentiable as an H1=2 -valued function of t and its derivative is the second component of x(t). From this property of T it follows that if x0 ∈ D(A), then u0 from (1.14) can also be expressed as 1 1 (1.16) u0 (t) = − Bu∗ Tt x0 = − [0 C1 ]Tt x0 : r r In this form, the formula for u0 looks like what we would expect, based on 1nite-dimensional optimal control theory (i.e., considering A0 ; B1 and C1 to be matrices). Remark 1.5. The semigroup T is called strongly stable if limt→∞ Tt x0 = 0 for all x0 ∈ X. Several equivalent conditions for the strong stability of semigroups with this structure are given in [12]. Suppose that T is strongly stable, so that (by point (4) above) for
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
every x0 ∈ X we have uopt = u0 . Then the formula J (x0 ; u0 ) = rx0 2 means that the optimal cost operator corresponding to our optimal control problem is P = rI , so that J (x0 ; uopt ) = Px0 ; x0 . It is easy to see that this P satis1es 1 Au∗ P + PAu + Bu Bu∗ = 2 PBu Bu∗ P Au∗ = −Au : r This is the algebraic Riccati equation that we would expect to hold based on the theory with bounded control and observation operators in Curtain and Zwart [5], or on the theory that allows unbounded operators in Lasiecka and Triggiani [7,8], even though the assumptions in [7,8] are not satis1ed. However, from this fact we cannot conclude directly that the feedback 1 1 u(t) = − 2 Bu∗ Px(t) = − y(t) r r leads to an optimal input function, because (as already mentioned) there is no Riccati equation theory that covers our ill-posed system u . Besides, the above Riccati equation with P = rI holds regardless if T is strongly stable (and regardless if (A0 ) has measure zero), so that it holds also for systems in our class where uopt = u0 and/or where P is not the optimal cost operator. It is trivial to 1nd examples where P = rI is not the optimal cost operator: take C1 = 0. 2. Reduction to another optimal control problem In this section we introduce a conservative linear system using the operators A0 ; B1 and C1 from description (1.1)–(1.3) of the original unstable system u . The input, state and output spaces remain U; X and U. We show that the optimal control problem for u is equivalent to an optimal control problem for . First we rewrite the cost J (x0 ; u) from (1.13) using the parallelogram identity: y(t)2 + r 2 u(t)2 = 12 [ru(t) − y(t)2 + ru(t) + y(t)2 ]: Thus, if we denote y1 (t) = ru(t) − y(t);
u1 (t) = ru(t) + y(t);
(2.1)
and if we regard y1 as the new output function and u1 as the new input function, then J (x0 ; u) = J1 (x0 ; u1 ) 1 ∞ = [y1 (t)2 + u1 (t)2 ] dt: 2 0
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In terms of the new signals y1 and u1 , Eqs. (1.1) and (1.3) become d2 d 1 1 z(t) + A0 z(t) + B1 C1 z(t) = B1 u1 (t); dt 2 r dt r (2.2) d C1 z(t) + u1 (t): (2.3) dt Now, we introduce the scaled versions of C1 and B1 de1ned by
2 2 C1 ; B0 = B1 ; C0 = r r so that B0 = C0∗ . Then (2.2) and (2.3) can be rewritten as d2 d 1 1 z(t) + A0 z(t) + B0 C0 z(t) = B0 √ u1 (t); dt 2 2 dt 2r 1 d 1 √ y1 (t) = − C0 z(t) + √ u1 (t): dt 2r 2r Finally, we introduce the scaled versions of y1 and u1 by 1 1 ˜ = √ u1 (t): (2.4) y(t) ˜ = √ y1 (t); u(t) 2r 2r Then the last two equations become d2 d 1 z(t) + A0 z(t) + B0 C0 z(t) = B0 u(t); ˜ (2.5) dt 2 2 dt d y(t) ˜ = − C0 z(t) + u(t); ˜ (2.6) dt and the cost function becomes J (x0 ; u) = J˜ (x0 ; u) ˜ ∞ 2 2 [y(t) ˜ + u(t) ˜ ] dt: (2.7) =r y1 (t) = −2
0
Transformations (2.1) and (2.4) are shown as a block diagram in Fig. 1. If we regard u˜ as the new input signal and y˜ as the new output signal, then this is a new system , with the same state and the same state space as for u . However, is much “nicer” because it is well-posed, as we shall see. It is important to note that the transformations (2.1) and (2.4) are reversible, as expressed in matrix form: u˜ r 1 u 1 =√ ; 2r r −1 y˜ y u 1 1 u˜ 1 =√ : (2.8) 2r r −r y y˜
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
Fig. 1. The conservative system with input u˜ and output y, ˜ as obtained from the possibly ill-posed system u (with input u and output y).
Thus, u; ˜ z and y˜ satisfy (2.5) and (2.6) (in the general sense of the equality of the Laplace transforms of the sides) if and only if the corresponding u; z and y satisfy (1.1) and (1.3) (again in the sense of Laplace transforms). Some properties of the system . The system described by (2.5) and (2.6) 1ts into the framework of the papers [16] and [12] by Tucsnak and the author. We know from [16, Theorem 1.1] that (2.5) and (2.6), together with (1.2) de1ne a conservative linear system with input and output space U and state space X. For the concept of a conservative linear system we refer to Arov and Nudelman [1], Weiss et al. [15,16]. The fact that is conservative implies, in particular, that is a well-posed linear system and for every ! ¿ 0 we have the balance equation ! 2 x(!)2 + y(t) ˜ dt 0
=x(0)2 +
0
!
2 u(t) ˜ dt:
(2.9)
Moreover, a similar balance equation holds for the dual system of . The semigroup generator of is A as de1ned before Theorem 1.4, see [16, Theorem 1.3], so that its semigroup is the contraction semigroup T appearing in Theorem 1.4. For every s ∈ C0 , the operator s2 I + A0 + (s=2)B0 C0 ∈ L(H1=2 ; H−1=2 ) is invertible, see [16, Proposition 5.3], and we denote −1 s V (s) = s2 I + A0 + B0 C0 2 ∈ L(H−1=2 ; H1=2 ):
(2.10)
We denote by X−1 the completion of X with respect to the norm x0 −1 = (I − A)−1 x0 : The semigroup T has a continuous extension to X−1 , whose generator is an extension of A, with domain
X. Hence, for every s ∈ (A); (sI − A)−1 can be extended to a bounded operator from X−1 to X. We use the same notation for the original operators and the extended ones. It has been proved in [5, Section 5] that H1=2 × H−1=2 (which obviously contains X) is a subspace of X−1 and on this subspace we have 1 [I − V (s)A ] V (s) 0 ; (sI − A)−1 = s (2.11) −V (s)A0 sV (s) for every s ∈ C0 . The control and observation operators of are given by 0 ; C = [0 − C0 ]: B= B0 Note that B maps into H1=2 × H−1=2 and hence, into X−1 . The domain of C is D(A). The input maps of are de1ned, as usual, by ! T!− Bu() ˜ d; (2.12) #! u˜ = 0
for all ! ¿ 0. We have #! ∈ L(L2 ([0; !]; U); X) with #! 6 1 for all ! ¿ 0, see [16, Proposition 6.1]. The state trajectory of corresponding to the initial state x0 ∈ X and the input function u˜ ∈ L2 ([0; ∞); U) is given by x(t) = Tt x0 + #t u˜
∀t ¿ 0:
This is a continuous and bounded X-valued function of t. If we denote its 1rst component by z(t), then its second component is z(t) ˙ and z satis1es (2.5) for almost every t ¿ 0 (as an equation in H−1=2 ), see [16, Theorem 1.1] for details. The extended output map of is de1ned, as usual, by ($x0 )(t) = CTt x0
∀x0 ∈ D(A); t ¿ 0;
and this operator has a unique continuous extension to X, denoted by the same symbol, so that $ ∈ L(X; L2 ([0; ∞); U)). Moreover, we have
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
$ 6 1, see [16, Proposition 6.2]. We have y = $x0 if and only if y(s) ˆ = C(sI − A)−1 x0 . We have seen in Section 1 that the (possibly ill-posed) system u from (1.1) to (1.3) has the following transfer function: Gu (s) = C1 s(s2 I + A0 )−1 B1 =
r C0 s(s2 I + A0 )−1 B0 ; 2
(2.13)
which is analytic on C0 , the open right half-plane. This means that if x0 = 0 and u ∈ D0 , then uˆ and y, ˆ the Laplace transforms of u and y, are related by ˆ y(s) ˆ = Gu (s)u(s)
∀s ∈ C0 :
The following proposition lists some properties of the transfer function of , denoted by G, which is related to Gu . Proposition 2.1. The transfer function of is given by G(s) = I − C0 sV (s)B0
∀s ∈ C0 ;
(2.14)
where V (s) is the operator de4ned in (2.10). We have G(s) = (rI − Gu (s))(rI + Gu (s))−1
∀s ∈ C0 : (2.15)
The function G satis4es G(s) 6 1 for all s ∈ C0 . If ! ∈ R is such that !2 ∈ (A0 ), then G has an analytic extension to a neighborhood of i! and G∗ (i!)G(i!) = G(i!)G∗ (i!) = I:
(2.16)
In particular, if (A0 ) has measure zero, then (2.16) holds for almost every ! ∈ R. Proof. Eq. (2.14) and the fact that G(s) 6 1 are contained in [16, Theorem 1.3], along with other properties of G. It is easy to check, using (2.1) and (2.4) (or using the block diagram in Fig. 1), that the transfer function of is also given by (2.15). For any ! ∈ R such that !2 ∈ (A0 ), it follows from (2.13) that Gu has an analytic continuation to a neighborhood of i!. For such !, we can factor Gu (i!) = iT (!), where T (!) is a self-adjoint operator in L(U). We have (rI −G(i!))∗ =rI +iT (!); (rI +G(i!))∗ =rI −iT (!),
335
so that using (2.15), G∗ (i!)G(i!) = (rI − iT (!))−1 (rI + iT (!)) ×(rI − iT (!))(rI + iT (!))−1 : Since the factors on the right-hand side commute, we obtain G∗ (i!)G(i!)=I . The proof of G(i!)G∗ (i!)= I is similar. Remark 2.2. A bounded analytic L(U)-valued function de1ned on C0 which satis1es G∗ (i!)G(i!)=I for almost every ! ∈ R is called inner. The values G(i!) are de1ned for almost every ! ∈ R by non-tangential strong limits, see [9, Theorem 4.5]. If the order of the factors G∗ (i!) and G(i!) is reversed, then G is called co-inner. Suppose that is a conservative linear system with semigroup T and transfer function G. It is not di3cult to prove that if T is strongly stable, then G is inner. Similarly, if T∗ is strongly stable, then G is co-inner. 3. Proof of the main results We continue to use the notation from Sections 1 and 2. The following proposition shows that u0 , our candidate optimal input function from Theorem 1.4, can be expressed using $, the extended output map of . Proposition 3.1. For every x0 ∈ X, the function ’ from Theorem 1.4 is in H1 (0; ∞; U), so that u0 = −(1=r)’˙ ∈ L2 ([0; ∞); U). This function u0 from (1.14) is also given by 1 u0 = √ $x0 : 2r
(3.1)
If the input function of u is u0 and its initial state is x0 , then the corresponding output function of u (see (1.10) or (1.12) is −ru0 . Proof. Denoting z(t) = [I 0]Tt x0 , it follows from [16, Theorem 1.3] that z is a solution of (2.5) corresponding to u˜ = 0. Now it follows from [16, Theorem 1.1] that C0 z ∈ H1 (0; ∞; U), and hence the same is true for ’. We see from (2.6) (with u˜ = 0) that ($x0 )(t) = −(d=dt)C0 z(t). Using that C1 = r=2C0 , we obtain (3.1).
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G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
If the system has input function u˜ = 0 and initial state x0 , then its output function is y˜ =$x0 . According to (2.8), the corresponding signals u and y are 1 u = √ y˜ = u0 ; 2r
1 y = − √ r y˜ = −ru0 : 2r
This proves the last statement in the proposition. We denote by F the extended input–output operator of . Thus, F is a bounded shift-invariant operator on L2 ([0; ∞); U) and y = Fu if and only if yˆ = Guˆ (see [16, Section 3]). If the input function of is u˜ ∈ L2 ([0; ∞); U) and its initial state is x0 ∈ X, then its output function is (as for any well-posed system) y˜ = $x0 + Fu; ˜
(3.2)
and y˜ ∈ L2 ([0; ∞); U). In the following proposition, we use F to describe Dx0 . Proposition 3.2. For every x0 ∈ X, the set Dx0 de4ned in (1.11) is described by Dx0 = u0 + (I + F)L2 ([0; ∞); U); where u0 is the function de4ned in (1.14). Proof. The last part of Proposition 3.1 together with (1.12) implies that − r uˆ 0 (s) = Bu∗ (sI − Au )−1 x0 + Gu (s)uˆ 0 (s):
(3.3)
Let x0 ∈ X and suppose that u ∈ L2 ([0; ∞); U) is of the form given in the proposition, i.e., u = u0 + (I + F)v, with v ∈ L2 ([0; ∞); U). Then uˆ = uˆ 0 + (I + G)v. ˆ Substituting this into (1.12) and using (3.3), we obtain that y, the corresponding output function of u , is given by yˆ = −r uˆ 0 + Gu (I + G)v: ˆ Note that (2.15) implies Gu (I + G) = r(I − G), so that yˆ = −r uˆ 0 + r(I − G)v; ˆ which shows that y ∈ L2 ([0; ∞); U) (because u0 ∈ L2 ([0; ∞); U)). Hence, u ∈ Dx0 . Conversely, let x0 ∈ X and suppose that u ∈ Dx0 . Then, by the de1nition of Dx0 , the corresponding output function y of u is also in L2 ([0; ∞); U). We see from (2.8) that u, ˜ the corresponding input function of , is also in L2 ([0; ∞); U). The corresponding output
function of ; y˜ ∈ L2 ([0; ∞); U) is given by (3.2). Using (2.8), then (3.2) and 1nally (3.1), we have 1 1 ˜ = √ [$x0 + (I + F)u] u = √ (u˜ + y) ˜ 2r 2r 1 ˜ = u0 + (I + F) √ u: 2r √ ˜ we see that u has the structure Denoting v = 1= 2r u, claimed in the proposition. Proof of Proposition 1.2. The fact that Dx0 is an in1nite-dimensional linear manifold follows from Proposition 3.2. Let x0 ∈ X be the initial state of u and let u ∈ Dx0 be its input function. Thus, the corresponding output function of u ; y is in L2 ([0; ∞); U). From (2.8) we see that u, ˜ the corresponding input function of , is in L2 ([0; ∞); U). The state trajectory x of u corresponding to x0 and u is the same as the ˜ Now state trajectory of corresponding to x0 and u. the properties of x and C1 z claimed in Proposition 1.2 follow from [16, Theorem 1.1]. The following proposition is a general result about conservative linear systems. It is simple and probably well known to specialists in conservative systems, but we do not know a good reference for it. Proposition 3.3. Let be a conservative linear system with input space U, state space X, semigroup T, extended output map $ and extended input–output map F. (a) If u ∈ L2 ([0; ∞); U) is such that Fu=u, then $∗ Fu = 0. (b) If x0 ∈ X is such that limt→∞ Tt x0 = 0, then F∗ $x0 = 0. Proof. We will need the input maps of , denoted (as usual) by #! , see (2.12). Let Y denote the output space of . Let P! denote the truncation operator which maps L2 ([0; ∞); U) onto L2 ([0; !]; U), and similarly for Y in place of U. We introduce $! = P! $;
F! = P! FP!
∀! ¿ 0;
which are the usual operators appearing in the de1nition of a well-posed linear system, see for example [14]. It is clear that we have, for any z0 ∈ X and any
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v ∈ L2 ([0; ∞); U), lim $! z0 = $z0 ;
!→∞
lim F! v = Fv:
!→∞
(3.4)
The fact that is conservative means that the operators T ! #! ! = $ ! F! are unitary, see [12,16,15] for details. From !∗ ! = I we see that T∗! #! + $!∗ F! = 0
or equivalently
#!∗ T! + F!∗ $! = 0:
(3.5)
Now we prove point (a). If Fu = u, then #! u converges to zero, because of the balance equation (2.9) rewritten for the initial state zero: #! u2 + F! u2 = P! u2 : Now we see from the 1rst equation in (3.5) and from the uniform boundedness of the operators T∗! that lim $!∗ F! u = 0:
t→∞
This implies that for any z0 ∈ X we have limt→∞ F! u; $! z0 = 0. From here, using (3.4) we see that Fu; $z0 = 0, which implies that $∗ Fu = 0. We proceed to the proof of (b). If T! x0 converges to zero as t → ∞, then we see from the second equation in (3.5) and from the uniform boundedness of the operators #!∗ that lim F!∗ $! x0 = 0:
!→∞
This implies that for any v ∈ L2 ([0; ∞); U) we have lim!→∞ $! x0 ; F! v = 0. From here, using (3.4) we see that $x0 ; Fv = 0, so that F∗ $x0 = 0. Proof of Theorem 1.4. (1) We know from Proposition 3.1 that indeed ’ ∈ H1 (0; ∞; U), so that u0 ∈ L2 ([0; ∞); U). From the same proposition we know that the output function of u corresponding to the input function u0 is −ru0 , which implies that u0 ∈ Dx0 . Substituting into (2.8) we see that u˜ = 0, so that according to (2.7) we obtain J (x0 ; u) = ry ˜ 2. Since (according to (3.2)) we have y˜ = $x0 , we get J (x0 ; u) = r$x0 2 . Since $ 6 1, we obtain that J (x0 ; u) 6 rx0 2 . (2) The optimal control problem for u has been reduced, via the transformations (2.8), to the optimal
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control problem for . This can be addressed using the techniques in Sta@ans [10] or Weiss and Weiss [17]. The paper [17] usually assumes that the system to be controlled is weakly regular, but in [17, Section 7] it is pointed out that for the results in that section, the regularity assumption is not needed. For subjective reasons, we will now use the terminology and a result from that section. The Popov function corresponding to with the cost (2.7) is ((i!) = r[G∗ (i!)G(i!) + I ];
(3.6)
which is positive and bounded from below. Hence, by [17, Proposition 7.2] there is a unique optimal input function corresponding to every initial state of , and this can be translated via (2.8) into a unique optimal input function for u . (3) If (A0 ) has measure zero, then (according to Proposition 2.1) G is inner. The Popov function from (3.6) becomes ((i!) = 2rI , which implies that the Toeplitz operator with symbol ( is also 2rI . Using the formula from [17, Proposition 7.2], we see that the optimal input function for is u˜ opt = − 12 F∗ $x0 : The fact that G is inner implies that we have Fu = u for all u ∈ L2 ([0; ∞); U). According to point (a) of Proposition 3.3 we have $∗ F =0, whence F∗ $ =0. Thus, the above formula for u˜ opt shows that in fact u˜ opt = 0. The corresponding output function of is of course $x0 . Using transformation (2.8) to compute the corresponding input of u , we obtain √ that the optimal input function of u is uopt = (1= 2r)$x0 . According to Proposition 3.1, this is the same as u0 from (1.4). (4) If x0 is such that Tt x0 converges to zero, then according to point (b) of Proposition 3.3 we have F∗ $x0 = 0. Using again the formula from [17, Proposition 7.2], we see that the optimal input function for is u˜ opt = 0. By the same argument as in the proof of (3), we obtain that the optimal input function of u is uopt = u0 . Since Tt x0 converges to zero, from the balance Eq. (2.9) with u=0 ˜ we see that $x0 =x0 . We have seen in the proof of (1) that J (x0 ; u) = r$x0 2 . Combining this with our earlier conclusion, we obtain that J (x0 ; u) = rx0 2 . (5) Closing feedback (1.15) around u (i.e., imposing relation (1.15) on u and y) is equivalent, according to (2.8), to imposing the restriction u˜ = 0 on . It is clear that this leads to a unique input function, state
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trajectory and output function for u . We know from the last part of Proposition 3.1 that the corresponding input function is u0 . It is clear that the state trajectory of corresponding to u˜ = 0 is x(t) = Tt x0 , and this is the same as the state trajectory of u with feedback (1.15). Thus, the closed-loop semigroup is T.
4. An example based on the wave equation We describe a challenging example of an unstable system u of the type introduced in Section 1. We assume that ) ⊂ Rn is a (possibly unbounded) domain with Lipschitz boundary * (such a boundary admits corners and edges). *0 and *1 are non-empty open subsets of * such that *0 ∩ *1 = ∅ and *0 ∪ *1 = *. V We assume We denote by x the space variable (x ∈ )). that the PoincarWe inequality holds for ) and *0 . This means that there exists a c ¿ 0 such that for every f ∈ H1 ()) with f|*0 = 0, 2 (∇f)(x) d x ¿ c |f(x)|2 d x: )
)
This holds, in particular, if ) is bounded. A function b ∈ L∞ (*1 ) is given, with b(x) = 0 for almost every x ∈ *1 . The equations of the system u are z(x; X t) = Yz(x; t) z(x; t) = 0
on ) × [0; ∞);
y(x; t) = b(x)z(x; ˙ t) z(x; 0) = z0 (x);
H1*0 ()) = {g ∈ H1 ()) | (I − R)0g = 0}; gH1 = ∇gH n : The Neumann trace 01 is an operator originally deV by 1ned on C 1 ()) 01 f =
@ f|*1 = ∇f; / |*1 ; @/
where / is the unit vector in the outward normal direction to *1 , which is de1ned almost everywhere on *1 . Thus, 01 is the outward normal derivative restricted to *1 . Using Green’s formula, it is possible to extend 01 to all those f ∈ H1*0 ()) for which Yf ∈ L2 ()) (Yf is computed in the sense of distributions on )). For the details we refer to [16, Section 7] (without any claim of originality). We put
A0 z = −Yz;
on *1 × [0; ∞); on );
(4.1)
where u is the input function and y is the output function. The functions z0 and w0 are the initial state of the system. We shall often write z(t) to denote a function of x, meaning that z(t)(x) = z(x; t), and similarly for other functions. To put Eqs. (4.1) into the framework (1.1)–(1.3) studied in this paper, we introduce the Hilbert spaces H = L2 ()) and U = L2 (*1 ). The Dirichlet trace opV erator 0 is initially de1ned for any function g ∈ C 1 ()) by 0g = g|* :
We call 00 g the Dirichlet trace of g on *1 . If we regard L2 (*1 ) as a subspace of L2 (*), then I − R is the restriction from L2 (*) onto L2 (*0 ) and we de1ne the Hilbert space
We de1ne the operator A0 : D(A0 ) ⊂ L2 ()) → L2 ()) by
on *1 × [0; ∞);
z(x; ˙ 0) = w0 (x)
00 g = R0g:
Z0 = {f ∈ H1*0 ()) | Yf ∈ L2 ()); 01 f ∈ bL2 (*1 )}:
on *0 × [0; ∞);
@ z(x; t) = b(x)u(x; t) @/
If we regard 0g as an element of L2 (*), then the operator 0 has a continuous extension to H1 ()). We denote by R the usual restriction operator mapping L2 (*) onto L2 (*1 ) and for all g ∈ H1 ()) we put
D(A0 ) = {z ∈ Z0 | 01 z = 0}:
Then A0 is self-adjoint, positive and boundedly invertible (the bounded invertibility of A0 follows from the PoincarWe inequality). The norms z and the spaces H , with ∈ R, are de1ned as in the Section 1. In particular, it can be checked (see [16, Section 7]) that H1=2 = D(A01=2 ) = H1*0 ()) and z21=2
=
A01=2 z2H
=
)
∇z(x)2 d x:
We de1ne the operator C1 ∈ L(H1=2 ; U ) by V 0: C1 = b0
G. Weiss / Systems & Control Letters 48 (2003) 329 – 340
Here, bV is the operator of pointwise multiplication with the complex conjugate of the function b introduced earlier. We put B1 = C1∗ , as in Section 1. An explicit description of B1 can be found in [16, Section 7]. We will also need √ √ C0 = 2C1 ; B0 = 2B1 ; to make it easier to follow [16], which is written in terms of C0 and B0 . It can be checked (see again [16, Section 7]) that we have Z0 = H1 + A−1 0 B0 U. We de1ne the operators G0 ; G1 : Z0 → U by G1 = b−1 01 ;
G0 =
√1 2
G1 :
Note that b−1 01 cannot be de1ned on the larger space of those f ∈ H1*0 ()) for which Yf ∈ L2 ()), but on Z0 , its de1nition makes sense because 01 f ∈ bL2 (*1 ). Clearly we have G0 H1 = {0} and it can be checked (see [16, Section 7]) that G0 A−1 0 B0 = I
or equivalently
G1 A−1 0 B1 = I:
(4.2)
We need one more operator: L0 : Z0 → H is de1ned by L0 = A0 − B0 G0 = A0 − B1 G1 . The fact that L0 maps indeed into H follows from (4.2), see [16, Section 6]. It is not di3cult to check that in fact L0 =−4, see [16, Section 7]. Now assuming that z(t) ∈ Z0 ; z(t) ˙ ∈ H1=2 and z(t) X ∈ H , we can rewrite (4.1) in the form z(t) X + L0 z(t) = 0; z(0) = z0 ;
G1 z(t) = u(t);
z(0) ˙ = w0 ;
y(t) = C1 z(t): ˙
(4.3)
Using formulas (4.2) and L0 = A0 − B1 G1 , it is easy to transform these into the equations (1.1)–(1.3). The transformations from (1.1)–(1.3) to (4.1) work also in the opposite way, if we assume again that z(t) ∈ Z0 ; z(t) ˙ ∈ H1=2 and z(t) X ∈ H. The state space X is de1ned, as in Section 1, by X = H1=2 × H , so that X = H1*0 ()) × L2 ()): It is known that for n ¿ 1, this system is ill-posed. In fact, using the notation from (1.6), Bu is not admissible for the unitary group generated by Au , see Lasiecka
339
and Triggiani [6]. We de1ne the cost function ∞ [y(t)2 + u(t)2 ] dt; J (x0 ; u) = 0
which corresponds to (1.13) with r = 1. Now we see that C0 and B0 de1ned above are the same as those de1ned in Section 2. The semigroup T is de1ned as in Section 1. It is proved in [12] that T is always strongly stable. Hence, we can apply point (4) of Theorem 1.4 to conclude that the optimal input function is generated by the feedback u = −y. Moreover, the optimal cost operator corresponding to this system with this cost function is P = I , and the closed-loop semigroup is T.
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