Lyapunov functions for boundary control systems€¦ · Lyapunov functions for boundary control...

Lyapunov functions for boundary control systems

Birgit Jacob, Andrii Mironchenko, Jonathan Partington, Fabian Wirth

Faculty of Mathematics and Computer ScienceUniversity of Passau

13th Elgersburg WorkshopElgersburg

25 February 2019

B. Jacob, A. Mironchenko, J. Partington and F. Wirth. Remarks on input-to-state stability and non-coercive

Lyapunov functions, CDC 2018.

Example: stabilization of a linear heat equation

∂x∂t (z, t) = ∂2x

∂z2 (z, t) + ax(z, t)x(0, t) = 0 ∀t ≥ 0x(1, t) = u(t) + d(t) ∀t ≥ 0.

If a > π, then this system is unstable, for u + d ≡ 0.

AimTo design a feedback controller

u(t) = p(x(·, t))

which uniformly globally asymptotically stabilizes the system, robustly w.r.t.actuator disturbances d .

Andrii Mironchenko Lyapunov functions for boundary control systems Elgersburg, 2019 2 / 28

Example: stabilization of a linear heat equation

∂x∂t (z, t) = ∂2x

∂z2 (z, t) + ax(z, t)x(0, t) = 0 ∀t ≥ 0x(1, t) = u(t) + d(t) ∀t ≥ 0.

−−−−−→Volterra tr.

∂w∂t (z, t) = ∂2w

∂z2 (z, t)w(0, t) = 0 ∀t ≥ 0w(1, t) = d(t) ∀t ≥ 0,

Σ1 is transformed into Σ2 by means ofw(z, t) = x(z, t) +

∫ z0 k(z, y)x(y , t)dy

u(t) = −∫ 1

0 k(1, y)x(y , t)dyk is a kernel of a Volterra transformation, which can be computed explicitly.

and we naturally come to the question of ISS of Σ2.

QuestionsHow to show that Σ2 is ISS?Can we derive some systematic methods for that?

Class of systemsDefinition

The triple Σ = (X ,U , φ) is called control system, if:

(Σ1) Forward-completeness: for every x ∈ X , u ∈ U and for all t ≥ 0 the valueφ(t , x , u) ∈ X is well-defined.

(Σ2) Continuity: for each (x , u) ∈ X × U the map t 7→ φ(t , x , u) is continuous.

(Σ3) Cocycle property: for all t , h ≥ 0, for all x ∈ X , u ∈ U we have

φ(h, φ(t , x , u), u(t + ·)) = φ(t + h, x , u).

Examples

Ordinary differential equations

Evolution Partial differential equations with Lipschitz nonlinearities

Broad classes of boundary control systems

Time-delay systems

Heterogeneous systems with distinct components

Class of systemsDefinition

The triple Σ = (X ,U , φ) is called control system, if:

(Σ1) Forward-completeness: for every x ∈ X , u ∈ U and for all t ≥ 0 the valueφ(t , x , u) ∈ X is well-defined.

(Σ2) Continuity: for each (x , u) ∈ X × U the map t 7→ φ(t , x , u) is continuous.

(Σ3) Cocycle property: for all t , h ≥ 0, for all x ∈ X , u ∈ U we have

φ(h, φ(t , x , u), u(t + ·)) = φ(t + h, x , u).

Examples

Ordinary differential equations

Evolution Partial differential equations with Lipschitz nonlinearities

Broad classes of boundary control systems

Time-delay systems

Heterogeneous systems with distinct components

Example (Linear systems with admissible control operators)

x(t) = Ax(t) + Bu(t), x(0) ∈ X , t > 0. (1)

A generates a C0-semigroup (T (t))t≥0 on a Banach space X

B ∈ L(U,X−1) for some Banach space U

X−1 is the completion of X w.r.t. ‖x‖X−1 := ‖(β − A)−1x‖X , for some β ∈ ρ(A).

T extends uniquely to T−1 on X−1 whose generator A−1 is an extension of A.

(1) is well-posed on X−1: ∀x0 ∈ X and ∀u ∈ L1loc([0,∞),U), the function

x : [0,∞)→ X−1,

x(t) := T (t)x0 +

0T−1(t − s)Bu(s)ds, t ≥ 0,

is called mild solution of (1).

That’s great, but the trajectory is now in X−1

Example (...Continue...)

B ∈ L(U,X−1) is called an q-admissible control operator for (T (t))t≥0, where1 ≤ q ≤ ∞, if ∀t ≥ 0 and ∀u ∈ Lq([0,∞),U):∫ t

0T−1(t − s)Bu(s)ds ∈ X .

Triple (X , L∞([0,∞),U), φ) defines a control system if:

B is∞-admissible

∀x0 ∈ X , ∀u ∈ L∞([0,∞),U) the mild solution φ(·, x0, u) is continuous.

Above 2 conditions are implied by each of the following conditions:

B is q-admissible for some q ∈ [1,∞)

B is∞-admissible, dim U <∞, X is a Hilbert space and A generates an analyticsemigroup which is similar to a contraction semigroup

Comparison functions

γ ∈ Pγ(s)

6 ⊃ γ ∈ Kγ(s)

6 ⊃ γ ∈ K∞γ(s)

lims→∞

γ(s) =∞

β ∈ KL︷︸︸︷K

β(s, ·)

β(·, r)

Input-to-state stability

Definition (E. Sontag, 1989, for ODEs)

ISS :⇔ ∃β ∈ KL, γ ∈ K∞: ∀t ≥ 0, ∀x ∈ X , ∀u ∈ U‖φ(t , x ,u)‖X ≤ β(‖x‖X , t) + γ(‖u‖U ).

β(‖x‖X , t)

‖x(t)‖X

(a) u ≡ 0

γ(‖u(·)‖U )

β(‖x‖X , t)

β(‖x‖X , t) + γ(‖u(·)‖U )

‖x(t)‖X

(b) u 6≡ 0

Motivation

Why ISS?1 Unified theory of internal and external stability

E. D. Sontag. Input to State Stability: Basic Concepts and Results. In Nonlinear andOptimal Control Theory, chapter 3, 2008.

2 Robust stabilization of nonlinear systemsM. Krstic, I. Kanellakopoulos, P. Kokotovic. Nonlinear and adaptive control design, Wiley,1995.

3 Design of robust nonlinear observersM. Arcak, P. Kokotovic. Nonlinear observers: a circle criterion design and robustnessanalysis, 2001.

4 Stability of networks of nonlinear control systemsZ.-P. Jiang, I. Mareels, Y. Wang. A Lyapunov formulation of the nonlinear small-gaintheorem for interconnected ISS systems, Automatica, 1996.S. Dashkovskiy, B. Rüffer, F. Wirth. Small Gain Theorems for Large Scale Systems andConstruction of ISS Lyapunov Functions, SICON, 2010.

5 . . .

MotivationInfinite-dimensional ISS theory: 2008 – now

Linear systems: Characterizations and applicationsF. Bribiesca Argomedo, B. Jacob, I. Karafyllis, M. Krstic, F. Mazenc, AM, R. Nabiullin,J. R. Partington, C. Prieur, J. Schmid, F. Schwenninger, F. Wirth, E. Witrant, H. Zwart,. . .

Nonlinear systems: Lyapunov theory, small-gain theoremsM. Ahmadi, A. Chaillet, Y. Chitour, S. Dashkovskiy, G. Is. Detorakis, M. Edalatzadeh, H.Ito, B. Jayawardhana, Z.-P. Jiang, I. Karafyllis, M. Krstic, H. Logemann, S. Marx, F.Mazenc, AM, K. Morris, S. Palfi, A. Papachristodoulou, A. Pisano, C. Prieur, E. P.Ryan, S. Senova, Y. Orlov, A. Tanwani, S. Tarbouriech, G. Valmorbida, J. Zheng, G.Zhu, . . .

Nonlinear systems: Characterizations, non-Lyapunov methodsB. Jacob, I. Karafyllis, M. Krstic, AM, J. Schmid, F. Schwenninger, F. Wirth, J. Zheng,G. Zhu, . . .

Almost 2/3 of papers appeared since 2016.

ISS Lyapunov functions

DefinitionV : X → R+ is a non-coercive ISS-Lyapunov function iff ∃ψ2, σ, α ∈ K∞:

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0(ii) Vu(x) ≤ −α(‖x‖X ) + σ(‖u(0)‖U) ∀x ∈ X , ∀u ∈ U ,

Vu(x) = limt→+0

1t(V (φ(t , x ,u))− V (x)

V is called a coercive ISS-Lyapunov function if

∃ψ1, ψ2 ∈ K∞ : ψ1(‖x‖X ) ≤ V (x) ≤ ψ2(‖x‖X ), ∀x 6= 0.

Theorem (Direct Lyapunov theorem)

∃ a coercive ISS Lyapunov function ⇒ ISS.

Converse Lyapunov Theorems for Lipschitz rhs

Theorem (Mironchenko, Wirth, SCL 2018)

Consider

x(t) = Ax(t) + f (x(t),u(t)), x(t) ∈ D(A) ⊂ X ,

U = PC(R+,U)

Ax = limt→+01t (T (t)x − x).

T is a C0-semigroup over Banach space X.f is a bi-Lipschitz continuous perturbation.

The following statements are equivalent:(i) Σ is ISS.(ii) There exists a coercive ISS Lyapunov function for Σ which is locally Lipschitz

continuous.

What we have right now

Using coercive ISS Lyapunov functions combined with integral inequalities, one cansuccessfully study ISS of PDEs with distributed disturbances

For ISS systems which are regular enough (bi-Lipschitz) a coercive ISS Lyapunovfunction always exists.

At the same time, study of PDEs with boundary disturbances calls for new methods forISS analysis.

Approaches, proposed to tackle this problem include

Novel constructions of coercive LFs

for systems of conservation lawsfor reaction-diffusion equations with Neumann and Robin boundary conditions

Monotonicity methods (for nonlinear reaction-diffusion equations with Dirichletboundary conditions)

Non-coercive ISS Lyapunov functions

Definition

Let Σ = (X ,U , φ) be given.We call 0 ∈ X an equilibrium point (of the undisturbed system) if φ(t ,0,0) = 0for all t ≥ 0.We say Σ has the CEP property, if 0 is an equilibrium and for every ε > 0 andfor any h > 0 there exists a δ = δ(ε,h) > 0, so that

t ∈ [0,h], ‖x‖X ≤ δ, ‖u‖U ≤ δ ⇒ ‖φ(t , x ,u)‖X ≤ ε. (2)

Σ has bounded reachability sets (BRS), if:

C > 0, τ > 0 ⇒ sup‖x‖X≤C, ‖u‖U≤C, t∈[0,τ ]

‖φ(t , x ,u)‖X <∞.

Definition

V : X → R+ is a non-coercive ISS-Lyapunov function iff ∃ψ2, σ, α ∈ K∞:

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0

(ii) Vu(x) ≤ −α(‖x‖X ) + σ(‖u(0)‖U) ∀x ∈ X , ∀u ∈ U ,

Vu(x) = limt→+0

1t(V (φ(t , x , u))− V (x)

Why non-coercive Lyapunov functions?

LFs for linear systems via Lyapunov equation are non-coercive.

Very useful for boundary control systems

May be of use for couplings of infinitely-many systems (for Lyapunov small-gaintheorems)

Definition

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0

Vu(x) = limt→+0

1t(V (φ(t , x , u))− V (x)

Non-coercive LFs are frequently used for linear systems.Next we show an essentially nonlinear result.

Definition

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0

Vu(x) = limt→+0

1t(V (φ(t , x , u))− V (x)

Theorem (Jacob, Mironchenko, Partington, Wirth, to be submitted, 2019)

Let Σ be a BRS control system, which is continuous at equlibrium.If ∃ a non-coercive ISS Lyapunov function for Σ ⇒ Σ is ISS.

Definition

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0

Vu(x) = limt→+0

1t(V (φ(t , x , u))− V (x)

We cannot use ’linear’ methods

We cannot use comparison principle

Definition

(i) 0 < V (x) ≤ ψ2(‖x‖X ) ∀x 6= 0

Vu(x) = limt→+0

1t(V (φ(t , x , u))− V (x)

We cannot use ’linear’ methods

We cannot use comparison principle

A proof relies deeply on characterizations of ISS, obtained inA.M. Local input-to-state stability: Characterizations and counterexamples. SCL, 2016.

A.M., F. Wirth. Characterizations of input-to-state stability for infinite-dimensional systems. IEEETAC, 2018.

Lyapunov functions for systems with admissible operators

Let A generate a C0-semigroup (T (t))t≥0 on a Hilbert space X.Assume that there is P ∈ L(X ) satisfying:

(i) P satisfies Re 〈Px , x〉X > 0, x ∈ X\{0}.

(ii) P satisfies Lyapunov inequality

〈(PA + A∗P)x , x〉X ≤ −〈x , x〉X , x ∈ D(A), (3)

(iii) It holds that Im(P) ⊂ D(A∗).

(iv) PA is bounded.

ThenV (x) := Re 〈Px , x〉X (4)

is a non-coercive ISS Lyapunov function for (1) with any∞-admissible input operatorB ∈ L(U,X−1), and thus (1) is ISS for such B.

When does P := −A−1 works?

Proposition (Jacob, Mironchenko, Partington, Wirth, to be submitted, 2019)

Σ : x = Ax + Bu.

A generate an exponentially stable C0-semigroup (T (t))t≥0 on a complex Hilbert spaceX

B ∈ L(U,X−1) be∞-admissible.

D(A) ⊆ D(A∗) and

∃δ < 1 : ∀x ∈ X Re 〈A∗A−1x , x〉X + δ‖x‖2X ≥ 0. (5)

Re 〈Ax , x〉X < 0 for every x ∈ D(A)\{0}.Then

V (x) := −Re 〈A−1x , x〉X (6)

is an ISS Lyapunov function.

Subnormal Operators

Definition

A densely defined operator (A,D(A)) : X → X on a Hilbert space X is said to be normal ifD(A) = D(A∗) and ‖Ax‖X = ‖A∗x‖X for all x ∈ D(A).

Definition

A closed, densely-defined operator A on a Hilbert space X is called subnormal, if there is aHilbert space Z containing X as a subspace and a normal operator (N,D(N)) : Z → Z sothat A = N|X (the restriction of N to X ) and X is an invariant subspace for N, that is,N(D(N) ∩ X ) ⊆ X .

Example

1 Clearly, every normal operator on a Hilbert space is subnormal.2 Symmetric operators on Hilbert spaces are subnormal.3 Isometries are subnormal

Subnormal Operators

Definition

Example

Subnormal Operators

Definition

Example

Definition

Corollary (Jacob, Mironchenko, Partington, Wirth, CDC, 2018)

Let A generate an exponentially stable analytic semigroup on a Hilbert space X andassume that A is subnormal and B ∈ L(Cn,X−1) be∞-admissible. Then

V (x) := −Re 〈A−1x , x〉X (7)

is a non-coercive ISS Lyapunov function satisfying

Vu(x) ≤ −c1‖x0‖2X + c2‖u‖2

for some constants c1, c2 > 0 and all x0 ∈ X and u ∈ L∞([0,∞),U).

Definition

Corollary (Jacob, Mironchenko, Partington, Wirth, CDC, 2018)

Let A generate an exponentially stable analytic semigroup on a Hilbert space X andassume that A is subnormal and B ∈ L(Cn,X−1) be∞-admissible. Then

V (x) := −Re 〈A−1x , x〉X (7)

is a non-coercive ISS Lyapunov function satisfying

Vu(x) ≤ −c1‖x0‖2X + c2‖u‖2

for some constants c1, c2 > 0 and all x0 ∈ X and u ∈ L∞([0,∞),U).

Understanding above results: Diagonal systems

x = Ax + Bu

Consider this system with:

X = l2(N) :={

x = {xk}∞k=1 : ‖x‖X =(∑∞

k=1 x2k

)1/2<∞

Aek = −λk ek , where ek is the k -th unity vector of l2(N) and λk ∈ R.Hence A is self-adjoint.λk < λk+1 for all k and λk →∞ as k →∞A generates an exponentially stable semigroup, i.e. there exists ε > 0: λk > εfor all k > 0.

Let first B be bounded.

We want to find an ISS Lyapunov function for this system

Diagonal systems with bounded B

Bounded B: First ISS LF constructionAbove method leads to the following ISS LF:

V1(x) = −⟨

A−1x , x⟩

=∞∑

12λk〈x ,ek 〉2 . (8)

Note: V1 is not coercive since λk →∞ as k →∞.

Bounded B: Second ISS LF constructionFor this example it is easy to find another ISS Lyapunov function V2:

V2(x) := 〈x , x〉 = ‖x‖2X , x ∈ X .

Note: V2 is a coercive ISS LF for any B ∈ L(U,X ).

Diagonal systems with bounded B

Bounded B: First ISS LF constructionAbove method leads to the following ISS LF:

V1(x) = −⟨

A−1x , x⟩

=∞∑

12λk〈x ,ek 〉2 . (8)

Note: V1 is not coercive since λk →∞ as k →∞.

Bounded B: Second ISS LF constructionFor this example it is easy to find another ISS Lyapunov function V2:

V2(x) := 〈x , x〉 = ‖x‖2X , x ∈ X .

Note: V2 is a coercive ISS LF for any B ∈ L(U,X ).

Can we characterize admissibility via Lyapunov equation?

Now assume that B is merely an∞-admissible operator.

X−1 :={{xk}∞k=1 :

∞∑k=1

|xk |2λ2

Lemma (Jacob, 2018)

B is∞-admissible for T ⇔∑∞

k=1|bk |2λ2

k<∞.

Let P := −A−1 (a solution of the Lyapunov equation). One can show:

Proposition

PB ∈ L(U,X ) ⇔ B is∞-admissible

Can we similarly characterize admissibility in a non-diagonal case?

X−1 :={{xk}∞k=1 :

∞∑k=1

|xk |2λ2

Lemma (Jacob, 2018)

k=1|bk |2λ2

k<∞.

Proposition

X−1 :={{xk}∞k=1 :

∞∑k=1

|xk |2λ2

Lemma (Jacob, 2018)

k=1|bk |2λ2

k<∞.

Proposition

X−1 :={{xk}∞k=1 :

∞∑k=1

|xk |2λ2

Lemma (Jacob, 2018)

k=1|bk |2λ2

k<∞.

Proposition

ISS LFs for unbounded B

After some further analysis one can show that

V1(x) = −⟨

A−1x , x⟩

=∞∑

12λk〈x ,ek 〉2 . (9)

is a non-coercive ISS LF for any∞-admissible B ∈ L(U,X−1).

At the same time,V2(x) := 〈x , x〉 = ‖x‖2

X , x ∈ X .

is not an ISS Lyapunov function for any unbounded admissible B ∈ L(U,X−1).

Do coercive ISS Lyapunov functions for diagonal systems with unbounded inputoperators exist?

Example: heat equation with Dirichlet boundary control

Consider the following system (a > 0):

xt (ξ, t) = axξξ(ξ, t), ξ ∈ (0,1), t > 0,x(0, t) = 0, x(1, t) = u(t), t > 0,x(ξ,0) = x0(ξ).

We choose X = L2(0,1), U = C,

Af := f ′′, f ∈ D(A) :={

f ∈ H2(0,1) | f (0) = f (1) = 0}, B := aδ′1.

No constructions of coercive ISS Lyapunov functions are available for this system.

Af := f ′′, f ∈ D(A) :={

f ∈ H2(0, 1) | f (0) = f (1) = 0}, B = aδ′1.

A is a self-adjoint operator on X generating an exponentially stable analyticC0-semigroup on X

B ∈ X−1 = L(U,X−1) is∞-admissible

∀x0 ∈ X , ∀u ∈ L∞(0,∞) the corresponding mild solution is continuous.

Further, the following ISS-estimates hold ∀x0 ∈ X , u ∈ L∞(0,∞):

‖x(t)‖L2(0,1) ≤ e−aπ2t‖x0‖L2(0,1) +1√3‖u‖L∞(0,t)

The corresponding non-coercive ISS Lyapunov function is:

V (x) = −〈A−1x , x〉X =

(∫ 1

(ξ − τ)x(τ)dτ)

x(ξ)dξ.

OverviewWe have discussed

Coercive and non-coercive Lyapunov functionsCEP ∧ BRS ∧ ∃ nc ISS-LF ⇒ ISSConstructions of nc ISS LF for boundary control systems, in particular:

for a heat equation with Dirichlet control.for diagonal systems.

Open problems

Do ISS Lyapunov functions (coercive or non-coercive always exist for linearISS systems with B ∈ L(U,X−1)?Do coercive ISS Lyapunov functions for diagonal analytic systems withunbounded input operators exist?Boundary control of coupled boundary control systems via Lyapunovsmall-gain theorems.

Open problems

Thank you for attention!Andrii Mironchenko Lyapunov functions for boundary control systems Elgersburg, 2019 26 / 28

Open problems

Papers and slides can be found atwww.mironchenko.com

Vision: Robust control of coupled∞-dim systems

Fundamentalsof ISS in ∞dimensions

Stability ofinterconnected∞-dim systems

ISS of non-linear time-

delay systems

Non-coerciveISS Lyapunov

functions

x = Ax + Buwith

unbounded B

ISS of PDEsinterconnectedvia boundary

Robust boundarycontrol of PDEs

Robust boundarycontrol of coupled

heterogeneous systems

Thank you for attention!Papers and slides can be found at

www.mironchenko.com

MCSS Topical Collection:Input-to-state stability for infinite-dimensional systems

Topics of interest

ISS for partial differential equations

ISS for boundary control systems

Lyapunov methods for input-to-state stability

Applications of ISS to robust control and observation of PDE systems

Important dates

Deadline for the initial submission of manuscripts: October 1, 2019.

Notification about the first decision: February 1, 2020.

Guest Editors

Birgit Jacob, University of Wuppertal

Andrii Mironchenko, University of Passau

Felix Schwenninger, University of Hamburg

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