Input-to-state stability of systems of partial differential - - PowerPoint PPT Presentation

Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Input-to-state stability of systems of partial differential equations

Andrii Mironchenko

Centre for Industrial Mathematics, University of Bremen, Germany

February 14, 2011, Elgersburg Workshop 2011, Elgersburg, Germany

joint work with Sergey Dashkovskiy

1 / 28 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections

Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Outline

1

Basic notions

2

Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions

3

Linearisation

4

Monotone control systems

5

Interconnections of control systems

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Semigroups and their Generators

Let X be a Banach space, and L(X) be the space of bounded

• perators, defined on X.

Definition (Strongly continuous semigroup)

A family of operators {T(t), t ≥ 0} ⊂ L(X), is called a strongly continuous semigroup (for short C0-semigroup), if it holds

1 T(0) = I 2 T(t + s) = T(t)T(s), ∀t, s ≥ 0. 3 For all x ∈ X function t → T(t)x belongs to C([0, ∞), X)

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Definition (Analytic semigroup)

The C0-semigroup is called analytic, if in addition it holds: T(t)x → x, when t → +0. t → T(t)x is real analytic on 0 < t < ∞ for every x ∈ X.

Definition (Generator of a C0-semigroup)

Linear operator L, defined by Lx = lim

t→+0

1 t (T(t)x − x) with domain D(L) = {x ∈ X : lim

t→+0

1 t (T(t)x − x) exists} is called an infinitesimal generator of a C0-semigroup T(t).

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Definition of control system

Let (X, · X) be a state space, (U, · U) be an input space and Uc be the set of admissible input functions: R+ → U.

Definition (Control system)

The triple Σ = (X, Uc, φ) is a control system, if: φ(t, t, x, ·) = x for all t ≥ 0. ∀ t ≥ r ≥ s ≥ 0, ∀x ∈ X, ∀u1 ∈ U[s,r]

c

, u2 ∈ U[r,t]

c

it holds φ(t, r, φ(r, s, x, u1), u2) = φ(t, s, x, u), where u(τ) := u1(τ), τ ∈ [s, r], u2(τ), τ ∈ [r, t]. ∀x ∈ X, u ∈ Uc the map t → φ(t, 0, x, u) is in C([0, ∞), X) φ is continuous in two last arguments.

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Stability notions

Let Σ = (X, Uc, φ) be time-invariant and φ(t, 0, 0, 0) ≡ 0.

Definition (Global asymptotic stability at zero)

Σ is globally asymptotically stable at zero (0-GAS), if ∃β ∈ KL: ∀φ0 ∈ X, ∀t ≥ 0 it holds φ(t, 0, φ0, 0)X ≤ β(φ0X , t).

Definition (Local input-to-state stability)

Σ is locally input-to-state stable (LISS), if ∃ρx, ρu > 0 and ∃β ∈ KL, γ ∈ K, such that ∀t ≥ 0, ∀φ0 : φ0X ≤ ρx and ∀u ∈ Uc: uUc ≤ ρu it holds φ(t, t0, φ0, u)X ≤ max{β(φ0X, t), γ(uUc)}.

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Definition (exponential LISS and ISS)

If β(r, t) = Meωtr, for some ω < 0, then (X, Uc, φ) is locally exponentially ISS If one can choose ρx = ρu = ∞, then (X, Uc, φ) is ISS

x(t) β(φ0, t) γ(u)

t Figure: Input-to-state stability in max-formulation

7 / 28 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections

Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Outline

1

Basic notions

2

Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions

3

Linearisation

4

Monotone control systems

5

Interconnections of control systems

8 / 28 Basic notions Lyapunov methods Linearisation Monotonicity Interconnections

Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

LISS-Lyapunov functions

Definition (Local ISS-Lyapunov function (LISS-LF))

A smooth function V : D → R+, D ⊂ X, 0 ∈ int(D) is LISS-LF for system (X, Uc, φ), if there exist ρx, ρu > 0, functions ψ1, ψ2 ∈ K∞, χ ∈ K and positive definite function α, such that: ψ1(xX) ≤ V(x) ≤ ψ2(xX), ∀x ∈ D and ∀x ∈ D : xX ≤ ρx, ∀u ∈ U : uU ≤ ρu it holds: xX ≥ χ(uU) ⇒ ˙ V(x) ≤ −α(xX), (1) where ˙ V(x) = lim

t→+0

1 t (V(φ(t, 0, x, u)) − V(x)). Function χ is called Lyapunov gain.

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Lyapunov characterisation of LISS

Theorem

Let Σ = (X, Uc, φ) be a time-invariant control system. If Σ possesses a LISS-Lyapunov function, then Σ is LISS.

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Lyapunov characterisation of LISS

Theorem

Let Σ = (X, Uc, φ) be a time-invariant control system. If Σ possesses a LISS-Lyapunov function, then Σ is LISS.

Example: semilinear heat equation

• ∂s

∂t = ∂2s ∂x2 − f(s) + u(x, t),

x ∈ (0, π), t > 0, s(0, t) = s(π, t) = 0. (2) We assume, that f is locally Lipschitz, monotonically increasing up to infinity, f(−r) = −f(r) for all r ∈ R and u(·, t) ∈ L2(0, π).

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Example: Formulation and Lyapunov function

We define: As = d2s dx2 with D(A) = H1

0(0, π) ∩ H2(0, π).

Operator A generates an analytic semigroup on L2(0, π). System (2) takes form ds dt = As − f(s) + u, t > 0. (3) Equation (3) defines the control system with state space X = H1

0(0, π) and input space U = L2(0, π).

The norm on H1

0(0, π) we define as sH1

0(0,π) =

π

0 s2 x(x)dx

1

2 .

V(s) = π

• 1

2s2

x(x) +

s(x) f(y)dy

• dx.

(4)

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Verification of the first property: s(x) f(y)dy ≥ 0 ⇒ V(s) ≥ π 1 2s2

x(x)dx = 1

2s2

H1

0(0,π)

The derivative of V along the trajectories is: ˙ V(s) = − π (sxx(x) − f(s(x)))2dx+ π (sxx(x) − f(s(x)))(−u)dx. Define I(s) := π (sxx(x) − f(s(x)))2dx. Using Cauchy-Schwarz inequality for the second term, we have: ˙ V(s) ≤ −I(s) +

• I(s) uL2(0,π).

(5)

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Lyapunov function

I(s) := π (sxx(x) − f(s(x)))2dx. One can prove directly: I(s) ≥ π s2

xx(x)dx.

For s ∈ H1

0(0, π) ∩ H2(0, π) it holds (a corollary of Friedrich’s

inequality), that: π s2

xx(x)dx ≥

π s2

x(x)dx.

Overall, we have: I(s) ≥ s2

H1

0(0,π).

(6)

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Gains

Now we choose the gain as χ(r) = ar, a > 1. If sH1

0(0,π) ≥ χ(uL2(0,π)), we obtain

˙ V(s) ≤ −I(s)+1 a

• I(s)sH1

0(0,π) ≤ −(1−1

a)I(s) ≤ −(1−1 a)s2

H1

0(0,π).

This proves, that V is ISS-Lyapunov function, and consequently, our control system (with X = H1

0(0, π),

U = L2(0, π)) is ISS.

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Outline

1

Basic notions

2

Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions

3

Linearisation

4

Monotone control systems

5

Interconnections of control systems

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Linear systems

Let A be a generator of a C0-semigroup T(t) and B ∈ L(U, X). ˙ s = As + Bu, s(0) = s0. (7) Uc = {g : R+ → U : g is locally Hölder continuous}. s(t) = T(t)s0 + t T(t − r)Bu(r)dr. (8) Let ∃ω0 : ∀ω > ω0 ∃Mω, such that T(t) ≤ Mωeωt. s(t)X ≤ Mωeωts0X + Mω |ω| BuUc.

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Linearisation

Let X be a Hilbert space with scalar product ·, ·, and A generates an analytic semigroup on X. Consider a system ˙ x = Ax + f(x, u), x(t) ∈ X, u(t) ∈ U. (9)

Theorem (Linearisation theorem)

Let for some B ∈ L(X) and C ∈ L(U, X) it holds f(x, u) = Bx + Cu + g(x, u). Let ∀w > 0 ∃ρ > 0, s.t. ∀x, u : xX ≤ ρ, uU ≤ ρ it holds g(x, u)X ≤ w(xX + uU). If the system ˙ x = Ax + Bx + Cu (10) is exponentially ISS, then (9) is LISS.

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Idea of the proof of linearisation theorem

R = A + B generates an exponentially stable analytic semigroup. V(x) = Px, x is Lyapunov function for linear system, where P ∈ L(X) is positive, and Rx, Px + Px, Rx = −x2

X,

∀x ∈ D(R). Assuming x ∈ D(R) ⊂ X, we compute derivative of V with respect to nonlinear system: ˙ V(x) ≤ −(1 − 2wP)x2

X + 2P(C + w)xXuU

Take χ(r) := √r. For xX ≥ χ(uU) we have: ˙ V(x) ≤ −(1 − 2wP)x2

X + 2P(C + w)x3

• X. (11)

Prove the same for x ∈ X\D(R).

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Outline

1

Basic notions

2

Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions

3

Linearisation

4

Monotone control systems

5

Interconnections of control systems

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Monotonicity of control systems

Definition (Positive cone)

Let X be Banach. A set K ⊂ X is called a positive cone in X, if: ∀a ∈ R+ aK ⊂ K, K + K ⊂ K, K ∩ (−K) = {0}

Definition (Ordered Banach space)

A pair (X, K), where X is Banach, and K ⊂ X is a positive cone, is called an ordered Banach space with an order ≤, given by x ≤ y ⇔ y − x ∈ K.

Definition (Monotonicity)

Control system S = (X, Uc, φ) is called monotone, if ∀t0 ∈ R+, for all t ≥ t0, u1, u2 ∈ Uc : u1 ≤ u2, ∀φ1, φ2 ∈ X : φ1 ≤ φ2 it holds φ(t, t0, φ1, u1) ≤ φ(t, t0, φ2, u2).

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ISS of Monotone Systems

    

∂s(x,t) ∂t

= c2∆s + f(s, u(x, t)), x ∈ G ⊂ Rp, t > 0, s (x, 0) = φ0 (x) , x ∈ G,

∂s ∂n

• ∂G×R≥0 = 0.

(12) Assume that f(0, 0) = 0 ∀t ≥ 0.

Theorem

Let system (12) be monotone. If the system ds(x,t)

dt

= f(s, u), x ∈ G, t > 0, s (0) = φ0 is ISS, then (12) is also ISS.

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Outline

1

Basic notions

2

Lyapunov methods LISS Lyapunov functions Lyapunov characterisation of LISS Example on construction of Lyapunov functions

3

Linearisation

4

Monotone control systems

5

Interconnections of control systems

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Centre for Industrial Mathematics Elgersburg Workshop 2011 Mironchenko

Interconnections of control systems

Let Xi, i = 1, . . . , n be Banach and Ai generate C0-semigroup

• n Xi.

Σ : Σi : ˙ xi = Aixi + fi(x1, . . . , xn, u), xi ∈ Xi i = 1, . . . , n (13) X = X1 × . . . × Xn is Banach with · X := · X1 + . . . + · Xn. A =      A1 . . . A2 . . . . . . . . . ... . . . . . . An      D(A) = D(A1) × . . . × D(An). We rewrite the system (13) in vector form: Σ : ˙ x = Ax + f(x, u) (14)

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˙ xi = Aixi + fi(x1, . . . , xn, u), xi ∈ Xi i = 1, . . . , n (15)

ISS-LF for i-th subsystem

A smooth function Vi : Xi → R+, is a ISS-Lyapunov function (ISS-LF) for i-th subsystem of (13), if ∃ψi1, ψi2 ∈ K∞, χij, χi ∈ K, j = 1, . . . , n, χii := 0 and positive definite function αi, such that: ψi1(xiXi) ≤ Vi(xi) ≤ ψi2(xiXi), ∀xi ∈ Xi, and ∀xi ∈ Xi it holds: Vi(xi) ≥ max{

n

max

j=1 χij(Vj(xj)), χi(uU)} ⇒

˙ Vi(xi) ≤ −αi(Vi(xi)), ˙ V(xi) = lim

t→+0

1 t (V(φi(t, 0, xi, u))) − V(xi)).

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Small-gain theorem

Let ΓM = (χij)i,j=1,...,n, χij ∈ K∞ ∪ 0 (gain matrix). Let us introduce the gain operator Γ : Rn

+ → Rn + defined by

Γ(s) :=

• n

max

j=1 χ1j(sj), . . . , n

max

j=1 χnj(sj)

• , s ∈ Rn

+.

(16)

Theorem (Small-gain theorem)

Let for all Σi there exist ISS-Lyapunov function Vi with corresponding gains χij. If Γ(s) ≥ s, ∀ s ∈ Rn

+\ {0}, then Σ is

ISS and possesses ISS-Lyapunov function, defined by V(x) := max

i

{σ−1

i

(Vi(xi))}. (17)

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Construction of the path σ

The path σ can be constructed as follows: σ(t) = Q(at), ∀t ∈ [0, ∞), ∀a > 0, where Q(x) = MAX{x, Γ(x), Γ2(x), . . . , Γn−1(x)}, and MAX for all ui ∈ Rn, i = 1, . . . , m is defined as z = MAX{u1, . . . , um} ∈ Rn, zi = max{u1i, . . . , umi}. The Lyapunov gain of the whole system is χ(r) := max

i

σ−1

i

(χi(r)).

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Conclusions

Main results

Verification of (L)ISS of distributed parameter systems

1 LISS-Lyapunov functions 2 Linearisation 3 Monotone control systems 4 Small-gain theorems

Possible directions of future work

Generalization of the known results from finite-dimensional theory. Definitions of stability for the inifinite-dimensional systems. Application of the obtained results.

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Supported by the German Research Foundation (DFG) as a part of Collaborative Research Centre 637 "Autonomous Cooperating Logistic Processes - A Paradigm Shift and its Limitations".

Thank you for attention!

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