Algebraic Properties of Riccati equations

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1 Algebraic Properties of Riccati equations Ruth Curtain University of Groningen, The Netherlands (collaboration with Amol Sasane, Royal Institute of Technology, Stockholm, Sweden)

2 Summary

3 Summary Algebraic properties of Riccati equations?

4 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems.

5 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC 1980.

6 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC New LQR result for commutative algebras.

7 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC New LQR result for commutative algebras. A class of symmetric algebras: Wiener algebras satisfying the G-R-S condition.

8 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC New LQR result for commutative algebras. A class of symmetric algebras: Wiener algebras satisfying the G-R-S condition. Applications to spatially invariant systems.

9 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC New LQR result for commutative algebras. A class of symmetric algebras: Wiener algebras satisfying the G-R-S condition. Applications to spatially invariant systems. False claim by Motee, Jadbabaie, IEEE 2008.

10 Summary Algebraic properties of Riccati equations? Motivation: Platoon-type spatially invariant systems. False claim by Chris Byrnes, CDC New LQR result for commutative algebras. A class of symmetric algebras: Wiener algebras satisfying the G-R-S condition. Applications to spatially invariant systems. False claim by Motee, Jadbabaie, IEEE The noncommutative case. Conclusions.

11 LQR Riccati equation

12 LQR Riccati equation Matrix Riccati equation result Let A C n n, B C m n, C C n p. If (A, B) is stabilizable and (A, C) is detectable, then a unique stabilizing solution P C n n of A P + PA PBB P + C C = 0. Stabilizing solution: P = P 0 and A BB P is stable.

13 LQR Riccati equation Matrix Riccati equation result Let A C n n, B C m n, C C n p. If (A, B) is stabilizable and (A, C) is detectable, then a unique stabilizing solution P C n n of A P + PA PBB P + C C = 0. Stabilizing solution: P = P 0 and A BB P is stable. Fundamental question Let A be a complex Banach algebra, with an involution. When will the following LQR Riccati equation have a unique exponentially stabilizing solution P A n n? A P + PA PBB P + C C = 0. Stabilizing solution: P = P and A BB P is exponentially stable, i.e., the semigroup e At on A n n is exponentially stable sup{re λ : λ σ(a)} < 0.

14 A positive result

15 A positive result An example of a Banach algebra with an involution is L(Z), where Z is a Hilbert space and the involution is the adjoint operation. It is not commutative in general, but it does have nice algebraic properties.

16 A positive result An example of a Banach algebra with an involution is L(Z), where Z is a Hilbert space and the involution is the adjoint operation. It is not commutative in general, but it does have nice algebraic properties. LQR Riccati equation Suppose that Z is a Hilbert space and A, B, C L(Z). If (A, B) is exponentially stabilizable and (A, C) is exponentially detectable, then there exists a unique nonnegative solution P L(Z), P = P 0 of the LQR control Riccati equation A P + PA PBB P + C C = 0. There also exists a unique nonnegative solution Q L(Z), Q = Q 0 of the LQR filter Riccati equation AQ + QA QC CQ + BB = 0. Moreover, A BB P and A QCC generate exponentially stable semigroups.

17 Complex Banach algebra A with an involution

18 Complex Banach algebra A with an involution A complex Banach space with a multiplication operation: a, b A = ab A a(bc) = (ab)c, ab a b.

19 Complex Banach algebra A with an involution A complex Banach space with a multiplication operation: a, b A = ab A a(bc) = (ab)c, ab a b. Involution is a map from A to itself: a a with the properties: (a + b) = a + b, (αb) = αb, (ab) = a b, (a ) = a. Commutative complex Banach algebra: ab = ba for all a, b A.

20 Example (Even-Weighted Wiener algebra W α (T)) Let α = (α n ) n Z be any sequence of positive real numbers satisfying W α (T) := α n+m α n α m, α n = α n (even). { f : f (z) = f n z n, z T = unit circle n Z and f Wα(T) := } α n f n <, n Z with pointwise operations.

21 Example (Even-Weighted Wiener algebra W α (T)) Let α = (α n ) n Z be any sequence of positive real numbers satisfying W α (T) := α n+m α n α m, α n = α n (even). { f : f (z) = f n z n, z T = unit circle n Z and f Wα(T) := } α n f n <, n Z with pointwise operations. W α (T) is a unital commutative Banach algebra contained in the commutative Banach algebra L (T) with possible involutions: f (z) := f (z) = n Z f n z n,

22 Example (Even-Weighted Wiener algebra W α (T)) Let α = (α n ) n Z be any sequence of positive real numbers satisfying W α (T) := α n+m α n α m, α n = α n (even). { f : f (z) = f n z n, z T = unit circle n Z and f Wα(T) := } α n f n <, n Z with pointwise operations. W α (T) is a unital commutative Banach algebra contained in the commutative Banach algebra L (T) with possible involutions: f (z) := f (z) = n Z f n z n, and f (z) := f (1/z) = n Z f n z n.

23 Motivation: Spatially Invariant Systems Consider the subalgebra of bounded convolution operators T : l 2 l 2 given by (Tx) l = r Z T r l x r.

24 Motivation: Spatially Invariant Systems Consider the subalgebra of bounded convolution operators T : l 2 l 2 given by (Tx) l = r Z T r l x r. Spatially invariant systems : Σ(A, B, C) with state space l 2 (Z; C n ). A, B, C are matrices whose entries are convolution operators and A, B, C L(l 2 (Z; C n )), a Banach algebra with the involution the adjoint operation.

25 Motivation: Spatially Invariant Systems Consider the subalgebra of bounded convolution operators T : l 2 l 2 given by (Tx) l = r Z T r l x r. Spatially invariant systems : Σ(A, B, C) with state space l 2 (Z; C n ). A, B, C are matrices whose entries are convolution operators and A, B, C L(l 2 (Z; C n )), a Banach algebra with the involution the adjoint operation. Under the Fourier transform F : l 2 (Z; C n ) L 2 (T; C n ). So the study of spatially invariant systems Σ(A, B, C) is transformed to the study of the isometrically isomorphic systems Σ(FAF 1, FBF 1, FCF 1 ):=Σ(Â, ˆB, Ĉ) on L 2 (T; C n ). Since Â, ˆB, Ĉ are multiplication operators on L (T; C n ) they are much easier to handle mathematically.

26 Example Take Â = 0, ˆB(z) = 10 z 1/z, Ĉ = 1. The LQR Riccati equation on L 2 (T) is Â(z) ˆP(z) + ˆP(z)Â(z) ˆP(z)ˆB(z)ˆB(z) ˆP(z) + Ĉ(z) Ĉ(z) = 0. This can be solved algebraically for each z T to obtain the unique positive solution ˆP(z) = = 1 10 z 1/z 1 4 δ k z k, 6 k Z where δ =

27 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ).

28 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ). Existence of Riccati solutions If (Â(z), ˆB(z), Ĉ(z)) is stabilizable and detectable for all z T the Riccati equation has a unique s.a stabilizing solution.

29 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ). Existence of Riccati solutions If (Â(z), ˆB(z), Ĉ(z)) is stabilizable and detectable for all z T the Riccati equation has a unique s.a stabilizing solution. Even for simple examples ˆP(z) = r Z p rz r L (T).

30 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ). Existence of Riccati solutions If (Â(z), ˆB(z), Ĉ(z)) is stabilizable and detectable for all z T the Riccati equation has a unique s.a stabilizing solution. Even for simple examples ˆP(z) = r Z p rz r L (T). But for an implementable control law you need to truncate and the truncation should be a good approximation of ˆP, i.e.,

31 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ). Existence of Riccati solutions If (Â(z), ˆB(z), Ĉ(z)) is stabilizable and detectable for all z T the Riccati equation has a unique s.a stabilizing solution. Even for simple examples ˆP(z) = r Z p rz r L (T). But for an implementable control law you need to truncate and the truncation should be a good approximation of ˆP, i.e., r T α rp r < for some (α r ) or ˆP must be in a Wiener algebra W α (T) L (T).

32 Control of Platoon-type spatially invariant systems Discrete spatially invariant operators are in L(l 2 (Z; C n )) L(L 2 (T; C n )) = L (T; C n n ). Existence of Riccati solutions If (Â(z), ˆB(z), Ĉ(z)) is stabilizable and detectable for all z T the Riccati equation has a unique s.a stabilizing solution. Even for simple examples ˆP(z) = r Z p rz r L (T). But for an implementable control law you need to truncate and the truncation should be a good approximation of ˆP, i.e., r T α rp r < for some (α r ) or ˆP must be in a Wiener algebra W α (T) L (T). Moreover, the larger the weights α r, the better the approximation will be; for example, an exponential weight α r = e r would be perfect.

33 Gelfand Representation Theorem for commutative Banach algebras

34 Gelfand Representation Theorem for commutative Banach algebras Every commutative Banach algebra is isomorphic to an algebra of continuous functions on its maximal ideal space M(A) (a compact Hausdorff space, equipped with the weak * topology). The Gelfand transform is a map ˇ : A C given by ǎ(ϕ) = ϕ(a), ϕ M(A), a A.

35 Gelfand Representation Theorem for commutative Banach algebras Every commutative Banach algebra is isomorphic to an algebra of continuous functions on its maximal ideal space M(A) (a compact Hausdorff space, equipped with the weak * topology). The Gelfand transform is a map ˇ : A C given by ǎ(ϕ) = ϕ(a), ϕ M(A), a A. Example (The commutative Banach algebra W α (T)) This has the maximal ideal space which is isomorphic to the annulus around T: A(ρ) = {z C : 1/ρ z ρ}, ρ = inf n>0 n αn = lim n n αn. For f (z) = k Z f kz k for z T the Gelfand transform is ˇf (z) = f k z k for z A(ρ), the whole annulus. k Z

36 Claim by Chris Byrnes, CDC 1980

37 Claim by Chris Byrnes, CDC 1980 The idea was to note that the Gelfand transforms (Ǎ(ϕ), ˇB(ϕ)), Č(ϕ)) are just complex matrices and so consider the isomorphic finite-dimensional Riccati equation for each ϕ Ǎ(ϕ) ˇP(ϕ)+ˇP(ϕ)Ǎ(ϕ) ˇP(ϕ))ˇB(ϕ)ˇB(ϕ) ˇP(ϕ))+Č(ϕ)) Č(ϕ)) = 0.

38 Claim by Chris Byrnes, CDC 1980 The idea was to note that the Gelfand transforms (Ǎ(ϕ), ˇB(ϕ)), Č(ϕ)) are just complex matrices and so consider the isomorphic finite-dimensional Riccati equation for each ϕ Ǎ(ϕ) ˇP(ϕ)+ˇP(ϕ)Ǎ(ϕ) ˇP(ϕ))ˇB(ϕ)ˇB(ϕ) ˇP(ϕ))+Č(ϕ)) Č(ϕ)) = 0. Theorem Suppose that A be a commutative, unital, complex Banach algebra, with an involution. Denote by Ǎ the Gelfand transform. Let A A n n, B A n m, C A p n be such that for all ϕ M(A), (Ǎ(ϕ), ˇB(ϕ)) is controllable and (Ǎ(ϕ), Č(ϕ)) is observable. Then there exists a solution P A n n such that A P + PA PBB P + C C = 0, (1) and A BB P is asymptotically stable. FALSE: The involution needs to match the complex conjugate.

39 New result by Amol Sasane (special case of SIAM 2011 paper) Let A be a commutative, unital, complex, semisimple Banach algebra. Suppose that A A n n, B A n m, C A p n satisfy the following for all ϕ M(A), (A1) (A ˇ )(ϕ) = (Ǎ(ϕ)), (B ˇ )(ϕ) = (ˇB(ϕ)), (C ˇ )(ϕ) = (Č(ϕ)). (A2) (Ǎ(ϕ), ˇB(ϕ)) is stabilizable, (A3) (Ǎ(ϕ), Č(ϕ)) is detectable. Then there exists a P A n n such that A P + PA PBB P + C C = 0, A BB P is exponentially stable, and P = P.

40 New result by Amol Sasane (special case of SIAM 2011 paper) Let A be a commutative, unital, complex, semisimple Banach algebra. Suppose that A A n n, B A n m, C A p n satisfy the following for all ϕ M(A), (A1) (A ˇ )(ϕ) = (Ǎ(ϕ)), (B ˇ )(ϕ) = (ˇB(ϕ)), (C ˇ )(ϕ) = (Č(ϕ)). (A2) (Ǎ(ϕ), ˇB(ϕ)) is stabilizable, (A3) (Ǎ(ϕ), Č(ϕ)) is detectable. Then there exists a P A n n such that A P + PA PBB P + C C = 0, A BB P is exponentially stable, and P = P. Note: condition (A1) on the involution: it needs to match complex conjugation: In general f (z) := f (z)h f (z), f (z) := f (1/z) f (z). Banach algebras satisfying (A1) = symmetric Banach algebras.

41 Example (Symmetric even-weighted Wiener algebras) Gelfand-Raikov-Shilov condition on the weights α n : ρ = inf n>0 n αn = lim n n αn = 1.

42 Example (Symmetric even-weighted Wiener algebras) Gelfand-Raikov-Shilov condition on the weights α n : ρ = inf n>0 n αn = lim n n αn = 1. Exponential weights do not satisfy this condition, but weights with ρ = 1 are subexponential weights: α n = e α n β, α > 0, 0 β < 1.

43 Example (Symmetric even-weighted Wiener algebras) Gelfand-Raikov-Shilov condition on the weights α n : ρ = inf n>0 n αn = lim n n αn = 1. Exponential weights do not satisfy this condition, but weights with ρ = 1 are subexponential weights: α n = e α n β, α > 0, 0 β < 1. When the Gelfand-Raikov-Shilov condition is satisfied, the annulus A(ρ) degenerates to the circle T, and for the Banach algebra W α (T) the involution reduces to f (z) := f (z) (z T), forf W α (T). With this involution W α (T) is a symmetric Banach algebra.

44 Example (Symmetric even-weighted Wiener algebras) Gelfand-Raikov-Shilov condition on the weights α n : ρ = inf n>0 n αn = lim n n αn = 1. Exponential weights do not satisfy this condition, but weights with ρ = 1 are subexponential weights: α n = e α n β, α > 0, 0 β < 1. When the Gelfand-Raikov-Shilov condition is satisfied, the annulus A(ρ) degenerates to the circle T, and for the Banach algebra W α (T) the involution reduces to f (z) := f (z) (z T), forf W α (T). With this involution W α (T) is a symmetric Banach algebra. Thus for matrices A, B, C with entries from W α (T) their involution is the usual Hermitian adjoint operation: A (z) = A(z) and assumption (A1) is always satisfied.

45 LQR control of spatially invariant systems The Sasane result solves the approximation problem of LQR controllers for spatially invariant systems for subexponential decay, but does not allow for exponential decay.

46 LQR control of spatially invariant systems The Sasane result solves the approximation problem of LQR controllers for spatially invariant systems for subexponential decay, but does not allow for exponential decay. Claim by Motee and Jadbabaie, IEEE 2008 They considered LQR control of a general class of spatially distributed systems. These include the platoon type spatially invariant systems whose operators have components in the even-weighted algebras W α (T). In particular, W τ (T) with the exponential weights α n = e τ n.

47 LQR control of spatially invariant systems The Sasane result solves the approximation problem of LQR controllers for spatially invariant systems for subexponential decay, but does not allow for exponential decay. Claim by Motee and Jadbabaie, IEEE 2008 They considered LQR control of a general class of spatially distributed systems. These include the platoon type spatially invariant systems whose operators have components in the even-weighted algebras W α (T). In particular, W τ (T) with the exponential weights α n = e τ n. Theorem: spatially invariant scalar case If A, B, C W τ (T) and the LQR Riccati equation A P + PA PBB P + C C = 0. has a unique positive definite solution P L(L 2 (T)), then P W τ (T).

48 LQR control of spatially invariant systems The Sasane result solves the approximation problem of LQR controllers for spatially invariant systems for subexponential decay, but does not allow for exponential decay. Claim by Motee and Jadbabaie, IEEE 2008 They considered LQR control of a general class of spatially distributed systems. These include the platoon type spatially invariant systems whose operators have components in the even-weighted algebras W α (T). In particular, W τ (T) with the exponential weights α n = e τ n. Theorem: spatially invariant scalar case If A, B, C W τ (T) and the LQR Riccati equation A P + PA PBB P + C C = 0. has a unique positive definite solution P L(L 2 (T)), then P W τ (T). Counterexample: Ruth Curtain, IEEE 2008.

49 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case.

50 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and

51 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space;

52 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space; A has the inverse-closed property :

53 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space; A has the inverse-closed property : D A, D 1 L(Z) = D 1 A;

54 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space; A has the inverse-closed property : D A, D 1 L(Z) = D 1 A; A A M A L(Z). If A, B, C A and (A, B, C) is exponentially stabilizable and detectable wrt Z, then P A, where P L(Z) is the unique nonnegative solution to the control Riccati equations: A P + PA PBB P + C C = 0.

55 The noncommutative case Spatially distributed systems correspond to noncommutative Banach algebras. Bunce 1985 proved a positive result for C -algebras, but these are very rare and do not cover the spatially distributed case. Recent result (Curtain 2011) Suppose that A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space; A has the inverse-closed property : D A, D 1 L(Z) = D 1 A; A A M A L(Z). If A, B, C A and (A, B, C) is exponentially stabilizable and detectable wrt Z, then P A, where P L(Z) is the unique nonnegative solution to the control Riccati equations: A P + PA PBB P + C C = 0. With recent results by Gröchenig & Leinert (2006) this result covers the spatially distributed LQR problem posed by Motee & Jadbabaie.

56 Curious An essential step in the proof uses the following result from Curtain & Opmeer MCSS, 2006: Let P, Q be the self-adjoint solutions to the control and filter Riccati equations A P + PA PBB P + C C = 0 QA + QA QC CQ + B B = 0. Then the solution to the following Lyapunov equation is X = P(I + PQ) 1. (A BB P)X + X(A BB P) = BB

57 Conclusions

58 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras.

59 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws.

60 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws. Similar techniques can be used to obtain algebraic properties of other Riccati equations, including H, positive-real and bounded-real type equations.

61 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws. Similar techniques can be used to obtain algebraic properties of other Riccati equations, including H, positive-real and bounded-real type equations. Recent results: algebraic properties of the LQR Riccati equation for inverse-closed noncommutative algebras.

62 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws. Similar techniques can be used to obtain algebraic properties of other Riccati equations, including H, positive-real and bounded-real type equations. Recent results: algebraic properties of the LQR Riccati equation for inverse-closed noncommutative algebras. The results have direct applications to spatially distributed systems.

63 Conclusions For commutative, unital, complex, semisimple Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws. Similar techniques can be used to obtain algebraic properties of other Riccati equations, including H, positive-real and bounded-real type equations. Recent results: algebraic properties of the LQR Riccati equation for inverse-closed noncommutative algebras. The results have direct applications to spatially distributed systems. Generalization to the algebraic properties of the LQR Riccati equation when A is an unbounded operator on A but it generates a semigroup on A.

64 Congratulations to Birgit and Roland! for organizing a most successful and enjoyable workshop!

65 Congratulations to Birgit and Roland! for organizing a most successful and enjoyable workshop! This does not happen without excellent support behind the scenes. In this case we extend our appreciation and thanks to Kerstin Scheibler.

66 Congratulations to Birgit and Roland! for organizing a most successful and enjoyable workshop! This does not happen without excellent support behind the scenes. In this case we extend our appreciation and thanks to Kerstin Scheibler. P.S. The morning and afternoon coffees and teas were enjoyed by all, especially the fresh fruit by those who missed breakfast and

67 Congratulations to Birgit and Roland! for organizing a most successful and enjoyable workshop! This does not happen without excellent support behind the scenes. In this case we extend our appreciation and thanks to Kerstin Scheibler. P.S. The morning and afternoon coffees and teas were enjoyed by all, especially the fresh fruit by those who missed breakfast and last but not least

68 Congratulations to Birgit and Roland! for organizing a most successful and enjoyable workshop! This does not happen without excellent support behind the scenes. In this case we extend our appreciation and thanks to Kerstin Scheibler. P.S. The morning and afternoon coffees and teas were enjoyed by all, especially the fresh fruit by those who missed breakfast and last but not least those delicious cakes!

Algebraic Properties of Riccati equations. Ruth Curtain University of Groningen, The Netherlands

Algebraic Properties of Riccati equations Ruth Curtain University of Groningen, The Netherlands Special Recognition Peter Falb Jan Willems Tony Pritchard Hans Zwart Riccati equation Ṗ(t) + A(t) P(t) +