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Stability of switching infinite-dimensional systems
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Automatica 41 (2005) 75 – 78 www.elsevier.com/locate/automatica
Brief paper
Stability of switching infinite-dimensional systems夡 Amol Sasane∗ Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, USA Received 7 August 2003; received in revised form 4 March 2004; accepted 23 July 2004
Abstract In this note, we generalize the results from Narendra and Balakrishnan (IEEE Trans. Automatic Control 39 (1994) 2469) to the infinite-dimensional system theoretic setting. The paper gives results on the stability of a switching system of the form x(t) ˙ = Ai x(t), i ∈ {1, 2}, when the infinitesimal generators A1 and A2 commute. In addition, the existence of a common quadratic Lyapunov function is demonstrated. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Stability; Lyapunov function; Infinite-dimensional switching systems
1. Introduction and preliminaries It was shown in Narendra and Balakrishnan (1994) that if one has a finite family of exponentially stable linear systems whose matrices commute, then the corresponding switched linear system is exponentially stable for arbitrary switching. Moreover, they gave an explicit expression for a common Lyapunov function for the individual linear systems and the switching system. In Gurvits (1995), the same problem of stability (of the so-called discrete linear inclusion) is considered in the discrete-time case. In this case the motivation for considering this problem arises from a number of different areas of mathematics such as control theory, wavelets and Markov chains. In this note we generalize the basic results from Narendra and Balakrishnan (1994) to the infinite-dimensional system theoretic setting. Let X be a separable Hilbert space and let L(X) denote the set of bounded linear operators on X. We use the
夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Ozbay under the direction of Editor I. Petersen. ∗ Tel.: +1 540 231 2185; fax: +1 540 231 9632. E-mail address: [email protected] (A. Sasane).
0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.07.013
notation (A) to denote the resolvent set of the operator A. For background material on infinite-dimensional systems we refer the reader to Curtain and Zwart (1995). Let A be the infinitesimal generator of a strongly continuous semigroup {T (t)}t 0 on a Hilbert space X. The semigroup {T (t)}t 0 is said to be (1) uniformly bounded if there exists an M such that T (t) M for all t ∈ [0, ∞), (2) strongly stable if for all x ∈ X, limt→∞ T (t)x = 0, (3) exponentially stable if there exists an M and an > 0 such that T (t) Me−t for all t ∈ [0, ∞). We note that exponential stability ⇒ strong stability ⇒ uniform boundedness. The last implication is a consequence of the uniform boundedness principle (see for instance, Kreyszig, 1978, Theorem 4.7-3, p. 249). We recall the following well-known result about Lyapunov equations that we will use (see for instance, Curtain & Zwart, 1995, Exercise 5.3, p. 252). Theorem 1.1. If A is the infinitesimal generator of an exponentially stable, strongly continuous semigroup {T (t)}t 0 on a Hilbert space X, then given any nonnegative Q = Q∗ ∈ L(X) the Lyapunov equation A∗ P x + P Ax = −Qx
for all x ∈ D(A)
76
A. Sasane / Automatica 41 (2005) 75 – 78
has a unique, bounded, self-adjoint and nonnegative solution P given by ∞ Px = T (t)∗ QT (t)x dt for all x ∈ X. 0
The definition of commutativity for unbounded operators is not easy because of the difficulty related with domains. However, for an infinitesimal generator A of a strongly continuous semigroup on a Hilbert space X, we know that if 0 denotes the growth bound of the semigroup, i.e. ∃ M < ∞ such that 0 = inf ∈ R , T (t) M et ∀t 0 then (0 , ∞) ⊂ (A) (see for instance, Curtain & Zwart, 1995, Lemma 2.1.11, p. 24). With this in mind, we adopt the following natural generalized definition of commutativity. Let A1 and A2 be the infinitesimal generators of strongly continuous semigroups on Hilbert space X, with growth bounds 0,1 and 0,2 , respectively. Then A1 commutes with A2 if (rI − A1 )−1 (rI − A2 )−1 = (rI − A2 )−1 (rI − A1 )−1 for all r ∈ max{0,1 , 0,2 }, ∞ . Note that in the case of bounded operators A1 , A2 ∈ L(X), this is equivalent with A1 A2 = A 2 A1 . Consider a family A = {Ai , i ∈ I } of infinitesimal generators of strongly continuous semigroups on the Hilbert space X, parameterized by some index set I. Let : [0, ∞) → I be a piecewise constant function of time, called a switching signal. A switched system is then given by d x(t) = A x(t), dt
t 0, x(0) = x0 ,
(1)
with initial condition x0 ∈ X. We consider mild solutions to (1). By this we mean the following: (1) x(·) ∈ C([0, ∞), X), (2) If a, b ∈ [0, ∞) are such that is constant (equal to, say, i∗ ∈ I) in [a, b], then for all g ∈ C([a, b], X), b−a x(t − a), g(t − a)dt 0 b−a ∗ − x(a), Ti∗ (t) g(t − a) dt = 0. 0
From Theorem 3.1.7 (p. 106) of Curtain and Zwart (1995), it can be proved that for every initial condition x0 ∈ X, there exists a unique mild solution to (1). We say that the switching system (1) is (1) uniformly bounded if for every switching function , there exists a positive constant M such that for every initial condition x0 ∈ X, the corresponding solution satisfies x(t) Mx0 for all t ∈ [0, ∞);
(2) strongly stable if for every switching function , the corresponding solution satisfies x(t) → 0 as t → ∞; (3) exponentially stable if for every switching function , there exists an M and an > 0 such that for every initial condition x0 ∈ X, the corresponding solution satisfies x(t) Me−t x0 for all t ∈ [0, ∞). A continuous function V : X → R is said to be a Lyapunov function for the system (1) if (1) V (x(t)) 0 for all t 0 and (2) V (x(t)) V (x()) if t 0. If is constant and equal to i∗ ∈ I , and V satisfies the above properties, then V is said to be a Lyapunov function for the system d x(t) = Ai∗ x(t), dt
t 0.
(2)
The connection between Lyapunov functions and stability can be found, for example, in the book by Guo, Luo, and Morgül (1999). In order to prove our main result in the next section, we will use to the following lemma, which says that commuting infinitesimal generators give rise to commuting semigroups. Lemma 1.2. Let A1 and A2 be the infinitesimal generators of strongly continuous semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space X. If A1 and A2 commute, then T1 (t1 )T2 (t2 ) = T2 (t2 )T1 (t1 ) for all t1 and t2 in [0, ∞). Proof. Fix any r ∈ max{0,1 , 0,2 }, ∞ and x ∈ X. From Lemma 2.1.11 (Curtain & Zwart, 1995, p. 24), it follows that the Laplace transforms of (rI − A2 )−1 T1 (t)x and of T1 (t)(rI − A2 )−1 x are (rI − A2 )−1 (sI − A1 )−1 x and (sI − A1 )−1 (rI − A2 )−1 x. But since A1 and A2 commute, and using the uniqueness of the Laplace transform (see for instance Curtain & Zwart, 1995, Property A.6.2.b, p. 636), it follows that (rI − A2 )−1 T1 (t)x = T1 (t)(rI − A2 )−1 x for all t ∈ [0, ∞). But again, in the above equation, considering the left- and right-hand sides of the equation (with t fixed) as the restriction of the Laplace transforms of T2 ()T1 (t)x and T1 (t)T2 ()x, respectively, (Curtain & Zwart, 1995, Lemma 2.1.11, p. 24) to the real axis, the analyticity of the Laplace transform (Curtain & Zwart, 1995, Property A.6.2.a, p. 636) and the uniqueness of the Laplace transform, it follows that T2 ()T1 (t)x = T1 (t)T2 ()x
for all t and ∈ [0, ∞).
Since this is true for all x ∈ X, the claim follows.
A. Sasane / Automatica 41 (2005) 75 – 78
2. Main results
is a Lyapunov function for the system
We begin this section with the following elementary theorem, which follows immediately from Lemma 1.2 of the previous section. This theorem gives conditions on the stability of A1 and A2 for the stability of the switched system. Theorem 2.1. Consider the switching system (1) with A = {A1 , A2 } such that A1 and A2 commute and are the infinitesimal generators of exponentially stable semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space X. Then we have the following The switched system is uniformly bounded strongly stable exponentially stable
if both A1 and A2 generate uniformly bounded semigroups strongly stable semigroups exponentially stable semigroups.
Proof. This is a consequence of Lemma 1.2.
In the following theorem we give a common Lyapunov function for the individual linear systems and the switching system. Theorem 2.2. Consider the switching system (1) with A = {A1 , A2 } such that A1 and A2 commute and are the infinitesimal generators of exponentially stable, strongly continuous semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space (X, ·, ·). If P0 = P0∗ ∈ L(X) is nonnegative, then let P1 and P2 be the unique bounded, self-adjoint and nonnegative operators on X such that A∗1 P1 x + P1 A1 x = −P0 x A∗2 P2 x + P2 A2 x = −P1 x
for all x ∈ D(A1 ), for all x ∈ D(A2 ).
(3) (4)
The function V : X → R defined by V (x) = P2 x, x is a common Lyapunov function for the two individual systems (2) and a Lyapunov function for the switching system (1). The operator P2 is given by ∞
∞ ∗ ∗ P2 x = T2 (t) T1 () P0 T1 () T2 (t) dt (5) 0
77
0
for all x ∈ X. Proof. Let V (x) = P2 x, x. If x ∈ D(A2 ), then using (4) we have d V (T2 (t)x) dt = P2 A2 T2 (t)x, T2 (t)x + P2 T2 (t)x, A2 T2 (t)x = −P1 x, x 0. So V (T2 (t)x) V (T2 ()x) for all t 0 and x ∈ D(A2 ). Since D(A2 ) is dense in X and V is continuous, it follows that V (T2 (t)x) V (T2 ()x) for all t 0 and x ∈ X. Thus V
d x(t) = A2 x(t), dt
t 0.
Since {T1 }t 0 and {T2 }t 0 are both exponentially stable, it follows that 0 ∈ (A1 ) ∩ (A2 ). Thus Eqs. (3) and (4) can be equivalently written as −1 ∗ −1 ∗ −1 P1 A−1 1 + (A1 ) P1 = −(A1 ) P0 A1 ,
P2 A−1 2
+ (A∗2 )−1 P2
= −(A∗2 )−1 P1 A−1 2 .
(6) (7)
Pre- and post-multiplying equation (6) by (A∗2 )−1 and A−1 2 , respectively, and using the fact that A1 and A2 commute, we obtain −1 −1 ∗ −1 ∗ −1 ∗ −1 (P2 A−1 1 + (A1 ) P2 )A2 + (A2 ) (P2 A1 + (A1 ) P2 ) −1 = (A∗2 )−1 (A∗1 )−1 P0 A−1 1 A2 .
But since A2 is stable and (A∗1 )−1 P0 A−1 1 is a bounded, selfadjoint and nonnegative operator, it follows from Theorem 1.1 that the bounded and self-adjoint operator P2 A−1 1 + (A∗1 )−1 P2 is nonpositive. Equivalently, A∗1 P2 x + P2 A1 x, x 0
for all x ∈ D(A1 ).
Thus we have d V (T1 (t)x) dt = P2 A1 T1 (t)x, T1 (t)x + P2 T1 (t)x, A1 T1 (t)x 0. So V (T1 (t)x) V (T1 ()x) for all t 0 and x ∈ D(A1 ). Since D(A1 ) is dense in X and V is continuous, it follows that V (T1 (t)x) V (T1 ()x) for all t 0 and x ∈ X. Thus V is a Lyapunov function for the system d x(t) = A1 x(t), dt
t 0.
Since V is a Lyapunov function for the two individual systems (2), it is clear that it is also a Lyapunov function for the switching system. Finally (5) follows from applications of Theorem 1.1 to Eqs. (3) and (4). Remarks. (1) It is easy to see that since ∞
∞ ∗ ∗ P2 x = T2 (t) T1 () P0 T1 () d T2 (t)x dt 0 0 ∞
∞ ∗ ∗ = T1 (t) T2 () P0 T2 () d T1 (t)x dt, 0
0
P2 is also the unique solution to A∗1 P2 x + P2 A1 x = −P3 x
for all x ∈ D(A1 ),
where P3 is the unique solution to A∗2 P3 x + P3 A2 x = −P0 x
for all x ∈ D(A2 ).
78
A. Sasane / Automatica 41 (2005) 75 – 78
(2) Similar results can be proved if for a switching system (1) comprising a finite family of pairwise commuting operators. 3. Conclusion In this paper we have given results on the stability of an infinite-dimensional switching system of the form x(t) ˙ = Ai x(t), i ∈ {1, 2}, with commuting infinitesimal generators A1 and A2 , and proved the existence of a common quadratic Lyapunov function. References Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinitedimensional systems theory. Berlin: Springer. Guo, B. Z., Luo, Z. H., & Morgül, O. (1999). Stability and stabilization of infinite-dimensional systems with applications. Berlin: Springer.
Gurvits, L. (1995). Stability of discrete linear inclusion. Linear Algebra and its Applications, 231, 47–85. Kreyszig, E. (1978). Introductory functional analysis with applications. New York: Wiley. Narendra, K. S., & Balakrishnan, J. (1994). A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on Automatic Control, 39, 2469–2471. Amol Sasane received the B.Tech. degree in Electrical Engineering from Indian Institute of Technology, Bombay in 1997 and the Ph.D. degree in Mathematics from the University of Groningen, The Netherlands in 2001. He has been Assistant Professor at the Department of Mathematics, University of Twente, The Netherlands and School of Mathematics, Tata Institute of Fundamental Research, India. He visited the MittagLeffler Institute, Sweden in 2003. He is currently working as a Senior Research Associate at Virginia Tech. His interests include infinite-dimensional control theory and behaviours.
Automatica 41 (2005) 75 – 78 www.elsevier.com/locate/automatica
Brief paper
Stability of switching infinite-dimensional systems夡 Amol Sasane∗ Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, USA Received 7 August 2003; received in revised form 4 March 2004; accepted 23 July 2004
Abstract In this note, we generalize the results from Narendra and Balakrishnan (IEEE Trans. Automatic Control 39 (1994) 2469) to the infinite-dimensional system theoretic setting. The paper gives results on the stability of a switching system of the form x(t) ˙ = Ai x(t), i ∈ {1, 2}, when the infinitesimal generators A1 and A2 commute. In addition, the existence of a common quadratic Lyapunov function is demonstrated. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Stability; Lyapunov function; Infinite-dimensional switching systems
1. Introduction and preliminaries It was shown in Narendra and Balakrishnan (1994) that if one has a finite family of exponentially stable linear systems whose matrices commute, then the corresponding switched linear system is exponentially stable for arbitrary switching. Moreover, they gave an explicit expression for a common Lyapunov function for the individual linear systems and the switching system. In Gurvits (1995), the same problem of stability (of the so-called discrete linear inclusion) is considered in the discrete-time case. In this case the motivation for considering this problem arises from a number of different areas of mathematics such as control theory, wavelets and Markov chains. In this note we generalize the basic results from Narendra and Balakrishnan (1994) to the infinite-dimensional system theoretic setting. Let X be a separable Hilbert space and let L(X) denote the set of bounded linear operators on X. We use the
夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Ozbay under the direction of Editor I. Petersen. ∗ Tel.: +1 540 231 2185; fax: +1 540 231 9632. E-mail address: [email protected] (A. Sasane).
0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.07.013
notation (A) to denote the resolvent set of the operator A. For background material on infinite-dimensional systems we refer the reader to Curtain and Zwart (1995). Let A be the infinitesimal generator of a strongly continuous semigroup {T (t)}t 0 on a Hilbert space X. The semigroup {T (t)}t 0 is said to be (1) uniformly bounded if there exists an M such that T (t) M for all t ∈ [0, ∞), (2) strongly stable if for all x ∈ X, limt→∞ T (t)x = 0, (3) exponentially stable if there exists an M and an > 0 such that T (t) Me−t for all t ∈ [0, ∞). We note that exponential stability ⇒ strong stability ⇒ uniform boundedness. The last implication is a consequence of the uniform boundedness principle (see for instance, Kreyszig, 1978, Theorem 4.7-3, p. 249). We recall the following well-known result about Lyapunov equations that we will use (see for instance, Curtain & Zwart, 1995, Exercise 5.3, p. 252). Theorem 1.1. If A is the infinitesimal generator of an exponentially stable, strongly continuous semigroup {T (t)}t 0 on a Hilbert space X, then given any nonnegative Q = Q∗ ∈ L(X) the Lyapunov equation A∗ P x + P Ax = −Qx
for all x ∈ D(A)
76
A. Sasane / Automatica 41 (2005) 75 – 78
has a unique, bounded, self-adjoint and nonnegative solution P given by ∞ Px = T (t)∗ QT (t)x dt for all x ∈ X. 0
The definition of commutativity for unbounded operators is not easy because of the difficulty related with domains. However, for an infinitesimal generator A of a strongly continuous semigroup on a Hilbert space X, we know that if 0 denotes the growth bound of the semigroup, i.e. ∃ M < ∞ such that 0 = inf ∈ R , T (t) M et ∀t 0 then (0 , ∞) ⊂ (A) (see for instance, Curtain & Zwart, 1995, Lemma 2.1.11, p. 24). With this in mind, we adopt the following natural generalized definition of commutativity. Let A1 and A2 be the infinitesimal generators of strongly continuous semigroups on Hilbert space X, with growth bounds 0,1 and 0,2 , respectively. Then A1 commutes with A2 if (rI − A1 )−1 (rI − A2 )−1 = (rI − A2 )−1 (rI − A1 )−1 for all r ∈ max{0,1 , 0,2 }, ∞ . Note that in the case of bounded operators A1 , A2 ∈ L(X), this is equivalent with A1 A2 = A 2 A1 . Consider a family A = {Ai , i ∈ I } of infinitesimal generators of strongly continuous semigroups on the Hilbert space X, parameterized by some index set I. Let : [0, ∞) → I be a piecewise constant function of time, called a switching signal. A switched system is then given by d x(t) = A x(t), dt
t 0, x(0) = x0 ,
(1)
with initial condition x0 ∈ X. We consider mild solutions to (1). By this we mean the following: (1) x(·) ∈ C([0, ∞), X), (2) If a, b ∈ [0, ∞) are such that is constant (equal to, say, i∗ ∈ I) in [a, b], then for all g ∈ C([a, b], X), b−a x(t − a), g(t − a)dt 0 b−a ∗ − x(a), Ti∗ (t) g(t − a) dt = 0. 0
From Theorem 3.1.7 (p. 106) of Curtain and Zwart (1995), it can be proved that for every initial condition x0 ∈ X, there exists a unique mild solution to (1). We say that the switching system (1) is (1) uniformly bounded if for every switching function , there exists a positive constant M such that for every initial condition x0 ∈ X, the corresponding solution satisfies x(t) Mx0 for all t ∈ [0, ∞);
(2) strongly stable if for every switching function , the corresponding solution satisfies x(t) → 0 as t → ∞; (3) exponentially stable if for every switching function , there exists an M and an > 0 such that for every initial condition x0 ∈ X, the corresponding solution satisfies x(t) Me−t x0 for all t ∈ [0, ∞). A continuous function V : X → R is said to be a Lyapunov function for the system (1) if (1) V (x(t)) 0 for all t 0 and (2) V (x(t)) V (x()) if t 0. If is constant and equal to i∗ ∈ I , and V satisfies the above properties, then V is said to be a Lyapunov function for the system d x(t) = Ai∗ x(t), dt
t 0.
(2)
The connection between Lyapunov functions and stability can be found, for example, in the book by Guo, Luo, and Morgül (1999). In order to prove our main result in the next section, we will use to the following lemma, which says that commuting infinitesimal generators give rise to commuting semigroups. Lemma 1.2. Let A1 and A2 be the infinitesimal generators of strongly continuous semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space X. If A1 and A2 commute, then T1 (t1 )T2 (t2 ) = T2 (t2 )T1 (t1 ) for all t1 and t2 in [0, ∞). Proof. Fix any r ∈ max{0,1 , 0,2 }, ∞ and x ∈ X. From Lemma 2.1.11 (Curtain & Zwart, 1995, p. 24), it follows that the Laplace transforms of (rI − A2 )−1 T1 (t)x and of T1 (t)(rI − A2 )−1 x are (rI − A2 )−1 (sI − A1 )−1 x and (sI − A1 )−1 (rI − A2 )−1 x. But since A1 and A2 commute, and using the uniqueness of the Laplace transform (see for instance Curtain & Zwart, 1995, Property A.6.2.b, p. 636), it follows that (rI − A2 )−1 T1 (t)x = T1 (t)(rI − A2 )−1 x for all t ∈ [0, ∞). But again, in the above equation, considering the left- and right-hand sides of the equation (with t fixed) as the restriction of the Laplace transforms of T2 ()T1 (t)x and T1 (t)T2 ()x, respectively, (Curtain & Zwart, 1995, Lemma 2.1.11, p. 24) to the real axis, the analyticity of the Laplace transform (Curtain & Zwart, 1995, Property A.6.2.a, p. 636) and the uniqueness of the Laplace transform, it follows that T2 ()T1 (t)x = T1 (t)T2 ()x
for all t and ∈ [0, ∞).
Since this is true for all x ∈ X, the claim follows.
A. Sasane / Automatica 41 (2005) 75 – 78
2. Main results
is a Lyapunov function for the system
We begin this section with the following elementary theorem, which follows immediately from Lemma 1.2 of the previous section. This theorem gives conditions on the stability of A1 and A2 for the stability of the switched system. Theorem 2.1. Consider the switching system (1) with A = {A1 , A2 } such that A1 and A2 commute and are the infinitesimal generators of exponentially stable semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space X. Then we have the following The switched system is uniformly bounded strongly stable exponentially stable
if both A1 and A2 generate uniformly bounded semigroups strongly stable semigroups exponentially stable semigroups.
Proof. This is a consequence of Lemma 1.2.
In the following theorem we give a common Lyapunov function for the individual linear systems and the switching system. Theorem 2.2. Consider the switching system (1) with A = {A1 , A2 } such that A1 and A2 commute and are the infinitesimal generators of exponentially stable, strongly continuous semigroups {T1 (t)}t 0 and {T2 (t)}t 0 , respectively, on the Hilbert space (X, ·, ·). If P0 = P0∗ ∈ L(X) is nonnegative, then let P1 and P2 be the unique bounded, self-adjoint and nonnegative operators on X such that A∗1 P1 x + P1 A1 x = −P0 x A∗2 P2 x + P2 A2 x = −P1 x
for all x ∈ D(A1 ), for all x ∈ D(A2 ).
(3) (4)
The function V : X → R defined by V (x) = P2 x, x is a common Lyapunov function for the two individual systems (2) and a Lyapunov function for the switching system (1). The operator P2 is given by ∞
∞ ∗ ∗ P2 x = T2 (t) T1 () P0 T1 () T2 (t) dt (5) 0
77
0
for all x ∈ X. Proof. Let V (x) = P2 x, x. If x ∈ D(A2 ), then using (4) we have d V (T2 (t)x) dt = P2 A2 T2 (t)x, T2 (t)x + P2 T2 (t)x, A2 T2 (t)x = −P1 x, x 0. So V (T2 (t)x) V (T2 ()x) for all t 0 and x ∈ D(A2 ). Since D(A2 ) is dense in X and V is continuous, it follows that V (T2 (t)x) V (T2 ()x) for all t 0 and x ∈ X. Thus V
d x(t) = A2 x(t), dt
t 0.
Since {T1 }t 0 and {T2 }t 0 are both exponentially stable, it follows that 0 ∈ (A1 ) ∩ (A2 ). Thus Eqs. (3) and (4) can be equivalently written as −1 ∗ −1 ∗ −1 P1 A−1 1 + (A1 ) P1 = −(A1 ) P0 A1 ,
P2 A−1 2
+ (A∗2 )−1 P2
= −(A∗2 )−1 P1 A−1 2 .
(6) (7)
Pre- and post-multiplying equation (6) by (A∗2 )−1 and A−1 2 , respectively, and using the fact that A1 and A2 commute, we obtain −1 −1 ∗ −1 ∗ −1 ∗ −1 (P2 A−1 1 + (A1 ) P2 )A2 + (A2 ) (P2 A1 + (A1 ) P2 ) −1 = (A∗2 )−1 (A∗1 )−1 P0 A−1 1 A2 .
But since A2 is stable and (A∗1 )−1 P0 A−1 1 is a bounded, selfadjoint and nonnegative operator, it follows from Theorem 1.1 that the bounded and self-adjoint operator P2 A−1 1 + (A∗1 )−1 P2 is nonpositive. Equivalently, A∗1 P2 x + P2 A1 x, x 0
for all x ∈ D(A1 ).
Thus we have d V (T1 (t)x) dt = P2 A1 T1 (t)x, T1 (t)x + P2 T1 (t)x, A1 T1 (t)x 0. So V (T1 (t)x) V (T1 ()x) for all t 0 and x ∈ D(A1 ). Since D(A1 ) is dense in X and V is continuous, it follows that V (T1 (t)x) V (T1 ()x) for all t 0 and x ∈ X. Thus V is a Lyapunov function for the system d x(t) = A1 x(t), dt
t 0.
Since V is a Lyapunov function for the two individual systems (2), it is clear that it is also a Lyapunov function for the switching system. Finally (5) follows from applications of Theorem 1.1 to Eqs. (3) and (4). Remarks. (1) It is easy to see that since ∞
∞ ∗ ∗ P2 x = T2 (t) T1 () P0 T1 () d T2 (t)x dt 0 0 ∞
∞ ∗ ∗ = T1 (t) T2 () P0 T2 () d T1 (t)x dt, 0
0
P2 is also the unique solution to A∗1 P2 x + P2 A1 x = −P3 x
for all x ∈ D(A1 ),
where P3 is the unique solution to A∗2 P3 x + P3 A2 x = −P0 x
for all x ∈ D(A2 ).
78
A. Sasane / Automatica 41 (2005) 75 – 78
(2) Similar results can be proved if for a switching system (1) comprising a finite family of pairwise commuting operators. 3. Conclusion In this paper we have given results on the stability of an infinite-dimensional switching system of the form x(t) ˙ = Ai x(t), i ∈ {1, 2}, with commuting infinitesimal generators A1 and A2 , and proved the existence of a common quadratic Lyapunov function. References Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinitedimensional systems theory. Berlin: Springer. Guo, B. Z., Luo, Z. H., & Morgül, O. (1999). Stability and stabilization of infinite-dimensional systems with applications. Berlin: Springer.
Gurvits, L. (1995). Stability of discrete linear inclusion. Linear Algebra and its Applications, 231, 47–85. Kreyszig, E. (1978). Introductory functional analysis with applications. New York: Wiley. Narendra, K. S., & Balakrishnan, J. (1994). A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on Automatic Control, 39, 2469–2471. Amol Sasane received the B.Tech. degree in Electrical Engineering from Indian Institute of Technology, Bombay in 1997 and the Ph.D. degree in Mathematics from the University of Groningen, The Netherlands in 2001. He has been Assistant Professor at the Department of Mathematics, University of Twente, The Netherlands and School of Mathematics, Tata Institute of Fundamental Research, India. He visited the MittagLeffler Institute, Sweden in 2003. He is currently working as a Senior Research Associate at Virginia Tech. His interests include infinite-dimensional control theory and behaviours.