Stability Results for the Continuity Equation

Systems & Control Letters 135 (2020) 104594

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Systems & Control Letters

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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 = +∞. 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 + (ρv) = 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 = +∞. 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

′ ρ (0) ∂v ˙ 0 Section3 is devoted to the application of the obtained results with ρs exp(b(0)) = ρ0(0), (0, 0) = −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 ∈ C (ℜ+ × [0, 1]; (0, +∞)) such results are provided in Section4. Finally, the concluding remarks that Eqs. (2.1), (2.2), (2.3) hold. Furthermore, the function ρ ∈ of the present work are given in Section5. 1 C (ℜ+ × [0, 1]; (0, +∞)) satisfies the following estimates for all −1 t ≥ 0, p ∈ (1, +∞), µ > 0 with µ > −p vmax(t) − vmin(t): Notation. Throughout the paper, we adopt the following nota- (∫ 1 ⏐ ( )⏐p )1/p tion. ⏐ ρ(t, x) ⏐ − ln ds ≤ exp (p 1v (t)t) ∗ ℜ+: = [0, +∞). Let u: ℜ+ × [0, 1] → ℜ be given. We use the ⏐ ⏐ max 0 ⏐ ρs ⏐ notation u[t] to denote the profile at certain t ≥ 0, i.e., (u[t])(x) = p 1/p p ( 1 )(∫ 1 ⏐ (ρ (x) )⏐ ) u(t, x) for all x ∈ [0, 1]. L (0, 1) with p ≥ 1 denotes the × − ⏐ 0 ⏐ [ ] → ℜ h t ⏐ln ⏐ ds equivalence class of measurable functions f : 0, 1 for vmin(t) 0 ⏐ ρs ⏐ 1/p (∫ 1 p ) ∞ ( −1 ) which ∥f ∥ = |f (x)| dx < +∞. L (0, 1) denotes the 1 µ + p vmax(t) p 0 + exp 1 + equivalence class of measurable functions f : [0, 1] → ℜ for which vmin(t) vmin(t) ′ ∥f ∥∞ = ess supx∈(0,1) (|f (x)|) < +∞. We use the notation f (x) (  ) ∂v  (2.4) for the derivative at x ∈ [0, 1] of a differentiable function f : [0, 1] × max [s] exp (−µ(t − s)) ( )   → ℜ. max 0,t− 1 ≤s≤t  ∂x p vmin(t) ∗ Let S ⊆ ℜn be an open set and let A ⊆ ℜn be a set that satisfies 1/p 0 ⎛ ( ) ⎞ S ⊆ A ⊆ cl(S). By C (A ; Ω), we denote the class of continuous exp pµ − 1 ⊆ ℜm k ; vmin(t) functions on A, which take values in Ω . By C (A Ω), + ⎝vmin(t) ⎠ where k ≥ 1 is an integer, we denote the class of functions on A ⊆ pµ ℜn, which takes values in Ω ⊆ ℜm and has continuous derivatives k × max (|b(s)| exp (−µ(t − s))) of order k. In other words, the functions of class C (A; Ω) are ( ) max 0,t− 1 ≤s≤t the functions which have continuous derivatives of order k in vmin(t) S = int(A) that can be continued continuously to all points in (⏐ (ρ(t, x) )⏐) ( 1 ) (⏐ (ρ (x) )⏐) ⏐ ⏐ ≤ − ⏐ 0 ⏐ ∂S ∩ A. When Ω = ℜ 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 : [0, 1] → ℜ (i.e. a function with 1 ( µ ) limy→x− (f (y)) = f (x) for all x ∈ (0, 1]) is called piecewise + exp 1 + C 1 on [0, 1] and we write f ∈ PC 1([0, 1]), if the following vmin(t) vmin(t) (  ) properties hold: (i) for every x ∈ [0, 1) the limits limy→x+ (f (y)), ∂v  ( −1 ) × max  [s] exp (−µ(t − s)) limh→0+,y→x+ h (f (y + h) − f (y)) exist and are finite, (ii) for ( ) max 0,t− 1 ≤s≤t  ∂x ∞ ( −1 ) vmin(t) every x ∈ (0, 1] the limit limh→0− h (f (x + h) − f (x)) exists and is finite, (iii) there exists a set J ⊂ (0, 1) of finite ( µ ) df ( −1 ) + exp max (|b(s)| exp (−µ(t − s))) cardinality, where (x) = lim → − h (f (x + h) − f (x)) = ( ) dx h 0 vmin(t) max 0,t− 1 ≤s≤t ( −1 ) vmin(t) limh→0+ h (f (x + h) − f (x)) holds for x ∈ (0, 1)\J, and (iv) the \ ∋ → df ∈ ℜ (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): = and continuous. 0 for s ≥ 0

2. Stability estimates for the continuity equation vmin(t) = min {v(s, x): s ∈ [0, t], x ∈ [0, 1]} {∂v } (2.6) We consider the continuity equation on a bounded domain, vmax(t) = max (s, x): s ∈ [0, t], x ∈ [0, 1] ∂x i.e., we consider the equation ≥ ∂ρ ∂ρ ∂v for t 0. (t, x) + v(t, x) (t, x) + ρ(t, x) (t, x) = 0, ∂t ∂x ∂x Remark 2.2. (a) Estimates(2.4),(2.5) are stability estimates for (t, x) ∈ ℜ+ × [0, 1] (2.1) in special state norms. Due to the positivity of the state, the ⏐ ( ρ(t,x) )⏐ where logarithmic norm of the state ρ appears, i.e., we have ⏐ln ⏐ ⏐ ρs ⏐ instead of the usual |ρ(t, x) − ρ | that appears in many stability • ρ is the state, required to be positive (i.e., ρ(t, x) > 0 for s estimates for linear PDEs. The logarithmic norm is a manifestation (t, x) ∈ ℜ+ × [0, 1]) and having a spatially uniform nominal of the nonlinearity of system(2.1),(2.2) and the fact that the state equilibrium profile ρ(x) ≡ ρ , where ρ > 0 is a constant, s s space is not a linear space but rather a positive cone: the state • v is an input which has a spatially uniform nominal profile space for system(2.1),(2.2) is the set X = C 1 ([0, 1]; (0, +∞)). v(x) ≡ v , where v > 0 is a constant. s s (b) The set of allowable inputs is not a linear space: it is the 1 1 set of all b ∈ C (ℜ+), and v ∈ C (ℜ+ × [0, 1]; (0, +∞)) with Eq.(2.1) is accompanied by the boundary condition ′ ρ (0) ∂v ˙ 0 ρs exp(b(0)) = ρ0(0), (0, 0) = −b(0) − v(0, 0) . Again this ρ(t, 0) = ρs exp(b(t)), for t ≥ 0 (2.2) ∂x ρ0(0) fact is a manifestation of the nonlinearity of system(2.1),(2.2). where b is an additional input (the boundary disturbance). (c) Estimate(2.4) is not an ISS estimate, due to the fact that ( ) For system(2.1),(2.2), we obtain the following result. vmax(t) the overshoot coefficient bounded by exp pv (t) (notice that ( ) ( min) ( −1 ) 1 vmax(t) Theorem 2.1. Consider the initial–boundary value problem (2.1), exp p vmax(t)t h t − ≤ exp for all t ≥ 0) vmin(t) pvmin(t) (2.2) with depends heavily on the input v. Estimates(2.4),(2.5) indicate that the gain coefficient of the boundary disturbance b also depends ρ(0, x) = ρ (x), for x ∈ (0, 1] (2.3) 0 on the input v. 1 where ρs > 0 is a constant, ρ0 ∈ C ([0, 1]; (0, +∞)), b ∈ (d) In the absence of disturbances, i.e., when b(t) ≡ 0 and 1 1 C (ℜ+), and v ∈ C (ℜ+ × [0, 1]; (0, +∞)) is a positive function v(t, x) ≡ vs > 0 both estimates(2.4),(2.5) indicate finite-time I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 3 stability. This is a well-known phenomenon to linear transport Theorem 2.3. Consider the initial–boundary value problem PDEs (see [20,21]). ∂w ∂w (e) There is another feature of the stability estimates(2.4),(2.5) (t, x) + v(t, x) (t, x) = a(t, x)w(t, x) + f (t, x) (2.7) ∂t ∂x that should be noticed: the feature of finite-time memory. Indeed, [ ( ) = ∈ ] − 1 w(0, x) ϕ(x), for x (0, 1 (2.8) only the input values in the time interval max 0, t v (t) , ] min w(t, 0) = b(t), for t ≥ 0 (2.9) t affect the state at time t ≥ 0 and this is manifested in esti- ∈ 1 [ ] ∈ 1 ℜ ∈ 1 ℜ × [ ] mates(2.4),(2.5) by the use of the operators max ( ) where ϕ PC ( 0, 1 ), b C ( +), f C ( + 0, 1 ), max 0,t− 1 ≤s≤t 1 1 vmin(t) a ∈ C (ℜ+ × [0, 1]) and v ∈ C (ℜ+ × [0, 1]; (0, +∞)) is a ( ∂v  ) [s] exp (−µ(t − s)) and max ( ) (|b(s)| exp ∈ [ = =  ∂x ∞ max 0,t− 1 ≤s≤t positive function. Let ξi 0, 1) (i 0,..., N) with ξ0 0 be the vmin(t) ∈ 1 [ ]\ = − − ( ∂v [ ] points for which ϕ C ( 0, 1 ξ0, . . . , ξN ). Let ri (i 0,..., N) ( µ(t s))) instead of the usual operators max0≤s≤t  ∂x s ∞ be the solutions of the initial value problems r˙i(t) = v(t, ri(t)) exp (−µ(t − s))) and max ≤ ≤ (|b(s)| exp (−µ(t − s))) that would 0 s t with r (0) = ξ and if there exists T > 0 with r (T ) = 1 then appear in a standard ISS estimate. i i i i i define ri(t) = 1 for all t > Ti. Then there exists a unique function (f) Applying the constant velocity v(t, x) ≡ vs, the constant 1 w: ℜ+ × [0, 1] → ℜ of class C (ℜ+ × [0, 1]\Ω), where Ω = boundary disturbance b(t) ≡ b and the constant initial condition 1 ∪i=0,...,N {(t, ri(t)): t ≥ 0, ri(t) < 1} with w[t] ∈ PC ([0, 1]) for ρ0(x) ≡ ρs exp(b), one can show that the gain of the boundary all t ≥ 0, such that (2.7) holds for all (t, x) ∈ ℜ+ × [0, 1]\Ω disturbance has to be greater or equal to 1 for every Lp(0 1) norm , and equations (2.8), (2.9) hold. Furthermore, the function w: ℜ × ∈ +∞] + with p (1, . On the other hand, using(2.4),(2.5) we obtain [0, 1] → ℜ satisfies the following estimates for all t ≥ 0, p ∈ that the gain of this specific boundary disturbance has to be less +∞ ≥ − − −1 − − ( ) 1/p (1, ), µ 0 with µ > A(t), µ > p vmax(t) A(t) vmin(t): ( pµ − ) exp v 1 than or equal to s for every Lp(0 1) norm with ( ) vs pµ , (( −1 ) ) 1 ∥w[t]∥p ≤ exp A(t) + p vmax(t) t h t − ∥ϕ∥p ( ) vmin(t) p ∈ (1, +∞) and less than or equal to exp µ for p = +∞, vs ( + −1 + ) 1 µ p vmax(t) A(t) ( pµ ) 1/p + + ( exp −1 ) exp 1 vs vmin(t) + = vmin(t) where µ > 0 is arbitrary. Since limµ→0 vs pµ × max (∥f [s]∥ exp (−µ(t − s))) 1, it follows that the estimation of the gain of the boundary ( ) p max 0,t− 1 ≤s≤t disturbance is optimal for this case. vmin(t) 1/p = + − = ⎛ ( p +A(t) ) ⎞ (g) The time-invariant velocities v1(x) 1 (θ 1)x and v2(x) exp (µ ) − 1 θ + (1 − θ)x with θ ∈ (0, 1), have equal minimum and maximum vmin(t)   + ⎝vmin(t) ⎠  ∂vi [ ] = − ∈ +∞] = p (µ + A(t)) values. Moreover, ∂x t 1 θ for all p (1, for i  p 1, 2. Applying these velocities, the constant boundary disturbance × max (|b(s)| exp (−µ(t − s))) vi(0) ( 1 ) b(t) ≡ 0 and the initial conditions ρ0,i(x) ≡ ρs for i = 1, 2, one max 0,t− ≤s≤t vi(x) vmin(t) ∂v can show that the corresponding gains of ∂x have to be greater (2.10) 1 than or equal to = 1 ∫ − ln 1 + ( − 1)x p dx and = γ1 1−θ 0 ( ( θ )) γ2 ( 1 ) 1 p ∥ [ ]∥ ≤ − ∥ ∥ 1 ∫ (ln (1 + (θ −1 − 1)x)) dx, respectively, for every Lp(0, 1) w t ∞ exp (tA(t)) h t ϕ ∞ 1−θ 0 vmin(t) norm with p ∈ (1, +∞). Since 1 < 1 + (θ −1 − 1)x for all 1+(θ−1)x 1 ( µ + A(t) ) x ∈ (0, 1), it follows that the gain of v2(x) = θ +(1−θ)x is strictly + exp 1 + vmin(t) vmin(t) greater than the gain of v1(x) = 1 + (θ − 1)x. Therefore, velocities × max (∥f [s]∥∞ exp (−µ(t − s))) that are increasing with respect to x (i.e., convection that speeds ( ) max 0,t− 1 ≤s≤t up downstream) add a greater bias to the solution profile than ve- vmin(t) locities that are decreasing with respect to x (i.e., convection that (µ + A(t) ) + exp max (|b(s)| exp (−µ(t − s))) slows down downstream). This is also apparent from the estima- ( ) vmin(t) 1 ∂v p max 0,t− ≤s≤t ∈ vmin(t) tion of the gain of ∂x from(2.4) for every L (0, 1) norm with p ( −1 ) +∞ 1 + µ+p vmax(t) (2.11) (1, ): the gain estimate v (t) exp 1 v (t) depends min min { on v (t) = max { ∂v (s, x): s ∈ [0, t], x ∈ [0, 1]} and indicates 1 for s < 0 max ∂x where h(s): = and that velocities that are increasing with respect to x (i.e., convec- 0 for s ≥ 0 tion that speeds up downstream with vmax(t) ≥ 0) add a greater vmin(t) = min {v(s, x): s ∈ [0, t], x ∈ [0, 1]} bias to the solution profile than velocities that are decreasing with { } respect to x (i.e., convection that slows down downstream for ∂v vmax(t) = max (s, x): s ∈ [0, t], x ∈ [0, 1] (2.12) which vmax(t) ≤ 0). ∂x (h) When the inputs are constant in time, i.e., b(t) ≡ b and A(t) = max {a(s, x): s ∈ [0, t], x ∈ [0, 1]} v(t, x) ≡ v(x), then the equilibrium profiles of system(2.1),(2.2) 0 are given by the equation ρ(x) = ρ exp(b) v(0) for x ∈ [0, 1]. for all t ≥ 0. Moreover, if b(0) = ϕ(0) and ϕ ∈ C ([0, 1]) then s v(x) ∈ 0 ℜ × [ ] ∈ 1 [ ] = ˙ + Consequently, it becomes clear that the velocity v acts as an w C ( + 0, 1 ). Finally, if ϕ C ( 0, 1 ), b(0) ϕ(0), b(0) ′ = + ∈ 1 ℜ × [ ] ‘‘equilibrium-shaping functional parameter’’. For instance, if v(x) v(0, 0)ϕ (0) a(0, 0)b(0) f (0, 0) then w C ( + 0, 1 ). is monotonically increasing, namely, if the convection speeds up The proof of Theorem 2.3 is provided in Section4 and is based downstream, the density equilibrium profile ρ(x) decreases — on a combination of different methodologies: and vice versa — and such non constant equilibrium profiles are finite-time stable (in logarithmic norm). • The exploitation of the superposition principle for the initial-value problem(2.7),(2.8),(2.9): the solution of(2.7), The proof of Theorem 2.1 relies on the following result which (2.8),(2.9) can be written as the sum of three functions: has its own interest. the solution of(2.7),(2.8),(2.9) with zero inputs f , b and 4 I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594

initial condition the given function ϕ ∈ PC 1 ([0, 1]), the Theorem 3.1. Consider the initial–boundary value problem (3.1), solution of(2.7),(2.8),(2.9) with zero initial condition, zero (3.2), (3.3), (3.4) with distributed input f and boundary input the given function 1 ρ(0, x) = ρ0(x), for x ∈ (0, 1] (3.5) b ∈ C (ℜ+), and the solution of(2.7),(2.8),(2.9) with 1 zero initial condition, zero boundary input f and distributed where ρs > 0 is a constant, ρ0 ∈ C ([0, 1]; (0, +∞)) and 1 1 input the given function f ∈ C (ℜ+ × [0, 1]). b ∈ C (ℜ+) is a bounded function with ρs exp(b(0)) = ρ0(0), ′ ( ) ρ (0) • The norms of the first two components of the solution are ˙ ∫ 1 0 b(0) = −λ ρ0(x)dx . Then there exists a unique function 0 ρ0(0) estimated by using the exact formulae of the solution on the 1 ρ ∈ C (ℜ+ ×[0, 1]; (0, +∞)) such that Eqs. (3.1), (3.2), (3.3), (3.4), characteristic curves of the PDE(2.7). 1 (3.5) hold. Furthermore, the function ρ ∈ C (ℜ+ × [0, 1]; (0, +∞)) • The norm of the third component of the solution is esti- satisfies the following estimates for all t ≥ 0, p ∈ (1, +∞), µ > 0: mated by using a Lyapunov analysis. 1/p 1/p (∫ 1 ⏐ ( ρ(t, x) )⏐p ) (∫ 1 ⏐ ( ρ (x) )⏐p ) ⏐ ⏐ ≤ − ⏐ 0 ⏐ 3. Feedback control of manufacturing systems ⏐ln ⏐ ds h (t r) ⏐ln ⏐ ds 0 ⏐ ρs ⏐ 0 ⏐ ρs ⏐ 1/p ( exp (pµr) − 1 ) Manufacturing systems with a high volume and a large num- + max (|b(s)| exp (−µ(t − s))) − ≤ ≤ ber of consecutive production steps (which typically number in pµr max(0,t r) s t the many hundreds) are often modelled by non-local PDEs of the (3.6) (⏐ ( )⏐) (⏐ ( )⏐) form (see [10–15]): ⏐ ρ(t, x) ⏐ ⏐ ρ0(x) ⏐ max ⏐ln ⏐ ≤ h (t − r) max ⏐ln ⏐ ∂ρ ∂ρ 0≤x≤1 ⏐ ρs ⏐ 0≤x≤1 ⏐ ρs ⏐ (3.7) (t, x) + λ (W (t)) (t, x) = 0, for (t, x) ∈ ℜ+ × [0, 1] (3.1) t x + exp (µr) max (|b(s)| exp (−µ(t − s))) ∂ ∂ max(0,t−r)≤s≤t ∫ 1 W (t) = ρ(t, x)dx, for t ≥ 0 (3.2) {1 for s < 0 where h(s): = and r is given in Box I. 0 0 for s ≥ 0 where Estimates(3.6),(3.7) guarantee robustness with respect to the • ρ(t, x) is the density of the processed material at time t ≥ 0 boundary uncertainty b. More specifically, estimates(3.6),(3.7) and stage x ∈ [0, 1], required to be positive (i.e., ρ(t, x) > 0 are very useful ISS-like estimates that guarantee all properties for (t, x) ∈ ℜ+ × [0, 1]) and having a spatially uniform that usual ISS estimates guarantee, such as the Bounded-Input- equilibrium profile ρ(x) ≡ ρs, where ρs > 0 is a constant Bounded-State property and the Converging-Input-Converging- (set point), and State property (see [31]). However, estimates(3.6),(3.7) show • λ ∈ C 1 ((0, +∞); (0, +∞)) is a nonlinear function that additional things that usual ISS estimates do not show: determines the production speed. (a) Estimates(3.6),(3.7) guarantee finite-time stability in the The model is accompanied by the influx boundary condition absence of uncertainties in every Lp norm (with p ∈ (1, +∞]) of the logarithmic deviation of the density. More- = ≥ ρ(t, 0)λ (W (t)) u(t), for t 0 (3.3) over, in every Lp norm (with p ∈ (1, +∞]) of the logarith- where u(t) ∈ (0, +∞) is the control input (the process influx mic deviation of the density, the overshoot coefficient is rate). Existence, uniqueness and related control problems for equal to 1. systems of the form(3.1),(3.2),(3.3) were studied in [12–14]. (b) Estimates(3.6),(3.7) show that the closed-loop system Here we want to address the feedback stabilization problem of (3.1),(3.2),(3.3),(3.4) has finite memory: only the input ∈ [ − ] the spatially uniform equilibrium profile ρ(x) ≡ ρ under the values b(s) for s max(0, t r), t affect the current value s [ ] feedback control law of the state ρ t . (c) Estimates(3.6),(3.7) guarantee that for every ε > 0, the u(t) = ρsλ (W (t)) exp(b(t)), for t ≥ 0 (3.4) gain of the input b(t) is less or equal to 1 + ε. This is a direct consequence of estimates(3.6),(3.7) and the fact where b(t) ∈ ℜ represents an uncertainty. The motivation for that µ > 0 is arbitrary. the feedback law(3.4) comes from the fact that in the absence of uncertainties the feedback law(3.4) combined with the boundary However, notice that the terminal time r given by(3.8) depends condition(3.3) gives the boundary condition ρ(t, 0) = ρs for on the initial condition and the boundary disturbance. Therefore, t ≥ 0, which guarantees finite-time stability for the linearization for certain initial conditions or for large boundary disturbances it of(3.1),(3.2). Moreover, the implementation of the feedback law may happen that the terminal time r is unacceptably large. For −1 1 (3.4) relies on the measurement of the total load in the produc- example, when λ(W ) = (1 + W ) and ρ0 ∈ C ([0, 1]; (0, +∞)) tion line W (t) which is a quantity that can be measured relatively is any function (its monotonicity does not play any role) then(3.8) ( ( )) easily. Finally, notice that the feedback law(3.4) guarantees that gives r: = 1+max ∥ρ0∥∞ , ρs exp supt≥0 (b(t)) . Consequently, u(t) is positive and for bounded disturbances as well as bounded if the initial density is large for some x ∈ [0, 1] or if the boundary functions λ ∈ C 1 ((0, +∞); (0, +∞)) (which is usually the case disturbance is large, then the terminal time r will be large. This for manufacturing systems) the control input u(t) is bounded is a possible disadvantage of the feedback law(3.4) and we do from above by a constant independent of the initial condition not know whether it is possible to achieve smaller terminal times (bounded feedback). It should be emphasized that the boundary (see also the discussion in Section 4 of [12] for the case λ(W ) = −1 input b(t) ∈ ℜ of the closed-loop system(3.1),(3.2),(3.3),(3.4) (1+W ) ). However, again when the terminal time r is very large appears as an actuator error of the nominal feedback controller then the gain of the uncertainty becomes very large. u(t) = ρsλ (W (t)) (that guarantees ρ(t, 0) = ρs for t ≥ 0) and does not coincide with the boundary input u(t) of the open-loop 4. Proofs of main results system(3.1),(3.2),(3.3). The proof of Theorem 2.1 is simply an application of Using Theorem 2.1, we are in a position to obtain the following ( ) Theorem 2.3 and the use of the transformation w(t, x)=ln ρ(t,x) result. ρs I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 5

1 r: = { ( ( )) ( ( ))} (3.8) min λ(s): min min0≤x≤1 (ρ0(x)) , ρs exp inft≥0 (b(t)) ≤ s ≤ max max0≤x≤1 (ρ0(x)) , ρs exp supt≥0 (b(t))

Box I.

(or its inverse ρ(t, x) = ρs exp(w(t, x))). Therefore, we next focus Definitions(4.3),(4.4) guarantee that w: ℜ+ × [0, 1] → ℜ is on the proof of Theorem 2.3. 1 a function of class C (ℜ+ × [0, 1]\Ω), where Ω = ∪i=0,...,N { ≥ } [ ] ∈ 1 [ ] ≥ 1 (t, ri(t)): t 0, ri(t) < 1 with w t PC ( 0, 1 ) for all t 0, Proof of Theorem 2.3. Extending v ∈ C (ℜ+ × [0, 1]; (0, +∞)) 1 ( 2) such that(2.7) holds for all ( t, x) ∈ ℜ+ × [0, 1]\Ω and Eqs.(2.8), so that v ∈ C ℜ we can define for all t0 ≥ 0, x0 ∈ [0, 1) (2.9) hold. Moreover, definitions(4.3),(4.4) guarantee that if the mapping X(s; t0, x0) ∈ [0, 1] as the unique solution of the 0 0 initial-value problem b(0) = ϕ(0) and ϕ ∈ C ([0, 1]) then w ∈ C (ℜ+ × [0, 1]) and 1 ˙ ′ dX if ϕ ∈ C ([0, 1]), b(0) = ϕ(0), b(0) + v(0, 0)ϕ (0) = a(0, 0)b(0) + (s) = v(t0 + s, X(s)), X(t0) = x0 (4.1) 1 ds f (0, 0) then w ∈ C (ℜ+ × [0, 1]). Uniqueness follows from a contradiction argument. Suppose Clearly, X(s; t0, x0) ∈ [0, 1] is defined for s ∈ [0, smax), where that there exist two functions ∈ C 1 (ℜ ; L2(0 1)) with smax ∈ (0, +∞] is the maximal existence time of the solution. w, w˜ + , [ ] [ ] ∈ 1 [ ] ≥ 2 Since v(t, x) > 0 for all t ≥ 0, x ∈ [0, 1], it follows that a finite w t , w˜ t PC ( 0, 1 ) for t 0, that satisfy(2.7) (in the L (0, 1) − ; = ; = smax implies that lims→s (X(s t0, x0)) X(smax t0, x0) 1. sense) and(2.8),(2.9). It then follows that the function e = w−w ; max = ≥ ˜ We also define X(0 t0, 1) 1 for all t0 0. Notice that the satisfies the following equations: mapping X(s; t0, x0) is increasing with respect to s ∈ [0, smax) and x0 ∈ [0, 1] and satisfies the following equations for all t0 ≥ 0, ∂e ∂e ∈ [ ∈ [ (t, x) + v(t, x) (t, x) = a(t, x)e(t, x), for t ≥ 0 (4.5) x0 0, 1) and s 0, smax): ∂t ∂x ∂X (∫ s ∂v ) (s; t0, x0) = exp (t0 + l, X(l; t0, x0))dl > 0 ∂x ∂x 0 0 e(0, x) = 0, for x ∈ (0, 1] (4.6) ∂X ; = + ; − (s t0, x0) v(t0 s, X(s t0, x0)) v(t0, x0) (4.2) = ≥ ∂t0 e(t, 0) 0, for t 0 (4.7) (∫ s ∂v ) ∫ 1 2 × exp (t0 + l, X(l; t0, x0))dl Using the functional V (t) = e (t, x)dx on [0, T ] for arbitrary 0 ∂x 0 T > 0, we have by virtue of(2.12),(4.5) and(4.7) for every ∂ It follows from(4.1) and(4.2) (which imply that X(t −t0; t0, 0) ∂t0 t ∈ [0, T ]: ( − ) = − ∫ t t0 ∂v + ; v(t0, 0) exp 0 ∂x (t0 l, X(l t0, 0))dl < 0) that ∫ 1 ∂e ∫ 1 ∂e V˙ (t) = 2 e(t, x) (t, x)dx = −2 v(t, x)e(t, x) (t, x)dx • for every t ≥ 0, x ∈ [0, 1] with x > r0(t) = X(t; 0, 0) 0 ∂t 0 ∂x the equation X(t; 0, x0) = x will be uniquely solvable with ∫ 1 2 respect to x0 ∈ [0, 1], and +2 a(t, x)e (t, x)dx 0 • for every t ≥ 0, x ∈ [0, 1] with x ≤ r0(t) = X(t; 0, 0) the ∫ 1 equation X(t − t0; t0, 0) = x will be uniquely solvable with ∂ = − v(t, x) (e2(t, x)) dx + 2A(T )V (t) respect to t0 ≥ 0. 0 ∂x Let x (t, x) ∈ [0, 1] and t (t, x) ≥ 0 be the solutions of the above ∫ 1 0 0 2 2 ∂v equations. By virtue of the implicit function theorem they are = −v(t, 1)e (t, 1) + e (t, x) (t, x)dx + 2A(T )V (t) 0 ∂x both C 1 on their domains. We define: ≤ (vmax(T ) + 2A(T )) V (t) (∫ t ) w(t, x): = exp a(s, X(s; 0, x (t, x)))ds ϕ(x (t, x)) 0 0 Gronwall’s lemma implies that V (t) ≤ exp ((vmax(T ) + 2A(T )) t) 0 ∫ t (∫ t ) V (0) for all t ∈ [0, T ] and consequently (using (4.6)), we get + exp a(s, X(s; 0, x0(t, x)))ds f (τ, X(τ; 0, x0(t, x)))dτ V (t) = 0 for all t ∈ [0, T ]. This equality in conjunction with the 0 τ [ ] [ ] ∈ 1 [ ] ≥ ≡ fact that w t , w˜ t PC ( 0, 1 ) for t 0, implies w w˜. for t ≥ 0, x ∈ [0, 1] with x > r0(t) = X(t; 0, 0) Using(4.3),(4.4), we next notice that (4.3) [t] = [t] + [t] + [t] for t ≥ 0 (4.8) and w w1 w2 w3 , (∫ t ) where w(t, x): = exp a(s, X(s − t0(t, x); t0(t, x), 0))ds b(t0(t, x)) t0(t,x) (∫ t ) t t ∫ (∫ ) w1(t, x): = 0, w2(t, x): = exp a(s, X(s; 0, x0(t, x)))ds ϕ(x0(t, x)), + exp a(s, X(s − t0(t, x); t0(t, x), 0))ds 0 t (t,x) τ ∫ t (∫ t ) 0 = ; ; × − ; w3(t, x): exp a(s, X(s 0, x0(t, x)))ds f (τ, X(τ 0, x0(t, x)))dτ f (τ, X(τ t0(t, x) t0(t, x), 0))dτ 0 τ for t ≥ 0, x ∈ [0, 1] with x ≤ r0(t) = X(t; 0, 0) for t ≥ 0, x ∈ [0, 1] with x > r0(t) = X(t; 0, 0) (4.4) (4.9) 6 I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 and follows from(4.10) that ∥w2[t]∥p = 0 for all p ∈ [1, +∞). Thus we obtain from(4.15) for all t ≥ 0 and p ∈ [1 +∞): (∫ t ) , w (t, x): = exp a(s, X(s − t (t, x); t (t, x), 0))ds 1 0 0 ( ) t0(t,x) (( −1 ) ) 1 ∥w2[t]∥p ≤ exp A(t) + p vmax(t) t h t − ∥ϕ∥p × b(t0(t, x)), w2(t, x): = 0, vmin(t) t ( t ) ∫ ∫ (4.16) w3(t, x): = exp a(s, X(s − t0(t, x); t0(t, x), 0))ds t (t,x) τ 0 {1 for s < 0 × − ; where h(s): = . f (τ, X(τ t0(t, x) t0(t, x), 0))dτ 0 for s ≥ 0 ≥ ∈ [ ] ≤ = ; p for t 0, x 0, 1 with x r0(t) X(t 0, 0) Next we estimate the L (0, 1) norm of w3 with p ∈ [1, +∞). ≡ (4.10) Notice that w3 is the solution of(2.7),(2.8),(2.9) with ϕ 0 and b ≡ 0. Let σ > 0 be a constant (to be selected) and define for p First we estimate the L (0, 1) norm of w1 with p ∈ [1, +∞). p ∈ (1, +∞), t ≥ 0: Notice that by virtue of(2.12),(4.1) and since t0(t, x) ≥ 0 solves the equation X(t − t ; t , 0) = x, we get for all t ≥ 0, x ∈ [0, 1] ∫ 1 0 0 V (t) = exp(−σ x) |w (t, x)|p dx (4.17) with x ≤ r (t) = X(t; 0, 0): 3 0 0 ( x ) Using the fact that is the solution of(2.7),(2.8),(2.9) with max 0, t − ≤ t0(t, x) ≤ t (4.11) w3 vmin(t) ϕ ≡ 0 and b ≡ 0, we get from(4.17) for all t ≥ 0 and s ∈ [0, t]: ≥ Therefore, we get from(4.10),(2.12),(4.11) for every µ ∫ 1 − ∈ [ +∞ ≥ ∈ [ ] ≤ = ˙ p−1 max(0, A(t)), p 1, ) and t 0, x 0, 1 with x r0(t) V (s) = −p exp(−σ x)sgn(w3(s, x)) |w3(s, x)| X(t; 0, 0): 0 ∂w3 ( ∫ t ) × v(s, x) (s, x)dx p p |w1(t, x)| = exp p a(s, X(s − t0(t, x); t0(t, x), 0))ds |b(t0)| ∂x 1 (4.18) t0(t,x) ∫ p p +p exp(−σ x)a(s, x) |w3(s, x)| dx ≤ exp (pA(t)(t − t0(t, x))) |b(t0(t, x))| 0 ( p ) ≤ exp (p (µ + A(t)) (t − t0(t, x))) max |b(s)| exp (−pµ(t − s)) ∫ 1 ≤ ≤ − t0(t,x) s t +p exp(−σ x)sgn(w (s, x)) |w (s, x)|p 1 f (s, x)dx ( ) 3 3 p (µ + A(t)) x 0 ≤ exp max (|b(s)|p exp (−pµ(t − s))) ( ) vmin(t) max 0,t− 1 ≤s≤t vmin(t) Integrating by parts, we obtain from(2.12),(4.17),(4.18) for all t ≥ 0 and s ∈ [0, t]: (4.12) ˙ ≤ − − | |p Using(4.9) and(4.12), we obtain for every µ ≥ 0 with µ > −A(t), V (s) exp( σ )v(s, 1) w3(s, 1) p ∈ [1, +∞) and t ≥ 0: ∫ 1 p ∂v + exp(−σ x) |w3(s, x)| (s, x)dx ( ) 1/p ∂x ⎛ p(µ+A(t)) − ⎞ 0 exp v (t) 1 1 ∥ ∥ min ∫ (4.19) w1[t] p ≤ ⎝vmin(t) ⎠ − − | |p + p (µ + A(t)) σ exp( σ x) w3(s, x) v(t, x)dx pA(t)V (s) 0 ∫ 1 × max (|b(s)| exp (−µ(t − s))) (4.13) p−1 ( ) +p exp(−σ x) |w3(s, x)| |f (s, x)| dx max 0,t− 1 ≤s≤t vmin(t) 0 p p Next we estimate the L (0 1) norm of with p ∈ [1 +∞). For p−1 p−1 p , w2 , Using the inequality |w(s, x)| |f (s, x)| ≤ ε p−1 |w(s, x)| + all t ≥ 0 with r (t) < 1 we get from(4.9) and(2.12): p 0 1 −p | |p p ε f (s, x) which holds for all ε > 0, we obtain from(2.12), (4.17),(4.19) for all t ≥ 0, ε > 0 and s ∈ [0, t]: |w2(t, x)| ≤ exp (A(t)t) |ϕ(x0(t, x))| , for x ∈ [0, 1] with x > r0(t)

(4.14) ( p ) − V˙ (s) ≤ − σ v (t) − v (t) − pA(t) − (p − 1)ε p−1 V (s)+ε p ∥f [s]∥p ( ) min max p ∂x0 ∫ t ∂v Using the fact that (t, x) = exp − (s, X(s; 0, x0(t, x)))ds ∂x 0 ∂x (4.20) (a consequence of(4.1),(4.2) and the fact that X(t; 0, x0(t, x)) = x), we get from(4.9),(4.10),(2.12),(4.14) and the substitution Using Lemma 2.12 in [31] in conjunction with(4.20) and using 1 ξ = x0(t, x) (allowable since ϕ ∈ PC ([0, 1])) for all p ∈ [1, +∞) the fact that V (0) = 0 (a consequence of(4.17) and the fact that and t ≥ 0 with r0(t) < 1: w3 is the solution of(2.7),(2.8),(2.9) with ϕ ≡ 0 and b ≡ 0), we obtain for all t ≥ 0, ε > 0 and µ ≥ 0: (∫ 1 )1/p ∥w [t]∥ ≤ exp (A(t)t) |ϕ(x (t, x))|p dx 2 p 0 ∫ t r0(t) −p ( V (t) ≤ ε exp −(t − s) (σ vmin(t) − vmax(t) 1 t 1/p (∫ (∫ ) ) 0 p ∂v ≤ exp (A(t)t) |ϕ(ξ)| exp (s, X(s; 0, ξ))ds dξ p )) p−1 ∥ ∥p 0 0 ∂x −pA(t) − (p − 1)ε f [s] p ds (( −1 ) ) ≤ exp A(t) + p vmax(t) t ∥ϕ∥p ∫ t −p ( ≤ ε exp −(t − s) (σ vmin(t) − pµ − vmax(t) (4.15) 0 p )) ≥ ∈ [ ] = p−1 (∥ ∥p ) Notice that the existence of t 0, x 0, 1 with x > r0(t) −pA(t) − (p − 1)ε dsmax f [s] p exp (−pµ(t − s)) 0≤s≤t X(t; 0, 0) in conjunction with(2.12) (which gives r0(t) ≥ tvmin(t)) implies that t < 1 . Consequently, when t ≥ 1 then it (4.21) vmin(t) vmin(t) I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 7

Exploiting(4.21) in conjunction with(4.17), we get for all t ≥ 0, Proof. Define: ε > 0 and µ ≥ 0: b(t): = ρ exp(b(t)), for t ≥ 0 (4.29) ( ) ˜ s ∥w3[t]∥p ≤ K(t) max ∥f [s]∥p exp (−µ(t − s)) (4.22) 0≤s≤t

vmin = min { λ(s): ρmin ≤ s ≤ ρmax } where (4.30) vmax = max { λ(s): ρmin ≤ s ≤ ρmax } (σ ) (∫ t = −1 − − − ( ) K(t): exp ε exp ( (t s) (σ vmin(t) pµ = ( ) p 0 ρmin min min (ρ0(y)) , inf ˜b(s) 0≤y≤1 s≥0 1/p p )) ) ( ) (4.31) p−1 − vmax(t) − pA(t) − (p − 1)ε ds (4.23) ( ) ρmax = max max (ρ0(y)) , sup ˜b(s) 0≤y≤1 s≥0 { ⏐ ′ ⏐ } Selecting Lλ = max ⏐λ (s)⏐ : ρmin ≤ s ≤ ρmax (4.32)

p−1 ( −1 −1 ) p ε = σ p vmin(t) − µ − p vmax(t) − A(t) It suffices to show that for every t1 > 0 there exists a unique 1 pµ + v (t) + pA(t) + pv (t) (4.24) solution ρ ∈ C ([0, t1] × [0, 1]; (0, +∞)) of the initial–boundary = max min σ value problem(4.27) with vmin(t) ∂ρ ∂ρ we obtain from(4.23) for all t ≥ 0 and µ ≥ 0 with µ > + = ∈ [ ] × [ ] −1 (t, x) λ (W (t)) (t, x) 0, for (t, x) 0, t1 0, 1 −p vmax(t) − A(t) − vmin(t): ∂t ∂x

( −1 ) (4.33) 1 µ + p vmax(t) + A(t) 1 ∥w3[t]∥p ≤ exp 1 + ∫ vmin(t) vmin(t) W (t) = u(t, x)dx, for t ∈ [0, t1] (4.34) ( ) 0 × max ∥f [s]∥p exp (−µ(t − s)) (4.25) 0≤s≤t ρ(t, 0) = ρs exp(b(t)), for t ∈ [0, t1] (4.35) [ ( Noticing that w3[t] depends only on f [s] with s ∈ max 0, t− which also satisfies the following estimate for all t ∈ [0, t1], 1 ) ] x ∈ [0 1]: , t and using a standard causality argument (when t > , vmin(t) 1 ) we obtain from(4.25) for all t ≥ 0 and µ ≥ 0 with ≤ ≤ vmin(t) ρmin ρ(t, x) ρmax (4.36) −1 µ > −p vmax(t) − A(t) − vmin(t): { ⏐ ′ ⏐ } Let Lρ ≤ max ρ (x) : x ∈ [0, 1] be the Lipschitz constant ( −1 ) 0 ⏐ 0 ⏐ 1 µ + p vmax(t) + A(t) { ⏐ ⏐ } ∥w [t]∥ ≤ exp 1 + for ρ and L ≤ max ⏐ d˜b (t)⏐ : t ∈ [0, t + 1] be the Lipschitz 3 p 0 ˜b ⏐ dt ⏐ 1 vmin(t) vmin(t) constant for b on [0, t + 1]. We first show that for every T ∈ (∥ ∥ ) ˜ 1 × max f [s] p exp (−µ(t − s)) (4.26) ( ( ( ))−1] ( ) Lb max 0,t− 1 ≤s≤t 0, 1 + Lλ Lρ + ˜ there exists a unique solution ρ ∈ vmin(t) 0 vmin 1 [ ]×[ ]; +∞ Using(4.8),(4.13),(4.16),(4.26) and the triangle inequality, we C ( 0, T 0, 1 (0, )) of the initial–boundary value problem obtain estimate(2.10). Estimate(2.11) is obtained by letting p → (4.27),(4.29),(4.30),(4.31) with t1 replaced by T , which satisfies ( ) ∈ [ ] ∈ [ ] +∞ and by using the facts limp→+∞ ∥w[t]∥p = ∥w[t]∥∞, estimate(4.36) for all t 0, T , x 0, 1 . Moreover, the ∥ϕ∥p ≤ ∥ϕ∥∞ for all p ∈ [1, +∞). Lipschitz constant Lρ[t] for the function ρ[t] satisfies: The proof is complete. ◁ ( L ) ≤ ˜b ∈ [ ] Lρ[t] max Lρ0 , , for all t 0, T (4.37) Theorem 3.1 is a consequence of Theorem 2.1 and the fol- vmin lowing existence/uniqueness result which also provides a useful Indeed, if we show the above implications then we can construct estimate. step-by-step the solution of the initial–boundary value problem (4.27),(4.29),(4.30),(4.31), first on the interval [0, T ], then on the Proposition 4.1. Consider the initial–boundary value problem (3.1), ( ( ( ) Lb interval [T , 2T ] and so on, with T = 1 + Lλ max Lρ , ˜ (3.2), (3.5) with 0 vmin − L )) 1 + ˜b [ ] v and we cover the interval 0, t1 . ρ(t, 0) = ρs exp(b(t)), for t ≥ 0 (4.27) min ( − ] ( ( L )) 1 Let arbitrary T ∈ 0 1 + L L + ˜b be given. 1 , λ ρ0 v where ρs > 0 is a constant, ρ0 ∈ C ([0, 1]; (0, +∞)) and b ∈ min 1 ˙ → C (ℜ+) is a bounded function with ρs exp(b(0)) = ρ0(0), b(0) = Consider the operator G: S S with ′ ( ) (0) ∫ 1 ρ0 −λ ρ0(x)dx . Then there exists a unique function ρ ∈ 0 ρ0(0) { } 1 0 C (ℜ+ × [0, 1]; (0, +∞)) such that Eqs. (3.1), (3.2), (3.5), (4.27) S: = v ∈ C ([0, T ]): vmin ≤ min (v(t)) ≤ max (v(t)) ≤ vmax t∈[0,T ] t∈[0,T ] 1 hold. Furthermore, the function ρ ∈ C (ℜ+ × [0, 1]; (0, +∞)) satisfies the following estimates for all t ≥ 0, x ∈ [0, 1]: (4.38) ( ( )) which maps the function v ∈ S to the function Gv = v ∈ S min min (ρ0(y)) , ρs exp inf (b(s)) ≤ ρ(t, x) 0≤y≤1 s≥0 defined by ( ( )) (∫ 1 ) ≤ max max (y) exp sup b(s) (4.28) (ρ0 ) , ρs ( ) (t) = (t x)dx for t ∈ [0 T ] (4.39) 0≤y≤1 s≥0 v λ ρv , , , 0 8 I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 where ρv: [0, T ] × [0, 1] → ℜ is the function defined by the implies that equation t ∥v − w∥∞ ≥ vmint0(t, x; w) (4.46) ⎧ ( ∫ t ) ∫ t ⎪ρ x − v(s)ds if 1 ≥ x > v(s)ds Using(4.45),(4.46), the compatibility condition ˜b(0) = ρ0(0) and ⎨⎪ 0 ρ (t, x) = 0 0 definition(4.40), we obtain v ∫ t ⎪ ; ≤ ≤ ⏐ ( ∫ t ) ⏐ ⎩⎪˜b (t0(t, x v)) if 0 x v(s)ds ⏐ ⏐ 0 |ρv (t, x) − ρw (t, x)| = ⏐ρ0 x − v(s)ds −˜b (t0(t, x; w))⏐ ⏐ ⏐ (4.40) 0 ⏐ ( ∫ t ) ⏐ ⏐ ⏐ ⏐ ⏐ ≤ ρ0 x − v(s)ds − ρ0 (0) + b(0) − b (t0(t, x; w)) where t0(t, x; v) ∈ [0, t] is the unique solution of the equation ⏐ ⏐ ⏐˜ ˜ ⏐ ⏐ 0 ⏐ ∫ t ≤ Lρ t ∥v − w∥∞ + L t0(t, x; w) x = v(s)ds for all (t, x) ∈ Ω 0 ˜b v t t (t,x;v) ≤ Lρ t ∥v − w∥∞ + L ∥v − w∥∞ 0 0 ˜b vmin { ∫ t } ( ) Lb : = (t, x) ∈ [0, T ] × [0, 1], x ≤ v(s)ds (4.41) ≤ T Lρ + ˜ ∥v − w∥∞ 0 vmin 0 (4.47) Notice that the mapping Ωv ∋ (t, x) → t0(t, x; v) is continu- 1 ous with |t0(t, x; v) − t0(t, y; v)| ≤ |x − y| and |t0(t, x; v) − A similar estimate holds for the case (t x) ∈ and (t x) ∈ . vmin , Ωv , / Ωw vmax t0(τ, x; v)| ≤ |t − τ| for all (t, x) ∈ Ωv,(τ, x) ∈ Ωv,(t, y) ∈ Thus by combining(4.42),(4.43),(4.44) and(4.37) we get: vmin ∫ t Ωv. Moreover, when x = v(s)ds ≤ 1 then t0(t, x; v) = 0. ( ) 0 L˜b ∥Gv − Gw∥∞ ≤ TL L + ∥v − w∥∞ (4.48) Therefore, by virtue of the compatibility condition b(0) = ρ0(0), λ ρ0 ˜ vmin the mapping ρv: [0, T ] × [0, 1] → ℜ defined by(4.40) is con- ( ( ( ))−1] tinuous and satisfies the estimate ρmin ≤ ρv(t, x) ≤ ρmax for all Lb Since T ∈ 0, 1 + Lλ Lρ + ˜ , estimate(4.48) implies t ∈ [0, T ], x ∈ [0, 1]. Moreover, due to the compatibility condition 0 vmin → ˜b(0) = ρ0(0), for every t ∈ [0, T ] the function ρv[t] is Lipschitz that the operator G: S S is a contraction. Consequently, by ( L ) ˜b virtue of Banach’s fixed point theorem, there exists a unique on [0, 1] with Lipschitz constant Lρ [t] ≤ max Lρ , , where v 0 vmin ⏐ ′ ⏐ v ∈ S with v = Gv. It follows from(4.29),(4.39),(4.40),(4.41) L ≤ max { ρ (x) : x ∈ [0, 1]} is the Lipschitz constant for ρ ρ0 ⏐ 0 ⏐ 0 and the compatibility conditions ρ exp(b(0)) = ρ (0), b˙(0) = ⏐ ⏐ ′ s 0 { db } ( ) ρ (0) and L ≤ max ⏐ ˜(t)⏐ : t ∈ [0, t + 1] is the Lipschitz constant ∫ 1 0 1 ˜b dt 1 −λ ρ0(x)dx that ρv ∈ C ([0, T ] × [0, 1]; (0, +∞)) is ⏐ ⏐ 0 ρ0(0) for ˜b on [0, t1 + 1]. a solution of the initial–boundary value problem(4.27),(4.29), Let two arbitrary functions v, w ∈ S be given. Using(4.32), (4.30),(4.31) with t1 replaced by T , which satisfies estimate(4.36) (4.39) we obtain for all t ∈ [0, T ], x ∈ [0, 1]. Uniqueness follows from the fact that any solution of the ∥Gv − Gw∥∞ ≤ Lλ max { |ρv(t, x) − ρw(t, x)| :(t, x) ∈ [0, T ] × [0, 1]} initial–boundary value problem(4.27),(4.29),(4.30),(4.31) with (4.42) t1 replaced by T , necessarily gives a function v ∈ S with v = Gv. The proof is complete. ◁ When (t, x) ∈ Ωv and (t, x) ∈ Ωw, definition(4.41) implies t t (t,x;w) that∫ (w(s) − v(s))ds = ∫ 0 v(s)ds. Consequently, we We are now ready to give the proof of Theorem 3.1. t0(t,x;w) t0(t,x;v) obtain T ∥v − w∥∞ ≥ vmin |t0(t, x; w) − t0(t, x; v)| which in conjunction with(4.40) gives Proof of Theorem 3.1. Every solution of the initial–boundary ⏐ ⏐ value problem(3.1),(3.2),(3.3),(3.4),(3.5) is a solution of the |ρv(t, x) − ρw(t, x)| = ⏐˜b (t0(t, x; v)) −˜b (t0(t, x; w))⏐ initial–boundary value problem(2.1),(2.2),(2.3) with v(t, x) = T λ(W (t)) for t ≥ 0, x ∈ [0, 1]. Estimates(2.4),(2.5) in conjunction ≤ ∥ − ∥ L˜b v w ∞ (4.43) vmin with estimate(4.28) imply estimates(3.6),(3.7). The proof is complete. ◁ When (t, x) ∈/ Ωv and (t, x) ∈/ Ωw, definition(4.40) implies that

⏐ ( ∫ t ) ( )⏐ 5. Concluding remarks | − | = ⏐ − − − ∫ t ⏐ ρv (t, x) ρw (t, x) ⏐ρ0 x v(s)ds ρ0 x 0 w(s)ds ⏐ ⏐ 0 ⏐ ⏐∫ t ∫ t ⏐ The present work provided a thorough study of stability of ≤ L ⏐ (s)ds − (s)ds⏐ ≤ L t ∥ − ∥ ≤ L T ∥ − ∥ ρ0 ⏐ v v ⏐ ρ0 v w ∞ ρ0 v w ∞ the 1-D continuity equation, which appears in many conservation ⏐ 0 0 ⏐ laws. We have considered the velocity to be a distributed input (4.44) and we have also considered boundary disturbances. Stability ∈ ∈ ∫ t ≥ estimates are provided in all Lp state norms with p > 1, including When (t, x) / Ωv and (t, x) Ωw, i.e., when 0 w(s)ds x > t the case p = +∞ (sup norm). However, in our Input-to-State ∫ v(s)ds then we get 0 Stability estimates, the gain and overshoot coefficients depend on ∫ t the velocity. Moreover, the logarithmic norm of the state appears ∥ − ∥ ≥ − t v w ∞ x v(s)ds > 0 (4.45) instead of the usual norm. 0 As remarked in the Introduction, the obtained results can be ∫ t ≥ ∫ t Moreover, the inequality 0 w(s)ds x > 0 v(s)ds implies used in the stability analysis of larger models that contain the t t t t (t,x;w) that x−∫ v(s)ds > ∫ v(s)ds−∫ v(s)ds = ∫ 0 v(s) continuity equation. In the present paper, our results were used t0(t,x;w) 0 t0(t,x;w) 0 t t ds which combined with x = ∫ w(s)ds = ∫ (w(s) − in a straightforward way for the stability analysis of non-local, t0(t,x;w) t0(t,x;w) t nonlinear manufacturing models under feedback control. Work- v(s))ds + ∫ v(s)ds (a consequence of definition(4.41)) gives t0(t,x;w) ing similarly, the obtained results can be used for the stability t t (t,x;w) ∫ (w(s) − v(s))ds > ∫ 0 v(s)ds. The previous inequality analysis of non-local traffic models. t0(t,x;w) 0 I. Karafyllis and M. Krstic / Systems & Control Letters 135 (2020) 104594 9

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