On the relation of delay equations to first-order hyperbolic partial differential equations

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DOI:10.1051/cocv/2014001 www.esaim-cocv.org

ON THE RELATION OF DELAY EQUATIONS TO FIRST-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Iasson Karafyllis

¹

and Miroslav Krstic

²

Abstract. This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System- theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.

Mathematics Subject Classification. 34K20, 35L04, 35L60, 93D20, 34K05, 93C23.

Received February 5, 2013. Revised July 14, 2013.

Published online June 13, 2014.

1. Introduction

The relation of first-order hyperbolic Partial Differential Equations (PDEs) with delay equations is well known. In many cases, a system of First-Order Hyperbolic PDEs (FOH-PDEs) can be transformed to a system described by Retarded Functional Differential Equations (see [4,10,11,25,27,28]), On the other hand, the recent works [14–16] have shown that systems of first-order hyperbolic PDEs can be utilized for the stabilization of delay systems.

However, recent works on the control of systems described by FOH-PDEs (see [1–3,5–7,14,21,23,24,30,31]) have led to the study of non-standard systems of FOH-PDEs with the following features: (i) the boundary conditions are given by functionals of the whole state profile; and (ii) the differential equations involve functionals of the whole state profile and not only point values of the states. It should be noted that for such systems there is no available existence-uniqueness theory similar to the theory of standard FOH-PDEs (see [17,26]): the researchers have utilized transformation arguments which can guarantee important system-theoretic properties.

Moreover, the existence of functionals of the whole state proﬁle in the mathematical description of such systems do not allow the answer to the following question:

Keywords and phrases.Integral delay equations, ﬁrst-order hyperbolic partial diﬀerential equations, nonlinear systems.

1 Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece.

iasonkar@central.ntua.gr

2 Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA.

krstic@ucsd.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2014

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“What class of delay systems can be used for the description of non-standard systems of FOH-PDEs?”

In this work, we answer the above question for a class of systems described by a single FOH-PDE. First, we show that the appropriate class of delay systems is a class which has been rarely studied,i.e., systems of the form:

η(t) =f(η_t, w_t)

η(t)∈Rⁿ, w(t)∈W ⊆R^m (1.1)

where W ⊆R^m is a non-empty closed set with 0∈W, r >0,η_t∈L^∞([−r,0);Rⁿ),w_t ∈L^∞([−r,0];W) are defined by (η_t) (s) = η(t+s) for s ∈ [−r,0), (w_t) (s) = w(t+s) for s ∈ [−r,0] and f : L^∞([−r,0);Rⁿ)× L^∞([−r,0];W)→Rⁿ is a mapping. We call the above class of delay systems a system described by Integral Delay Equations (IDEs). Systems of the form (1.1) have been studied in [11,18–20,22]. It should be noted at this point that systems described by IDEs have been utilized for a long time in the stabilization of finite- dimensional control systems with input delays (see [13,15,16]): every predictor feedback is a system described by an IDE, whose input is the state of the finite-dimensional control system. Indeed, a predictor feedback for the finite-dimensional system

˙

y(t) =g(y(t), η(t−r) +v(t)) y(t)∈R^k, η(t)∈Rⁿ, v(t)∈Rⁿ

where we usexto denote the control input andv∈L^∞_loc(R₊;Rⁿ) denotes the control actuator error, results in a static feedback law of the form:

η(t) =f(η_t, y(t) +u(t))

where u∈L^∞_loc(R+;Rⁿ) denotes the measurement error. Clearly, the above closed-loop system is the feedback interconnection of an time-invariant system described by ODEs and an integral delay system of the form (1.1).

Therefore, the study of systems described by IDEs is important on its own. Important system-theoretic properties for the class of systems described by IDEs are provided in Section 2 of the present paper.

Secondly, we show that a class of systems described by a non-standard single FOH-PDE, namely systems of the form:

∂ x

∂ t(t, z) +c∂ x

∂ z(t, z) =a(p(t), z)x(t, z) +g(z)p(t), f or t >0, z∈(0,1) (1.2) p_i(t) =K_i(w(t), x_t), fort >0, i= 1, . . . , N andp(t) = (p₁(t), . . . , p_N(t)) ∈R^N (1.3)

x(t,0) =G(w(t), x_t), fort≥0 (1.4)

where N > 0 is an integer, x(t, z) ∈ R, w(t) ∈ W ⊆ R^m, W ⊆ R^m is a closed set, c > 0 is a constant, g(z) = (g₁(z), . . . , g_N(z)), G : W ×L^∞((0,1];R) → R, K_i : W ×L^∞((0,1];R) → R (i = 1, . . . , N) are functionals,a(p, z),g_i(z) (i= 1, . . . , N) are suﬃciently regular scalar functions andx_tis the state proﬁle (i.e., (x_t) (z) = x(t, z) for all t ≥ 0 andz ∈ [0,1]), are equivalent to systems described by IDEs. The equivalence allows the development of important system-theoretic results for the class of systems described by (1.2), (1.3) and (1.4) (Section 3 of the present paper). The obtained results can allow the study of measurable inputs w(t)∈W ⊆R^m, which is an important feature because in many cases boundary conditions of the form (1.4) result from the implementation of stabilizing feedback laws (see [1–3,5,7,14,23,24]). However, the control action comes together with control actuator errors and measurement errors, which are typically modeled by measurable inputs.

Finally, the obtained results can allow the development of a methodology for the solution of control problems for systems described by a non-standard single FOH-PDE, namely systems of the form (1.2), (1.3), (1.4).

The methodology is presented in Section 4 of the present paper by means of three illustrative examples. The examples show many important features of the proposed methodology and a comparison is made with other existing methodologies.

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I. KARAFYLLIS AND M. KRSTIC

Notation.Throughout this paper, we adopt the following notation:

* R₊:= [0,+∞).

* LetI⊆Rbe an interval andU ⊆R^mbe a set. ByL^∞(I;U) we denote the space of measurable and essentially bounded functions u(·) deﬁned on I and taking values in U ⊆ R^m. By L^μ(I;U), where μ ∈ [1,+∞), we denote the space of measurable functions u(·) deﬁned on I and taking values in U ⊆ R^m for which

I|u(s)|^μds < +∞. By L^∞_loc(I;U) we denote the space of measurable and locally essentially bounded functions u(·) deﬁned on I and taking values in U ⊆R^m, i.e., for every compact intervalJ ⊆ I it holds thatu∈L^∞(J;U). For everyx∈L^∞(I;Rⁿ), whereI ⊆R is an interval, we denote byx the essential supremum ofx,i.e., x= sup

s∈I|x(s)|. Forx∈L^∞(I;Rⁿ), whereI⊆Ris an interval with [a, b)⊆I, a < b, we deﬁnex_[a,b)= sup

a≤s<b|x(s)|.

* For a vectorx∈Rⁿ, we denote byx its transpose and by|x|its Euclidean norm.

2. Systems described by integral delay equations

We consider system (1.1) under the following assumptions:

(H1) There exist non-decreasing functions a: R₊ →R₊,M :R₊ →R₊,N : R₊ →R₊ such that for every R >0 and for every η, y∈L^∞([−r,0);Rⁿ),w∈L^∞([−r,0];W)with η ≤R, y ≤R,w ≤R the following inequalities hold:

|f(η, w)−f(y, w)| ≤N(R)h sup

−h≤s<0|η(s)−y(s)|+M(R) sup

−r≤s<−h|η(s)−y(s)|, f or all h∈(0, r) (2.1)

|f(η, w)| ≤a(R) (2.2)

(H2) For every δ > 0, η ∈ L^∞([−r, δ);Rⁿ), w ∈ L^∞([−r, δ];W), the function z : [−r, δ) → Rⁿ defined by z(t) =η(t)for t∈[−r,0) a.e. andz(t) =f(η_t, w_t) fort∈[0, δ)a.e. satisfiesz∈L^∞([−r, δ);Rⁿ).

Systems of the form (1.1) can represent integral delay equations of the form:

η(t) = N i=1

p_i(η(t−τ_i), w(t−τ_i)) + _r

0 q(s, η(t−s), w(t−s)) ds

wherer >0, 0< τ₁ ≤τ₂≤. . .≤τ_N ≤rare constants,p_i :Rⁿ×W →Rⁿ (i= 1, . . . , N) are locally Lipschitz mappings,W ⊆R^m is a non-empty closed set andq:R×Rⁿ×W →Rⁿ is a locally Lipschitz mapping. It can be shown that assumptions (H1), (H2) hold for this class of systems. The fact that assumption (H2) holds for this class of systems is a direct consequence of Lemma 1 on page 4 in [8].

The following result shows us that system (1.1) can be regarded as a time-invariant system with inputs, whose state space isL^∞([−r,0);Rⁿ) and whose input set is L^∞([−r,0];W). System (1.1) satisﬁes the classical semigroup property, the Boundedness-Implies-Continuation (BIC) property and the property of Lipschitz dependence on the initial conditions (see [12]).

Theorem 2.1. Consider system(1.1)under assumptions(H1),(H2). Then for everyη₀∈L^∞([−r,0);Rⁿ),w∈ L^∞_loc([−r,+∞);W) there exists t_max =t_max(η₀, w) ∈(0,+∞] with t_max > ^r

1+2rN

5a

max

η0, sup

−r≤s≤r|w(s)|

and a unique mapping η∈L^∞_loc([−r, tmax);Rⁿ)satisfyingη(t) =f(η_t, w_t)for t∈[0, t_max) a.e. andη(t) =η₀(t) for t ∈ [−r,0) a.e.. Moreover, if t_max < +∞ then lim sup

t→t⁻max

ηt = +∞. Finally, there exist non-decreasing

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functions P : R₊ → R₊, G : R₊ → R₊ such that for every η₀ ∈ L^∞([−r,0);Rⁿ), y₀ ∈ L^∞([−r,0);Rⁿ), w∈L^∞_loc([−r,+∞);W)it holds that:

η_t−y_t ≤G(s(t)) exp(P(s(t))t)η₀−y₀, f or all t∈[0, δ) (2.3) whereδ= min (t_max(η₀, w), t_max(y₀, w)),s(t) := max

0≤τ≤tsup η_τ, sup

0≤τ≤ty_τ, sup

0≤τ≤tw_τ

and

η ∈ L^∞_loc([−r, t_max(η₀, w));Rⁿ), y ∈ L^∞_loc([−r, t_max(y₀, w));Rⁿ) are the unique mappings satisfying η(t) = f(η_t, w_t) for t∈[0, t_max(η₀, w)) a.e., y(t) =f(y_t, w_t) for t∈[0, t_max(y₀, u))a.e. and η(t) =η₀(t),y(t) =y₀(t) for t∈[−r,0) a.e..

Proof. Let arbitraryη₀∈L^∞([−r,0);Rⁿ),w∈L^∞_loc([−r,+∞);W). Deﬁnes:= max

η0, sup

−r≤t≤r|u(t)|

and select arbitraryp∈Rⁿ with|p| ≤s. Deﬁne η¹(t) =η₀(t) fort∈[−r,0) a.e. andη¹(t) =pfort∈[0, r). Clearly, η¹∈L^∞([−r, r);Rⁿ) with sup

0≤τ <r

η¹_τ ≤s.

Without loss of generality we may assume that the non-decreasing functiona:R+ →R+ involved in (2.2) satisﬁesa(t)≥tfor allt≥0. Let δ∈(0, r) be such that:

2N(R)δ <1, forR= 5a(s). (2.4)

Using induction and assumption (H2), we can guarantee that the sequence of functionsη^(k): [−r, δ)→Rⁿ for k ≥ 1 deﬁned by: η^(k)(t) =η₀(t) for t ∈ [−r,0) a.e. and η^(k)(t) = f(η_t^(k−1), w_t) for t ∈ [0, δ) a.e. for k ≥ 2 satisﬁesη^(k)∈L^∞([−r, δ);Rⁿ) fork≥1.

Notice that (2.2) in conjunction with the facts that sup

0≤τ <r

η¹_τ ≤sand sup

0≤τ≤ruτ ≤ simplies that

0≤τ <δsup

η¹_τ ≤a(s) and sup

0≤τ <δ

η_τ² ≤a(s). (2.5)

We next claim that

l=1,...,kmax

0≤τ <δsup

η^(l)_τ

≤5a(s) for allk≥1. (2.6)

The proof of the claim is made by induction. We ﬁrst notice that by virtue of (2.5) the claim is true fork= 1 and k= 2. Next suppose that the claim is true for certaink≥2. Notice that, by virtue of (2.1) and the fact thatR= 5a(s), the following inequality holds for alll= 2, . . . , kand for almost allt∈[0, δ):

η^(l+1)(t)−η^(l)(t)=f(η^(l)_t , w_t)−f(η_t^(l−1), w_t)

≤N(R)δ sup

−δ≤s<0

η^(l)_t

(s)−

η^(l−1)_t

(s)+M(R) sup

−r≤s<−δ

η^(l)_t

(s)−

η_t^(l−1)

(s)

=N(R)δ sup

−δ≤s<0

η^(l)(t+s)−η^(l−1)(t+s)+M(R) sup

−r≤s<−δ

η^(l)(t+s)−η^(l−1)(t+s)

=N(R)δ η^(l)−η^(l−1)

[−r,δ). (2.7)

Since (2.7) holds for almost allt∈[0, δ) and sinceη^(l+1)(t) =η^(l)(t) =η₀(t) for almost allt∈[−r,0) and for alll≥1, we get:

η^(l+1)−η^(l)

[−r,δ)≤N(R)δ η^(l)−η^(l−1)

[−r,δ), for alll= 2, . . . , k. (2.8)

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Inequality (2.8) implies that:

η^(2+m)−η^(1+m)

[−r,δ)≤(N(R)δ)^m η²−η¹

[−r,δ), for allm= 0, . . . , k−1. (2.9) Using the triangle inequality repeatedly and the fact thatN(R)δ≤1/2 (a consequence of (2.4)), we get for all l= 2, . . . , k:

η^(l+1)−η⁽¹⁾

[−r,δ)≤_l−1

m=0 η^(m+2)−η^(m+1)

[−r,δ)

≤ η⁽²⁾−η⁽¹⁾

[−r,δ)

_l−1

m=0(N(R)δ)^m= η⁽²⁾−η⁽¹⁾

[−r,δ)1−(N(R)δ)^l 1−N(R)δ

≤2 η⁽²⁾−η⁽¹⁾

[−r,δ).

(2.10)

It follows from (2.5) and (2.10) that

0≤τ <δsup

η_τ^(k+1) = η^(k+1)

[−r,δ)≤5a(s) (2.11)

and therefore we have proved (2.6).

The existence of η ∈ L^∞([−r, δ);Rⁿ) satisfying η(t) = f(η_t, w_t) for t ∈ [0, δ) a.e. and η(t) = η₀(t) for t ∈ [−r,0) a.e. follows closely the proof of Banach’s ﬁxed point theorem on the closed set L^∞([−r, δ);B), where B ⊆ Rⁿ denotes the closed ball of radius 5a(s). Since (2.9) holds for all m ≥ 0, the sequence η^(k) ∈ L^∞([−r, δ);B) is a Cauchy sequence. Therefore, there exists a unique limitη ∈ L^∞([−r, δ);B). Assumption (H1) (inequality (2.1)) implies that the right hand side of the inequality

|η(t)−f(η_t, w_t)| ≤η(t)−η^(k)(t)+f(η_t, w_t)−f(η_t^(k−1), w_t)

can be made arbitrarily small for suﬃciently largek≥1 and for almost allt∈[0, δ) and consequently, we obtain η(t) =f(η_t, w_t) fort∈[0, δ) a.e.

In order to prove uniqueness, we use a contradiction argument. Suppose that there existsy∈L^∞([−r, δ);Rⁿ) satisfyingy(t) =f(y_t, w_t) fort∈[0, δ) a.e. andy(t) =η₀(t) fort∈[−r,0) a.e. with sup

−r≤s<δ|y(s)−η(s)|>0. Let p≥0 be the least upper bound of all t∈[0, δ) with sup

−r≤s<t|y(s)−η(s)|= 0. A contradiction argument shows that p < δ. Since p∈ [0, δ) is the least upper bound of all t ∈ [0, δ) with sup

−r≤s<t|y(s)−η(s)| = 0, it follows that sup

−r≤s<t|y(s)−η(s)| >0 for allt∈ (p, δ). A contradiction argument shows that sup

−r≤s<p|y(s)−η(s)|= 0.

Deﬁne R= max( sup

−r≤s<δ|y(s)|, sup

−r≤s<δ|η(s)|, sup

−r≤s≤δ|w(s)|). It follows from (2.1) for almost allt∈(p, δ) and h=t−p:

|η(t)−y(t)|=|f(η_t, w_t)−f(y_t, w_t)| ≤N(R)(t−p) sup

p≤s<t|η(s)−y(s)|. (2.12)

Inequality (2.12) implies sup

p<s<t|η(s)−y(s)| ≤N(R)(t−p) sup

p≤s<t|η(s)−y(s)|for allt∈(p, δ) and since sup

p<s<t|η(s)−y(s)|= sup

p≤s<t|η(s)−y(s)|, we obtain sup

p≤s<t|η(s)−y(s)|= 0 for allt <min

δ, p+_N(R)+1¹

,i.e.,

−r≤s<tsup |y(s)−η(s)|= 0 for allt <min

δ, p+_N(R)+1¹

, a contradiction.

Since the unique functionη∈L^∞([−r, δ);Rⁿ) satisfyingη(t) =f(η_t, w_t) fort∈[0, δ) a.e. andη(t) =η₀(t) for t∈[−r,0) a.e. is bounded, it follows (by repeating the arguments above) that there existsδ > δ and a unique functionη ∈L^∞([−r, δ);Rⁿ) satisfyingη(t) =f(η_t, w_t) fort∈[0, δ) a.e. andη(t) =η₀(t) fort ∈[−r,0) a.e..

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More speciﬁcally, by virtue of (2.4),δ> δ satisﬁes

δ+r≥δ≥δ+ r

1 + 2rN

5a

max

sup

−r≤s<δ|η(s)|, sup

−r≤s≤δ+r|w(s)|

.

(2.13)

Let t_max = t_max(η₀, w) ∈ (0,+∞] be the least upper bound of all δ > 0 for which there exists function η ∈ L^∞([−r, δ);Rⁿ) satisfying η(t) = f(η_t, w_t) for t ∈ [0, δ) a.e. and η(t) = η₀(t) for t ∈ [−r,0) a.e.. If t_max <+∞(i.e., is ﬁnite) then for everyε >0 there exists δ > t_max−ε and a functionη ∈L^∞([−r, δ);Rⁿ) satisfyingη(t) =f(η_t, w_t) fort∈[0, δ) a.e. andη(t) =η₀(t) fort∈[−r,0) a.e.. It follows in any case (t_max<+∞

or t_max= +∞) that there exists a functionη ∈L^∞_loc([−r, t_max);Rⁿ) satisfyingη(t) =f(η_t, w_t) fort∈[0, t_max) a.e. andη(t) =η₀(t) fort∈[−r,0) a.e..

For the caset_max<+∞we notice that (2.13) implies the inequality

t_max≥δ+ r

1 + 2rN

5a

max

−r≤s<δsup |η(s)|, sup

−r≤s≤δ+r|w(s)|

or

N

5a

max

−r≤s<δsup |η(s)|, sup

−r≤s≤δ+r|w(s)|

≥ r−ε 2rε

for all ε > 0 and δ > t_max−ε. The above inequality cannot hold for all ε > 0 if sup

−r≤s<tmax

|η(s)| < +∞.

Therefore, ift_max< +∞then lim sup

t→t⁻max

ηt= +∞.

Finally, we proceed to the constructive proof of estimate (2.3). Let η₀ ∈ L^∞([−r,0);Rⁿ), y₀ ∈ L^∞([−r,0);Rⁿ),w∈L^∞_loc([−r,+∞);W) and letη∈L^∞_loc([−r, t_max(η₀, w));Rⁿ),y∈L^∞_loc([−r, t_max(y₀, w));Rⁿ) be the unique mappings satisfying η(t) = f(η_t, w_t) for t ∈ [0, t_max(η₀, w)) a.e., y(t) = f(y_t, w_t) for t ∈ [0, t_max(y₀, w)) a.e. andη(t) =η₀(t),y(t) =y₀(t) for t∈[−r,0) a.e.. Deﬁne δ= min (t_max(η₀, w), t_max(y₀, w)).

For all t ∈ (0, δ) we deﬁne s(t) := max

0≤τ≤tsup ητ, sup

0≤τ≤tyτ, sup

0≤τ≤twτ

and h = _N^t, where N = 1 +

t(1+2rN(s(t))) min(r,t)

and

denotes the integer part of the realt(1+2rN(s(t)))

min(r,t) . Inequality (2.1) implies for almost allv∈[0, h):

|η(v)−y(v)|=|f(η_v, w_v)−f(y_v, w_v)| ≤ (2.14) N(s(t))h sup

−h≤q<0|η(v+q)−y(v+q)|+M(s(t))η₀−y₀ Notice that sinceN(s(t))h≤1/2 we get from (2.14):

−r≤q<hsup |η(q)−y(q)| ≤(1 + 2M(s(t)))η0−y₀. (2.15) Using induction and similar arguments we can prove that the following inequality holds fori= 1, . . . , N

−r≤q<ihsup |η(q)−y(q)| ≤(1 + 2M(s(t)))ⁱη0−y₀. (2.16) SinceN = 1 +

≤1 +t(1+2rN(s(t)))

min(r,t) ≤2 + 2rN(s(t)) +^t(1+2rN_r^(s(t))), we obtain from (2.16) for i=N:

−r≤q<tsup |η(q)−y(q)| ≤(1 + 2M(s(t)))^2+2rN^(s(t))exp

2N(s(t)) +r⁻¹

ln(1+2M(s(t)))t

η₀−y₀. (2.17)

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Inequality (2.17) implies (2.3) withG(s) := (1 + 2M(s))^2+2rN(s) andP(s) :=

2N(s) +r⁻¹

ln(1 + 2M(s)).

The proof is complete.

Having proved Theorem 2.1, we are justiﬁed for arbitraryη₀ ∈L^∞([−r,0);Rⁿ), w∈L^∞_loc([−r,+∞);W) to name the unique mappingη ∈L^∞_loc([−r, t_max);Rⁿ) satisfyingη(t) =f(η_t, w_t) fort∈[0, t_max) a.e.,η(t) =η₀(t) fort∈[−r,0) a.e. for certain t_max=t_max(η₀, w)∈(0,+∞] as “the solutionη(t)of (1.1)with initial condition η(t) =η₀(t)for t∈[−r,0) a.e., corresponding to input w∈L^∞_loc([−r,+∞);W)”, without explicit reference to the maximal existence time of the solutiont_max=t_max(η₀, w)∈(0,+∞].

A number of system-theoretic properties can be proved by using the results of Theorem 2.1. More speciﬁcally, we study system (1.1) under assumptions (H1), (H2) and the following assumption:

(H3)w= (d, u), whered∈D⊆R^m¹is a compact set,u∈U ⊆R^m²is a closed set with0∈Uandm=m₁+m₂. Moreover, there existsb∈K_∞such that for everyη∈L^∞([−r,0);Rⁿ),d∈L^∞([−r,0];D),u∈L^∞([−r,0];U), the following inequality holds:

|f(η, d, u)| ≤b

max

η, sup

−r≤s≤0|u(s)|

. (2.18)

Assumption (H3) means that the system’s right-hand side is uniformly bounded with respect to the distur- bancedand the right-hand side is zero when the stateηand inputuare zero, irrespective of the disturbanced.

This means, in particular, that the origin is an equilibrium when the inputuis zero, irrespective of the value of the disturbance d, which excludes, for example, systems that have an additive disturbance in the model’s right-hand side. In Example 2.7 we provide a simple system that captures some of the essence of assumption (H3) and the system’s dependence ond.

We are ready to give a list of properties for system (1.1) that are derived from the results of Theorem 2.1.

Property 1:Robustness of the equilibrium point.

Assumption (H3) guarantees that 0∈L^∞([−r,0);Rⁿ) is an equilibrium point for system (1.1), whenuis zero and for anyd. However, in order to study the robust stability properties of the equilibrium point of (1.1) we need a stronger assertion, namely that 0∈L^∞([−r,0);Rⁿ) is a robust equilibrium point for the system (1.1) with inputu∈U ⊆R^m² (see [12]).

Theorem 2.2. Consider system (1.1)under assumptions(H1),(H2),(H3). The function0∈L^∞([−r,0);Rⁿ) is a robust equilibrium point for the system(1.1)with inputu∈U ⊆R^m²,i.e., for allε >0andT≥0there exists δ:=δ(ε, T)>0 such that for everyη₀∈L^∞([−r,0);Rⁿ), d∈L^∞([−r,+∞);D), u∈L^∞_loc([−r,+∞);U)with η0+ sup

t≥0ut< δ there existst_max=t_max(η₀, d, u)∈(T,+∞]and a unique mappingη ∈L^∞_loc([−r, tmax);Rⁿ) satisfyingη(t) =f(η_t, d_t, u_t)fort∈[0, t_max)a.e.,η(t) =η₀(t)fort∈[−r,0)a.e. andηt< εfor all t∈[0, T].

Proof. SinceD⊆R^m¹ is compact, we are in the position to deﬁne Q:= max

d∈D|d|. Without loss of generality, we may assume that the function b ∈K_∞ involved in (2.18) satisﬁes b(s)≥s for all s≥0. Let arbitraryε >0, T ≥0 be given and deﬁne:

δ(ε, T) :=κ⁻¹(ε/2), κ:=g^(l)=g ◦. . .◦g

l times

, l= T

ρ

+ 1 andρ= r

1 + 2rN(5b(Q+ε)) (2.19) whereg(s) := 5b(s) for alls≥0,

T ρ

is the integer part of^T_ρ andN :R+→R+is the function involved in (2.1).

Let arbitraryη₀ ∈ L^∞([−r,0);Rⁿ), d∈ L^∞([−r,+∞);D), u∈ L^∞_loc([−r,+∞);U) withη₀+ sup

t≥0ut < δ and consider the unique mapping η ∈ L^∞_loc([−r, t_max);Rⁿ) satisfying η(t) = f(η_t, d_t, u_t) fort ∈ [0, t_max) a.e.,

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η(t) =η₀(t) fort∈[−r,0) a.e. for certaint_max=t_max(η₀, d, u)∈(0,+∞]. Using (2.18), the fact that sup

t≥0ut<

δ, the fact that δ ≤ ε (a direct consequence of deﬁnition (2.19)), the fact that Q := max

d∈D|d| and proceeding exactly as in the proof of Theorem 2.1, we can show that:

“F or all t₀∈[0, t_max)with η_t₀ ≤ε it holds that η_t ≤g(max (η_t₀, δ)), f or all t∈[t₀, t₀+ρ]” (2.20) We next claim thatηiρ ≤g⁽ⁱ⁾(δ), for alli= 1, . . . , l(ηiρis the normηtfort=iρ). The claim is a direct consequence of property (2.20) and deﬁnition (2.19). Therefore, we get ηt ≤ κ(δ) for all t ∈ [0, lρ]. Since lρ≥T (a direct consequence of the fact thatl =

T ρ

+ 1; recall (2.19)) andδ=κ⁻¹(ε/2) (recall (2.19)), we

obtainηt< εfor allt∈[0, T]. The proof is complete.

Property 2:Robust global asymptotic stability and input-to-state stability

Using the results contained in [12], we are in a position to deﬁne the notion of (Uniform) Robust Global Asymptotic Stability and Input-to-State Stability (ISS) for system (1.1) under assumptions (H1), (H2), (H3).

The notion of ISS for systems described by IDEs is completely analogous to the corresponding notion introduced by E. D. Sontag for ﬁnite-dimensional systems in [29].

Definition 2.3. Consider system (1.1) under assumptions (H1), (H2), (H3) and assume that for every η₀ ∈ L^∞([−r,0);Rⁿ),d∈L^∞([−r,+∞);D),u∈L^∞_loc([−r,+∞);U), there existsη∈L^∞_loc([−r,+∞);Rⁿ) satisfying η(t) = f(η_t, d_t, u_t) for t ∈ [0,+∞) a.e., η(t) = η₀(t) for t ∈ [−r,0) a.e.. We say that system (1.1) is Input- to-State Stable (ISS) from the input u∈ U ⊆R^m² uniformly in d∈ D ⊆R^m¹, if there exists a continuous, non-decreasing function γ:R₊→R₊ (called the gain function of the inputu∈L^∞_loc([−r,+∞);U)) such that the following properties hold:

Robust Lagrange Stability:For everyε >0 it holds that sup

η_t − sup

0≤s≤tγ(u_s) : t≥0,η₀ ≤ε , d∈L^∞([−r,+∞);D), u∈L^∞_loc([−r,+∞);U)

<+∞. Robust Lyapunov Stability:For every ε >0there existsδ:=δ(ε)>0such that

sup

ηt − sup

0≤s≤tγ(us) : t≥0,η0 ≤δ, d∈L^∞([−r,+∞);D), u∈L^∞_loc([−r,+∞);U)

< ε.

Uniform Robust Attractivity: For everyε >0 andR≥0 there existsτ :=τ(ε, R)>0 such that sup

ηt − sup

0≤s≤tγ(us) : t≥τ ,η0 ≤R , d∈L^∞([−r,+∞);D), u∈L^∞_loc([−r,+∞);U)

< ε.

IfU ={0}then we say that0∈L^∞([−r,0);Rⁿ)is (Uniformly) Robustly Globally Asymptotically Stable (RGAS) for (1.1).

Lemma 2.1, Lemma 4.2 in [12] and Theorem 2.2 in [12] give us the following result.

Theorem 2.4. Consider system (1.1)under assumptions(H1),(H2),(H3). Then the following statements are equivalent:

(a) System (1.1)is ISS from the inputu∈U ⊆R^m² uniformly ind∈D⊆R^m¹.

(b) There exists σ ∈ KL and a continuous, non-decreasing function γ : R+ → R+ such that for every η₀∈L^∞([−r,0);Rⁿ), d∈L^∞([−r,+∞);D), u∈L^∞_loc([−r,+∞);U), there existsη ∈L^∞_loc([−r,+∞);Rⁿ) satisfying η(t) = f(η_t, d_t, u_t) for t ∈ [0,+∞) a.e., η(t) = η₀(t) for t ∈ [−r,0) a.e. and ηt ≤ σ(η₀ , t) + sup

0≤s≤tγ(u_s) for allt≥0.

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Furthermore, if U ={0}then the following statement is equivalent to statements (a), (b):

(c) System (1.1)is Robustly Forward Complete,i.e., for every T ≥0 andR≥0it holds that sup{ η_t : t∈[0, T],η₀ ≤R , d∈L^∞([−r,+∞);D)}<+∞

where η(t) denotes the solution of (1.1) with η(t) = η₀(t) for t ∈ [−r,0) a.e., corresponding to input d ∈ L^∞([−r,+∞);D), and the property of Uniform Robust Attractivity of Definition 2.3 holds.

Property 3:Lyapunov characterization of RGAS.

Theorem 3.4 in [12] and the results of Theorems 2.1, 2.2 allow us to obtain a complete Lyapunov characterization for the RGAS property for system (1.1).

Theorem 2.5. Consider system (1.1) under assumptions (H1), (H2), (H3) and assume that U = {0}.

The equilibrium point 0 ∈ L^∞([−r,0);Rⁿ) is RGAS for (1.1) if and only if there exists a functional V : L^∞([−r,0);Rⁿ) → R₊, a non-decreasing function Q : R₊ → R₊ and functions a₁, a₂ ∈ K_∞ such that the following inequalities hold:

a₁(η)≤V(η)≤a₂(η), f or all η∈L^∞([−r,0);Rⁿ) (2.21)

|V(η)−V(y)| ≤Q(max (η,y))η−y, f or all η, y∈L^∞([−r,0);Rⁿ). (2.22)

V(η_t)≤exp (−t)V(η₀), f or all t≥0, η₀∈L^∞([−r,0);Rⁿ), d∈L^∞([−r,+∞);D) (2.23) where η_t denotes of (1.1) with initial condition η(t) = η₀(t) for t ∈ [−r,0) a.e., corresponding to input d ∈ L^∞([−r,+∞);D).

Inequality (2.22) guarantees that the functional V :L^∞([−r,0);Rⁿ)→R+ is Lipschitz on bounded sets of the state space L^∞([−r,0);Rⁿ). However, inequality (2.22) does not guarantee Frechet diﬀerentiability of the functional V : L^∞([−r,0);Rⁿ)→ R+ nor that the limit lim

t→0⁺t⁻¹(V(η_t)−V(η₀)) exists for the solution η(t) of (1.1) with initial condition η(t) = η₀(t) fort ∈ [−r,0) a.e., corresponding to inputd ∈L^∞([−r,+∞);D).

Notice that inequality (2.23) guarantees that for everyη₀∈L^∞([−r,0);Rⁿ),d∈L^∞([−r,+∞);D), the solution η(t) of (1.1) with initial conditionη(t) =η₀(t) fort∈[−r,0) a.e., corresponding to inputd∈L^∞([−r,+∞);D) satisﬁes lim sup

t→0⁺ t⁻¹(V(η_t)−V(η₀))≤ −V(η₀).

Property 4:Suﬃcient conditions for stability properties.

Theorem 2.5 is not the most convenient way of proving RGAS or ISS for (1.1). For practical purposes we can use the following result, which is an extension of the classical Razumikhin theorem for time delay systems (see [10,12]).

Theorem 2.6. Consider system (1.1)under assumptions(H1),(H2),(H3). Assume that there exists a contin- uous, positive definite and radially unbounded function W : Rⁿ →R+, a continuous, non-decreasing function

˜

γ : R+ → R+ and a constant λ ∈ (0,1) such that the following inequality holds for all η ∈ L^∞([−r,0);Rⁿ), d∈L^∞([−r,0];D),u∈L^∞([−r,0];U):

W(f(η, d, u))≤λ sup

−r≤s<0W(η(s)) + ˜γ(u). (2.24) Then system (1.1) is ISS from the input u ∈ U ⊆ R^m² uniformly in d ∈ D ⊆ R^m¹ with gain function γ(s) :=a⁻¹₁

8

(1−λ)²(1+λ)˜γ(s)

for all s≥0, wherea₁∈K_∞ satisfiesa₁(|η|)≤W(η), for allη∈Rⁿ.

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Proof. Deﬁne the functional:

V(η) := sup

−r≤s<0exp (σ s)W(η(s)), for allη∈L^∞([−r,0);Rⁿ) (2.25) where σ ∈

0,¹_rln₁

λ

is a constant. Since W : Rⁿ → R+ is a continuous, positive deﬁnite and radially unbounded function, there exist functionsa₁, a₂∈K_∞ such that:

a₁(|η|)≤W(η)≤a₂(|η|), for allη∈Rⁿ (2.26) Using (2.25) and (2.26) we conclude that the following inequality holds:

˜

a₁(η)≤V(η)≤a₂(η), for allη∈L^∞([−r,0);Rⁿ) (2.27) where ˜a₁(s) := exp (−σ r)a₁(s), for alls≥0.

Let arbitrary η₀ ∈L^∞([−r,0);Rⁿ), d∈L^∞([−r,+∞);D),u∈L^∞([−r,+∞);U) and consider the unique solution η(t) of (1.1) with initial condition η(t) = η₀(t) for t ∈ [−r,0) a.e., corresponding to inputs d ∈ L^∞([−r,+∞);D),u∈ L^∞_loc([−r,+∞);U). Let arbitraryt ∈(0, t_max(η₀, d, u)), wheret_max(η₀, d, u)>0 is the maximal existence time of the solution and leth∈(0, t_max(η₀, d, u)−t). Using deﬁnition (2.25) we get:

V(η_t+h) := sup

−r≤s<0exp (σ s)W(η(t+h+s))

= max

−r≤s<−hsup exp (σ s)W(η(t+h+s)), sup

−h≤s<0exp (σ s)W(η(t+h+s))

= max

exp (−σ h) sup

−r+h≤s<0exp (σ s)W(η(t+s)), sup

−h≤s<0exp (σ s)W(η(t+h+s))

≤max

exp (−σ h)V(η_t), sup

0≤s<hW(η(t+s))

.

(2.28)

Using (2.24) we get for almost allq∈[0, h):

W(η(t+q)) =W(f(η_t+q, d_t+q, u_t+q))≤λmax

−r+q≤s<0sup W(η(t+s)), sup

0≤s<qW(η(t+s))

+ ˜γ(u) (2.29) whereu= sup

t≥0ut. Deﬁnition (2.25) and inequality (2.29) gives:

0≤q<hsup W(η(t+q))≤λmax

exp (σr)V(η_t), sup

0≤s<hW(η(t+s))

+ ˜γ(u) which directly implies:

0≤q<hsup W(η(t+q))≤λexp (σr)V(η_t) + 1

1−λγ˜(u). (2.30)

Sinceλexp (σr)≤exp (−σh) for allh≤ ¹_σln₁

λ

−r(notice thatσ∈

0,¹_rln₁

λ

), we get from (2.28), (2.30) for allh∈(0, t_max(η₀, d, u)−t) withh≤ _σ¹ln₁

λ

−r:

V(η_t+h)≤exp (−σ h)V(η_t) + 1

1−λγ˜(u). (2.31)

Using the Boundedness-Implies-Continuation property, inequalities (2.27), (2.31) and a standard contradiction argument, we conclude thatt_max(η₀, d, u) = +∞. Therefore, we conclude that (2.31) holds for allt≥0 and h≥0 withh≤ ¹_σln₁

λ

−r.

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Using induction and (2.31) we show thatV(η_ih)≤exp (−iσ h)V(η₀) +_1−λ¹ γ˜(u)_i−1

j=0exp (−jσ h), for all integersi≥1 andh= ¹_σln₁

λ

−r. Consequently, the previous inequality in conjunction with (2.31) shows that V(η_ih+q)≤exp (−σ(q+ih))V(η₀) +_1−λ¹ γ˜(u)^{2−exp(−σh)}_{1−exp(−σh)}, for all integersi≥0,h= ¹_σln₁

λ

−rand for all q∈

0,_σ¹ln₁

λ

−r

. Since, for everyt≥0 there exists integeri≥0 andq∈

0,¹_σln₁

λ

−r

witht=ih+q, we obtain:

V(η_t)≤exp (−σ t)V(η₀) + 1 1−λ

2−λexp(σr)

1−λexp(σr)˜γ(u), for allt≥0. (2.32) Inequality (2.32) in conjunction with (2.27) gives for allε >0:

˜

a₁(ηt)≤max

(1 +ε) exp (−σ t)a₂(η0),1 +ε⁻¹ 1−λ

2−λexp(σr) 1−λexp(σr)˜γ(u)

, for allt≥0. (2.33) Inequality (2.33) with ε = 1 and σ = ¹_rln

2 1+λ

shows that there exists σ ∈ KL such that for every η₀ ∈ L^∞([−r,0);Rⁿ), d ∈ L^∞([−r,+∞);D), u ∈ L^∞([−r,+∞);U), there exists η ∈ L^∞_loc([−r,+∞);Rⁿ) satisfyingη(t) =f(η_t, d_t, u_t) fort∈[0,+∞) a.e.,η(t) =η₀(t) fort∈[−r,0) a.e. andη_t ≤σ(η₀ , t)+γ(u) for all t ≥ 0 with γ(s) := a⁻¹₁ (_(1−λ)⁸2(1+λ)γ˜(s)) for all s ≥ 0. A standard causality argument shows that the estimate ηt ≤ σ(η₀, t) + sup

0≤s≤tγ(us) holds for all t ≥ 0 and u ∈ L^∞_loc([−r,+∞);U). Therefore, Theorem 2.4 implies that system (1.1) is ISS from the inputu∈U ⊆R^m² uniformly ind∈D⊆R^m¹ with gain functionγ(s) :=a⁻¹₁ (_(1−λ)⁸₂_(1+λ)γ˜(s)) for all s≥0. The proof is complete.

Example 2.7. Consider the linear system described by the single IDE:

η(t) =d(t)₀

−1q(s)η(t+s)ds+u(t)

η(t)∈R, d(t)∈[−1,1], u(t)∈R (2.34)

where q : [−1,0] → R is a continuous function. Using Theorem 2.6 with W(η) = |η|, we can conclude that system (2.34) is ISS from the input u ∈ R under the assumption that there exists λ ∈ (0,1) such that ₀

−1|q(s)|ds ≤ λ. The gain function of the input u ∈ R is estimated by Theorem 2.6 to be the linear func- tionγ(s) := _(1−λ)⁸₂_(1+λ)sfor alls≥0.

3. Systems described by a single first-order hyperbolic partial differential equation

Consider the system described by a single FOH-PDE of the form (1.2), (1.3), (1.4) under the following assumption:

(A1) There exist non-decreasing functions L :R+ →R+,σ : R+ → R+ such that the functionals G :R^m× L^∞((0,1];R)→R,K_i:R^m×L^∞((0,1];R)→R(i= 1, . . . , N) satisfy the following inequalities for all R≥0:

|G(w, x)−G(w, y)|+

N i=1

|Ki(w, x)−K_i(w, y)|≤L(R)h sup

0<z≤h(|x(z)−y(z)|)+L(R) sup

h<z≤1(|x(z)−y(z)|),(3.1) f or all h∈(0,1), w∈W with |w| ≤R and x, y∈L^∞((0,1]; [−R, R])

|G(w, x)|+ N

i=1

|Ki(w, x)| ≤σ(R), f or all w∈W with |w| ≤R and x∈L^∞((0,1]; [−R, R]). (3.2) We show next how to transform system (1.2), (1.3), (1.4) under assumption (A1) to an equivalent system described by IDEs.