Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints

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1 Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints Joseph Winkin Namur Center of Complex Systems (naxys) and Dept. of Mathematics, University of Namur, Belgium Joint work with Bouchra Abouzaïd, Elarbi Achhab (University Chouaïb Doukkali, El Jadida, Morocco) and Vincent Wertz (Université Catholique de Louvain, Belgium) Gipsa-lab, Grenoble (F) Automatic Control Seminar 8 April 2014

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3 Stabilizing a positive dynamical systems and simultaneously maintaining its positivity can be incompatible objectives. Unstable and positive (finite-dimensional) system: dx 1 dt (t) = x2(t) + u(t), dx 2 (t) = x1(t) dt It is not possible to stabilize positively this system, i.e. one can t find a state feedback u(t) = Kx(t) such that [ ] k1 1 + k 2 A + BK = 1 0 generates an exponentially stable and positive closed-loop system. However... Need for some tradeoff. Gipsa-lab Seminar 2014 (Grenoble, F) 3

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5 Semi- Results: groups A real vector space X is called an ordered vector space if a partial order is defined in X such that x y in X x + z y + z for all z X, and λx λy for all 0 λ R. Given such a partial order, the positive cone of X is defined by X + = {x X x 0} [X + is a cone: αx + βy X + whenever x, y X + and 0 α, β R. X + ( X + ) = {0}, so X + is proper] Conversely, given a proper cone K in X, a partial order in X is defined by setting x y whenever y x K, and then (X, ) is an ordered vector space with positive cone X + = K. Gipsa-lab Seminar 2014 (Grenoble, F) 5

6 Semi- Results: groups A real Banach space (X, ) is called an ordered Banach space if X is an ordered vector space such that X + is norm closed, i.e. closed in the strong topology. From now on we assume that X is an ordered Banach space with positive cone X +. Gipsa-lab Seminar 2014 (Grenoble, F) 6

7 Semi- Results: groups A family (T (t)) t 0 in L(X) is called a C 0- if T (0) = I, T (t + s) = T (t)t (s), t, s 0 lim T (t)x x = 0, x X t 0 + The infinitesimal generator A of a C 0- (T (t)) t 0 is defined by T (t)x x Ax = lim t 0 + t on Definition D(A) = {x X T (t)x x lim exists in X} t 0 + t (T (t)) t 0 is said to be positive if all the operators T (t), t 0, are positive, i.e. T (t)x + X + for all t 0 Gipsa-lab Seminar 2014 (Grenoble, F) 7

8 Semi- Results: groups Proposition A C 0- (T (t)) t 0 is positive if and only if where its resolvent R(λ, A) := (λi A) 1 is positive for all λ > ω 0, log T (t) log T (t) ω 0 := inf = lim t>0 t t t is the growth constant of (T (t)) t 0. Gipsa-lab Seminar 2014 (Grenoble, F) 8

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10 Algebraic Conditions of Positivity: Characterization of the positivity of a C 0- in terms of its generator A : Definition A linear operator A : D(A) X is said to have the Off-Diagonal (POD) property if whenever where Au, φ 0 0 u D(A) and φ (X ) + with u, φ = 0 (X ) + = {φ X x, φ 0, x X + } Gipsa-lab Seminar 2014 (Grenoble, F) 10

11 Algebraic Conditions of Positivity: Theorem Let A be the infinitesimal generator of a C 0- (T (t)) t 0 in an ordered Banach space X with int(x + ). The following assertions are equivalent: (i) (T (t)) t 0 is a positive C 0-. (ii) A satisfies the POD property. Moreover, if one of the two assertions above hold, then where s(a) = inf{λ R Au λu for some u D(A) int(x + )} denotes the spectral bound of A. s(a) = sup{re(λ) λ σ(a)} Gipsa-lab Seminar 2014 (Grenoble, F) 11

12 Algebraic Conditions of Positivity: Algebraic conditions of positivity for systems defined on a space whose positive cone has an empty interior? Fact a) The positive cone l 2 + of l 2 has an empty interior. b) The positive cone of any infinite-dimensional separable Hilbert space (e.g. L 2 ) has an empty interior. Indeed: every x = (x n) l = for any ball B = B(x, ɛ), there exists a sequence y = (y n) which belongs to B but not to l 2 +. In this case, the POD property of the generator is still necessary but not sufficient for the positivity of the. Gipsa-lab Seminar 2014 (Grenoble, F) 12

13 Algebraic Conditions of Positivity: Let Z be an ordered Banach space such that int(z + ) =. Let {e n} n 1 be a positive Schauder basis of Z, i.e. each element z of Z has a unique representation of the form such that the linear functional z = α ne n n=1 z α n =: z, e n is bounded where α n := the nth coordinate of z with respect to the basis {e n} n 1 and the positive cone is given by { } Z + = z = α ne n α n 0, n. n=1 Gipsa-lab Seminar 2014 (Grenoble, F) 13

14 Algebraic Conditions of Positivity: Consider a closed linear operator A : D(A) Z Z. Assume that: {e n} n 1 D(A), A is the infinitesimal generator of a C 0- (T A(t)) t 0. Definition 1) The operator A is said to be Metzler if a nk = Ae k, e n 0, n k. 2) The system ż(t) = Az(t) is said to be positive if Z + is T A(t)-invariant, i.e. Proposition T A(t)Z + Z +, t 0 If the system ż(t) = Az(t) is positive on Z, then A satisfies the POD property. Gipsa-lab Seminar 2014 (Grenoble, F) 14

15 Algebraic Conditions of Positivity: The POD property of the generator of a C 0- guarantees the positivity of the latter on invariant finite-dimensional subspaces. Theorem Assume that Z N := span{e 1, e 2,..., e N } (where N < ) is T A(t)-invariant for all t 0. If A is Metzler and a nk > 0 for all n k such that 1 n, k N, then the system ż(t) = Az(t) is positive on Z N, i.e. T A(t)Z + N Z + N, t 0 where Z + N = Z N Z + := the positive cone of Z N Gipsa-lab Seminar 2014 (Grenoble, F) 15

16 Algebraic Conditions of Positivity: Theorem Assume that and Then A is Metzler Z N is T A(t)-invariant for all t 0. the system ż(t) = Az(t) is positive on Z N. Hint: Consider A ɛ := A + B ɛ where B ɛz := z, e k B ɛe k and k=1 B ɛe k, e n = ɛ > 0 for all n, k N and B ɛe k, e n = 0 for all n, k such that n or k > N. Gipsa-lab Seminar 2014 (Grenoble, F) 16

17 Algebraic Conditions of Positivity: Corollary Assume that and Then A has the POD property Z N is T A(t)-invariant for all t 0. the system ż(t) = Az(t) is positive on Z N. Indeed: for all n, z z, e n is a positive bounded linear functional such that, for all k n, e k, e n = 0 (where 0 e k D(A)). It follows by the POD property that A is a Metzler operator. Gipsa-lab Seminar 2014 (Grenoble, F) 17

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19 Systems Consider the infinite dimensional linear system (Σ) described by the following abstract differential equation { ż(t) = Az(t) + Bu(t), where z(0) = z 0 D(A), A is the infinitesimal generator of a C 0- (T A(t)) t 0 on an ordered Banach space Z with positive cone Z +, B is a bounded linear operator from U to Z, U = {u : R + U, continuous} and U is a control ordered Banach space with a positive cone U +. Gipsa-lab Seminar 2014 (Grenoble, F) 19

20 Systems Definition The system (Σ), i.e. the pair (A, B), is said to be positive if for every z 0 Z + and all inputs u U +, i.e. u U such that u(t) U +, t 0, the state trajectories z(t) remain in Z + for all t 0. Definition The system (Σ), i.e. the pair (A, B), is positively stabilizable if there exists a state feedback control law K L(Z, U) such that the C 0- generated by A BK is an exponentially stable positive. Conditions of existence and computation of a state feedback such that the closed loop system is exponentially stable and positive? Gipsa-lab Seminar 2014 (Grenoble, F) 20

21 Systems Theorem The system (Σ) is positive A is the infinitesimal generator of a positive C 0- and B is a positive operator. Gipsa-lab Seminar 2014 (Grenoble, F) 21

22 Systems A functional p : X R is said to be sublinear if for all x, y X and all λ R + p(x + y) p(x) + p(y) and p(λx) = λp(x). Observe that max(p(x), p( x)) 0 for all x X. Moreover, the continuity of p implies that there exists a constant C > 0 such that p(x) C x. A continuous sublinear functional p is called a half-norm if max(p(x), p( x)) > 0 whenever x 0, and then defines a norm on X, namely x p= max(p(x), p( x)). In general, the two norms p and are not equivalent. Gipsa-lab Seminar 2014 (Grenoble, F) 22

23 Systems Given a closed proper cone X + on X p(x) := dist( x, X + ) = inf{ x + y y X + } defines a half-norm on X such that X p = X +. This half-norm is called the canonical half-norm on the ordered Banach space (X, X + ). In addition, 0 p(x) x p C x, x X. Recall that the positive cone X + is normal if there exists a constant M > 0 such that is M-monotone, i.e. x M y whenever 0 x y in X. Proposition The positive cone X + in an ordered Banach space X is normal if and only if the norm induced by the canonical half-norm p is equivalent to the given norm in X. Gipsa-lab Seminar 2014 (Grenoble, F) 23

24 Systems The next theorem characterizes the generators of positive C 0-s and it can be considered as an adaptation of the Hille-Yoshida theorem. Theorem Let X be an ordered Banach space with normal positive cone X +. Assume that the operator norm in L(X) is positively attained, i.e. S = sup{ Sx x X +, x 1}, S L(X). A linear operator A : D(A) X is the generator of a positive C 0- (T (t)) t 0 satisfying T (t) Me ωt if and only if (i) A is closed and densely defined, (ii) (ω, ) ρ(a) and p(r(λ, A) n x) x X, λ > ω, n 1. where p is the canonical half-norm in X. M (λ ω) n p(x) Gipsa-lab Seminar 2014 (Grenoble, F) 24

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26 Decomposition Assume that the state space Z is an ordered Hilbert space and that: (H1) δ > 0 such that the set σ(a) {s C Re(s) > δ} contains only a finite number of elements of the spectrum σ(a), and (H2) A satisfies the spectrum assumption at δ. Then the spectrum of A can be decomposed as follows: The spectral projection σ + δ (A) = σ(a) {λ C Re(λ) > δ}, σ δ (A) = σ(a) {λ C Re(λ) δ}. P δ z = 1 2πj Γ δ (λi A) 1 zdλ induces a of the state space Z = Z u Z s, Z u := P δ Z, Z s := (I P δ )Z, where Z u := P δ Z is finite-dimensional. Gipsa-lab Seminar 2014 (Grenoble, F) 26

27 Decomposition Using the subscript notations u for unstable and s for stable, one can write the operators A and B as: [ ] Au 0 A = where A 0 A u := A Zu, A s := A Zs, s with σ(a u) := σ + δ (A), σ(as) := σ δ (A), [ ] Bu B = where B u := P δ B and B s := (I P δ )B. B s Gipsa-lab Seminar 2014 (Grenoble, F) 27

28 Decomposition The spectrum assumption is valid for a wide class of infinite-dimensional systems: e.g. systems whose generator is a Riesz-spectral operator, parabolic systems and systems described by delay differential equations. A u may have some stable eigenvalues. A s is the infinitesimal generator of an exponentially stable C 0-. Proposition (A, B) is exponentially stabilizable (A u, B u) is exponentially stabilizable Gipsa-lab Seminar 2014 (Grenoble, F) 28

29 Decomposition Let Z + u = Z u Z + and Z + s = Z s Z + Z + u and Z + s are proper cones and therefore define an order on Z u and Z s. Clearly: Z + u Z + s Z + Lemma If A is the infinitesimal generator of a positive C 0-, then A u and A s are infinitesimal generators of positive C 0-s. If, in addition, the converse holds, i.e. Z + u Z + s = Z + T A(t)Z + Z +, t 0 { TAu (t)z + u Z + u, t 0 T As (t)z + s Z + s, t 0. Gipsa-lab Seminar 2014 (Grenoble, F) 29

30 Decomposition Theorem Assume that A is the infinitesimal generator of a positive C 0- and (A u, B u) is positively stabilizable such that there exists a state feedback K u L(Z u, U) such that the operator Then B sk u L(Z u, Z s) is positive. (A, B) is positively stabilizable, i.e. there exists a state feedback K L(Z, U) such that A BK is the infinitesimal generator of an exponentially stable and positive C 0- with respect to the cone Z + u Z + s. Gipsa-lab Seminar 2014 (Grenoble, F) 30

31 Example: Heat Diffusion Heat diffusion model with Neumann boundary conditions: z t (x, t) = 2 z (x, t) + b1u(t) x2 z z (0, t) = 0 = (1, t). x x Described on Z = L 2 (0, 1) by: where Az = d2 z dx 2 ż(t) = Az(t) + Bu(t), z(0) = z 0 D(A), is defined on its domain D(A) = {z L 2(0, 1) z, dz are absolutely continuous, dx d 2 z dx and B L(R, L 2(0, 1)) is given by dz L2(0, 1) and 2 dx dz (0) = (1) = 0}, dx Bu = b 1u, where b 1 L 2(0, 1) Gipsa-lab Seminar 2014 (Grenoble, F) 31

32 Example: Heat Diffusion A has a pure point spectrum σ(a) which consists of the simple eigenvalues λ n = n 2 π 2, n 0, and the corresponding eigenvectors ϕ 0 = 1 and ϕ n(x) = 2 cos(nπx), n 1, form an orthonormal basis of L 2(0, 1). So A is the Riesz spectral operator given by Az = (nπ) 2 z, ϕ n ϕ n, n=0 for z D(A) and is the infinitesimal generator of the C 0-: T A(t)z 0 = z 0, 1 + 2e (nπ)2t z 0, cos nπ( ) cos nπ( ) n=1 Gipsa-lab Seminar 2014 (Grenoble, F) 32

33 Example: Heat Diffusion (T A(t)) t 0 is a positive C 0-, i.e. where T A(t)(L 2(0, 1)) + (L 2(0, 1)) +, t 0 L 2(0, 1)) + = {h L 2(0, 1) h 0 almost everywhere}. A satisfies the spectrum assumption, so w.l.g. : [ ] Au 0 A =, where A 0 A u = A, L u s 2 (0,1) As = A L s 2 (0,1) where L u 2 (0, 1) = span{ϕ 0}={the constant functions} L s 2(0, 1) = span{ϕ n, n 1} Gipsa-lab Seminar 2014 (Grenoble, F) 33

34 Example: Heat Diffusion T Au (t) = 1, t 0, is a positive unstable C 0- on L u 2 (0, 1) and T As (t) is positive on L s 2(0, 1). Indeed: let z s (L s 2(0, 1)) + = L s 2(0, 1) (L 2(0, 1)) +. Then z s, 1 = 0. It follows that T As (t)z s( ) = 2e nπ2t z s(.), cos nπ( ) cos nπ( ) Hence, for all t 0, n=1 = T A(t)z s( ) L s 2(0, 1) (L 2(0, 1)) + (L s 2(0, 1)) +. T As (t)(l s 2(0, 1)) + (L s 2(0, 1)) + Gipsa-lab Seminar 2014 (Grenoble, F) 34

35 Example: Heat Diffusion Let b 1 = α be a strictly positive constant function. Then B uu = αu is a positive operator from R to L u 2 (0, 1) and B s = 0. So, k u R 0 +, A u B uk u is the infinitesimal generator of the positive exponentially stable C 0- given by T Au B uk u (t)z u = e αkut z u, z u L u 2 (0, 1) Hence (A, B) is positively stabilizable. Gipsa-lab Seminar 2014 (Grenoble, F) 35

36 Example: Heat Diffusion Moreover, for all K = [ k 0 ] L(L 2(0, 1), R), with k R [ ] 0 +, Au B uk 0 A BK = is the infinitesimal generator of a 0 A s positive exponentially stable C 0-, or equivalently, the closed loop system z t (x, t)(t) = 2 z (x, t) b1(x)k ϕ0, z(, t) x2 z z (0, t) = (1, t) = 0 x x is a positive exponentially stable system for all k R 0 + with respect to the cone (L u 2 (0, 1)) + (L s 2(0, 1)) +. Gipsa-lab Seminar 2014 (Grenoble, F) 36

37 Example: Heat Diffusion z(x,t) x t Figure: z 0 (x) = (x(x 1)) Gipsa-lab Seminar 2014 (Grenoble, F) 37

38 Example: Heat Diffusion z(x,t) x t Figure: z 0 = characteristic function of [0.4,0.6] Gipsa-lab Seminar 2014 (Grenoble, F) 38

39 Contents Gipsa-lab Seminar 2014 (Grenoble, F) 39

40 Nonnegative Input Infinite dimensional systems described by abstract linear differential equations: { ẋ(t) = Ax(t) + Bu(t) x(0) = x 0 D(A), (6.1) where A is a Riesz spectral operator on an ordered (separable) Hilbert space X with positive cone X +, which is the infinitesimal generator of a positive unstable C 0- T (t) on X such that there exist constants M 1 and ω IR such that for all t 0, T (t) Me ωt, (6.2) the operator B L(IR m, X) is positive, i.e. B is a bounded linear operator from IR m to X such that B(IR m + ) X +, (6.3) and the input u( ) is assumed to be any locally square integrable function, i.e. u( ) L 2 ([0, T ], IR m ) for all T > 0. Gipsa-lab Seminar 2014 (Grenoble, F) 40

41 Nonnegative Input Assumptions: (C1) the system (6.1), i.e. the pair (A, B), is (exponentially) stabilizable; (C2) the C 0- T (t) has a compact resolvent, i.e. the resolvent operator R(λ, A) := (λi A) 1 L(X) is compact for all λ > ω. It follows from condition (C1) that the unstable part of the dynamics is a (totally) unstable finite-dimensional system corresponding to all the eigenvalues of the operator A that belong to the closed right-half plane. In the sequel, we will assume that the number of such eigenvalues (counting multiplicities) is m. Gipsa-lab Seminar 2014 (Grenoble, F) 41

42 Nonnegative Input Definition The system (6.1), i.e. the pair (A, B), is said to be partially positively stabilizable if there exist a subset S X and a state feedback control law u = F x, where the feedback operator F is in L(X, IR m ), such that the resulting closed-loop system is stable and positive on S, i.e. the C 0- T F (t) generated by A + BF is exponentially stable and T F (t) is positive on the subset S, i.e. the set X + S is T F (t)-invariant: whence, in particular, (A, B) is stabilizable. T F (t)(x + S) X + S ; (6.4) Gipsa-lab Seminar 2014 (Grenoble, F) 42

43 Nonnegative Input Theorem: Consider an infinite-dimensional system described by (6.1)-(6.3) and assume that conditions (C1) and (C2) hold. Under these conditions : a) Consider a feedback operator F L(X, IR m ) of full rank m. Then the set C(F ) given by C(F ) := F 1 (IR m + ) := { x X : F x IR m + } is T F (t)-invariant, i.e. T F (t) C(F ) C(F ), if and only if there exists a Metzler matrix H such that i.e. F (A + BF ) HF = 0 on D(A) (6.5) F (A + BF )x = HF x for all x D(A). Gipsa-lab Seminar 2014 (Grenoble, F) 43

44 Stabilinewzation: Nonnegative Input b) If there exists a stabilizing feedback operator F L(X, IR m ) of full rank m, such that condition (6.5) holds for some Metzler matrix H, then the system (6.1), i.e. the pair (A, B), is partially positively stabilizable and, in particular, for every initial state x 0 X + C(F ) and for all t 0, x(t) = T F (t)x 0 X + C(F ) such that, for some constants µ 1 and σ > 0, x(t) µ e σt, for all t 0. Gipsa-lab Seminar 2014 (Grenoble, F) 44

45 Nonnegative Input Consider the system { ż(t) = Az(t) + Bu(t), z(0) = z 0 D(A), where A is a Riesz operator spectral which is the infinitesimal generator of a positive unstable C 0- on a separable Hilbert space and B is a positive operator on U = R m. Aim: find an operator F L(Z, R m ), which is necessarily non positive, such that the control law u(t) = F x(t) is positive for some initial conditions and stabilizes the system. Method: link the positive stabilization problem with the existence of a convenient solution of the algebraic equation for a well-chosen m m matrix H. F (A + BF ) = HF Gipsa-lab Seminar 2014 (Grenoble, F) 45

46 Nonnegative Input Assume that: 1 A has m unstable eigenvalues. 2 H R m m is such that σ(a) σ(h) = (6.6) σ(h) = {λ i C Re(λ i) < 0, 1 i m} (6.7) the eigenvectors (θ i) i=1,m of H are linearly independent. (6.8) Gipsa-lab Seminar 2014 (Grenoble, F) 46

47 Nonnegative Input Let F L(Z, R m ) such that σ(a + BF ) = σ(h) {the stable eigenvalues of A} (6.9) Let (ξ i) i 1 be the set of eigenvectors of A + BF. Assume that F is solution of the following equation F (A + BF ) = HF. (6.10) Consider λ i σ(h), with the associated vectors θ i and ξ i such that Hθ i = λ iθ i and (A + BF )ξ i = λ iξ i, i = 1,.., m. It follows from equation (6.10) that H(F ξ i) = λ i(f ξ i), i = 1,.., m. Since λ i is also an eigenvalue of the matrix H, we should obtain F ξ i = µθ i, for µ IR, i = 1, 2. (6.11) Gipsa-lab Seminar 2014 (Grenoble, F) 47

48 Nonnegative Input Assume wlg that µ = 1, whence F ξ i = θ i. (6.12) By pluging (6.12) in the identity Aξ i + BF ξ i = λ iξ i, one gets ξ i = R(λ i, A)Bθ i, i = 1,.., m. (6.13) Now consider λ j {the stable eigenvalues of A} and ξ j the associated eigenvectors of (A + BF ), then F (A + BF )ξ j = HF ξ j, j m + 1, which is equivalent to H(F ξ j) = λ j(f ξ j). Since λ j is not an eigenvalue of the matrix H, this implies that F ξ j = 0, j m + 1. (6.14) Hence, the eigenvectors ξ j of (A + BF ) for j m + 1 are also eigenvectors of A. Gipsa-lab Seminar 2014 (Grenoble, F) 48

49 Nonnegative Input Lemma For a given matrix H R m m satisfying (6.6)-(6.8) and (6.9), there exists a unique solution to equation (6.10) if (ξ i) i 1 is a basis of Z. Proof. Let F L(Z, R m ) given by F ξ i = θ i, for all 1 i m and F ξ i = 0, i m + 1. It follows that { λif ξ i = λ iθ i = Hθ i, 1 i m λ if ξ i = 0, i m + 1 replacing λ iξ i by (A + BF )ξ i for all i 1 and θ i by F ξ i for all 1 i m, we get { F (A + BF )ξi = HF ξ i, 1 i m F (A + BF )ξ i = 0 = HF ξ i, i m + 1 Since (ξ i) i 1 is a basis of Z, then F (A + BF ) = HF. Gipsa-lab Seminar 2014 (Grenoble, F) 49

50 Nonnegative Input Assume that there exist F 1, F 2 two solution of (6.10) satisfying (6.9), then F 1ξ i = F 2ξ i = θ i, 1 i m and F 1ξ i = F 2ξ i = 0, i m + 1. That implies (F 1 F 2)ξ i = 0, i 1, since (ξ i) i 1 is a basis of Z, we deduce that F 1 = F 2. Now, let H R m m be a Metzler matrix and F L(Z, R m ) solution of the equation (6.10) such that (6.9) holds, i.e. u(t) = F x(t) is a stabilizing state feedback. The control dynamics is given by u(t) = F ẋ(t) = F (A + BF )x(t) = HF x(t) = Hu(t) H is a Metzler matrix implies that the control remains positive if it is initially positive. Gipsa-lab Seminar 2014 (Grenoble, F) 50

51 Nonnegative Input How to determine F solution of the algebraic equation (6.10) such that F is not positive and the set D 0 = {x 0 Z + u 0 = F x 0 R m + } is nonempty? 1 A is a Riesz spectral operator, hence R(λ, A)x = x, ϕ k ϕ k, λ λ k n=1 for all λ ρ(a) and x Z, where (λ k, ϕ k ) k 1 is the eigenset of A. By (6.12)-(6.13) and (6.14), { F R(λi, A)Bθ i = θ i, 1 i m (6.15) F ϕ k = 0, k m + 1 Then one can determine F by solving the following system of m equations and m unknowns F ϕ i, i = 1, m : k=1 1 λ λ k Bθ i, ϕ k F ϕ k = θ i, 1 i m. (6.16) Gipsa-lab Seminar 2014 (Grenoble, F) 51

52 Contents Gipsa-lab Seminar 2014 (Grenoble, F) 52

53 and perspectives Necessary and sufficient conditions for the positivity of controlled systems. stabilization by approximation: positivity preserving numerical schemes. Sufficient conditions based on spectral, for the existence of a stabilizing state feedback such that the closed loop system remains positive: conservative. stabilization by spectral assignment and nonnegative input: promising; numerical implementation. Gipsa-lab Seminar 2014 (Grenoble, F) 53

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55 Abouzaïd B.; Winkin, J.J.; Wertz, V., stabilization of infinite-dimensional linear systems, Proceedings of 49th IEEE Conference on Decision and Control (CDC), 2010, Dec. 2010, pp Achhab M. E.; Winkin J. J., Stabilization of Infinite Dimensional Systems by State Feedback with Positivity Constraints, Proceedings of MTNS 2014; to appear. Abouzaïd B.; Achhab M. E.; Winkin J. J.; Wertz V., stabilization of infinite-dimensional linear systems by state feedback, in preparation. Gipsa-lab Seminar 2014 (Grenoble, F) 55