Systems & Control Letters 135 (2020) 104594 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Stability results for the continuity equation ∗ Iasson Karafyllis a, , Miroslav Krstic b a Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece b Department of Mechanical and Aerospace Eng., University of California, San Diego, La Jolla, CA 92093-0411, USA article info a b s t r a c t Article history: We provide a thorough study of stability of the 1-D continuity equation, which models many physical Available online 13 December 2019 conservation laws. In our system-theoretic perspective, the velocity is considered to be an input. An Keywords: additional input appears in the boundary condition (boundary disturbance). Stability estimates are p Transport PDEs provided in all L state norms with p > 1, including the case p D C1. However, in our Input-to- Hyperbolic PDEs State Stability estimates, the gain and overshoot coefficients depend on the velocity. Moreover, the Input-to-State Stability logarithmic norm of the state appears instead of the usual norm. The obtained results can be used in Boundary disturbances the stability analysis of larger models that contain the continuity equation. In particular, it is shown that the obtained results can be used in a straightforward way for the stability analysis of non-local, nonlinear manufacturing models under feedback control. ' 2019 Elsevier B.V. All rights reserved. 1. Introduction which also describes the evolution of the velocity profile. In other words, the continuity equation does not have the velocity as The continuity equation is the conservation law of every quan- an independent input but is accompanied by at least one more tity that is transferred only by means of convection. It arises in equation: a differential equation (momentum balance) in fluid many models in mathematical physics and for the 1-D case takes mechanics and traffic flow (see also [9] for oil drilling), or a the form non-local equation in manufacturing models (see [10–15]). @ρ @ In this paper we study the stability properties of the continuity C .ρv/ D 0 (1.1) equation on its own. We adopt a system-theoretic perspective, @t @x where v is an input. When the velocity profile is given, the where t denotes time, x is the spatial variable, ρ is the density continuity equation falls within the framework of transport PDEs, of the conserved quantity and v is the velocity of the medium. which are studied heavily in the literature (see for instance [4,16– Eq.(1.1) is used in fluid mechanics (conservation of mass; see 29]). In this framework, the continuity equation is a bilinear Chapter 13 in [1] and [2]), in electromagnetism (conservation of transport PDE. Using the characteristic curves and a Lyapunov charge; see Chapter 13 in [1]), in traffic flow models (conservation analysis we are in a position to establish stability estimates that of vehicles; see Chapter 2 in [3–5] and references therein) as well look like Input-to-State Stability (ISS) estimates with respect to all as in many other cases where ρ is not necessarily the density of a inputs: the input v as well as boundary inputs (Theorem 2.1). The conserved quantity (e.g., in shallow water equations the continu- stability estimates are provided in all Lp state norms with p > 1, ity equation is obtained as a consequence of the conservation of including the case p D C1. However, the obtained estimates mass with the fluid height in place of ρ; see [6]). In many cases, are not precisely ISS estimates since both the gain and overshoot the conserved quantity can only have positive values (e.g., mass coefficients depend on the input v. Moreover, the logarithmic density, vehicle density) and Eq.(1.1) comes together with the norm of the state appears instead of the usual norm; this is positivity requirement ρ > 0. common in many systems where the state variable is positive Although the continuity equation is used extensively in many (see [24]). mathematical models, its stability properties have, surprisingly, The study of the stability properties of the continuity equa- not been studied in detail (but see [7,8] for other aspects of the tion can be useful in the stability analysis of larger models (by continuity equation). This is largely because the continuity equa- using small-gain arguments; see [30]). Indeed, we show that the tion usually appears as a part of a larger mathematical model, obtained results can be used in a straightforward way for the stability analysis of non-local, nonlinear manufacturing models ∗ Corresponding author. (Theorem 3.1). E-mail addresses: [email protected], [email protected] The structure of the paper is as follows. In Section2, we (I. Karafyllis), [email protected] (M. Krstic). present the stability estimates for the continuity equation. https://doi.org/10.1016/j.sysconle.2019.104594 0167-6911/' 2019 Elsevier B.V. All rights reserved. 2 I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 0 ρ (0) @v P 0 Section3 is devoted to the application of the obtained results with ρs exp(b(0)) D ρ0(0), (0; 0) D −b(0) − v(0; 0) . Then @x ρ0(0) to non-local, nonlinear manufacturing models. The proofs of all 1 there exists a unique function ρ 2 C (<C × T0; 1UI (0; C1)) such results are provided in Section4. Finally, the concluding remarks that Eqs. (2.1), (2.2), (2.3) hold. Furthermore, the function ρ 2 of the present work are given in Section5. 1 C (<C × T0; 1UI (0; C1)) satisfies the following estimates for all −1 t ≥ 0, p 2 (1; C1), µ > 0 with µ > −p vmax(t) − vmin(t): Notation. Throughout the paper, we adopt the following nota- (Z 1 ⏐ ( )⏐p )1=p tion. ⏐ ρ(t; x) ⏐ − ln ds ≤ exp (p 1v (t)t) ∗ <C: DT0; C1). Let u: <C × T0; 1U ! < be given. We use the ⏐ ⏐ max 0 ⏐ ρs ⏐ notation uTtU to denote the profile at certain t ≥ 0, i.e., (uTtU)(x) D p 1=p p ( 1 )(Z 1 ⏐ (ρ (x) )⏐ ) u(t; x) for all x 2 T0; 1U. L (0; 1) with p ≥ 1 denotes the × − ⏐ 0 ⏐ T U ! < h t ⏐ln ⏐ ds equivalence class of measurable functions f : 0; 1 for vmin(t) 0 ⏐ ρs ⏐ 1=p (R 1 p ) 1 ( −1 ) which kf k D jf (x)j dx < C1. L (0; 1) denotes the 1 µ C p vmax(t) p 0 C exp 1 C equivalence class of measurable functions f : T0; 1U ! < for which vmin(t) vmin(t) 0 kf k1 D ess supx2(0;1) .jf (x)j/ < C1. We use the notation f (x) ! @v (2.4) for the derivative at x 2 T0; 1U of a differentiable function f : T0; 1U × max TsU exp .−µ(t − s)/ ( ) ! <. max 0;t− 1 ≤s≤t @x p vmin(t) ∗ Let S ⊆ <n be an open set and let A ⊆ <n be a set that satisfies 1=p 0 0 ( ) 1 S ⊆ A ⊆ cl(S). By C (A I Ω), we denote the class of continuous exp pµ − 1 ⊆ <m k I vmin(t) functions on A, which take values in Ω . By C (A Ω), C @vmin(t) A where k ≥ 1 is an integer, we denote the class of functions on A ⊆ pµ <n, which takes values in Ω ⊆ <m and has continuous derivatives k × max .jb(s)j exp .−µ(t − s)// of order k. In other words, the functions of class C (AI Ω) are ( ) max 0;t− 1 ≤s≤t the functions which have continuous derivatives of order k in vmin(t) S D int(A) that can be continued continuously to all points in (⏐ (ρ(t; x) )⏐) ( 1 ) (⏐ (ρ (x) )⏐) ⏐ ⏐ ≤ − ⏐ 0 ⏐ @S \ A. When Ω D < then we write C 0(A) or C k(A). max ⏐ln ⏐ h t max ⏐ln ⏐ 0≤x≤1 ⏐ ρs ⏐ vmin(t) 0≤x≤1 ⏐ ρs ⏐ ∗ A left-continuous function f : T0; 1U ! < (i.e. a function with 1 ( µ ) limy!x− .f (y)/ D f (x) for all x 2 (0; 1U) is called piecewise C exp 1 C C 1 on T0; 1U and we write f 2 PC 1(T0; 1U), if the following vmin(t) vmin(t) ( ) properties hold: (i) for every x 2 T0; 1) the limits limy!xC .f (y)/, @v ( −1 ) × max TsU exp .−µ(t − s)/ limh!0C;y!xC h .f (y C h) − f (y)/ exist and are finite, (ii) for ( ) max 0;t− 1 ≤s≤t @x 1 ( −1 ) vmin(t) every x 2 (0; 1U the limit limh!0− h .f (x C h) − f (x)/ ex- ists and is finite, (iii) there exists a set J ⊂ (0; 1) of finite ( µ ) df ( −1 ) C exp max .jb(s)j exp .−µ(t − s)// cardinality, where (x) D lim ! − h .f (x C h) − f (x)/ D ( ) dx h 0 vmin(t) max 0;t− 1 ≤s≤t ( −1 ) vmin(t) limh!0C h .f (x C h) − f (x)/ holds for x 2 (0; 1)nJ, and (iv) the n 3 ! df 2 < (2.5) mapping (0; 1) J x dx (x) is continuous. Notice that 1 we require a piecewise C function to be left-continuous but not {1 for s < 0 where h(s): D and continuous. 0 for s ≥ 0 2. Stability estimates for the continuity equation vmin(t) D min fv(s; x): s 2 T0; tU; x 2 T0; 1Ug {@v } (2.6) We consider the continuity equation on a bounded domain, vmax(t) D max (s; x): s 2 T0; tU; x 2 T0; 1U @x i.e., we consider the equation ≥ @ρ @ρ @v for t 0.