Nonlinear bilateral output-feedback control for a class of viscous Hamilton¿Jacobi PDEs

Nonlinear Bilateral Output-Feedback Control for a Class of Viscous Hamilton-Jacobi PDEs Nikolaos Bekiaris-Liberis a,1and Rafael Vazquez b aDepartment of Production Engineering & Management, Technical University of Crete, Chania, 73100, Greece. bDepartment of Aerospace Engineering, University of Seville, Seville 41092, Spain. Abstract We tackle the boundary control and estimation problems for a class of viscous Hamilton-Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. First, we solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. Second, in order to guarantee trajectory tracking with an arbitrary decay rate, we construct nonlinear, full-state feedback laws employed at the two boundary ends. Third, a nonlinear observer is constructed, using measurements from both boundaries, which is combined with the full-state feedback designs into an observer-based output-feedback law. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation (which we introduce) with backstepping. Due to the fact that the linearizing transformation is locally invertible, only regional stability results are established, which are, nevertheless, accompanied with region of attraction estimates. Our stability proofs are based on the utilization of the linearizing transformation together with the employment of backstepping transformations, suitably formulated to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results. 1 Introduction 1.1 Motivation Contrary to linear parabolic Partial Differential Equations (PDEs), for which explicit boundary control and estimation designs are now largely available, see, for instance, [31], [35], in the nonlinear case, the design of explicit boundary control and estimation schemes is a more challenging problem. In addition, specific engineering applications, such as, for example, vehicular traffic [26], [46], plasma systems [10], [11], fluids [6], [12], chemical reactors [35], heat exchangers [35], and litium-ion batteries [44], [45], to name only a few, call for the development of systematic control and estimation design methodologies that, besides being able to efficiently exploit the capabilities of the available actuators and sensors, they can also be made fault tolerant. Motivated by scalar, conservation law models for vehicular traffic flow that include a viscous term, in order to account for drivers’ look-ahead ability [26], [46], we consider the problems of boundary control and estimation of a certain class of viscous Hamilton-Jacobi (HJ) PDEs, which constitutes an alternative macroscopic description of traffic flow dynamics [14], Email addresses: [email protected] (Nikolaos Bekiaris-Liberis), [email protected] (Rafael Vazquez). 1Corresponding author. [39]. In particular, we consider the case in which actuation and sensing is available at both boundaries (which we refer to as “bilateral” in our control and estimation approaches), aiming at constructing control and estimation schemes capable of utilizing efficiently both the available actuators and the available measurements. 1.2 Literature Arguably, the most relevant results to the ones presented here are those dealing with the controller and observer designs for viscous Burgers-type PDEs, which may be viewed as conservation law counterparts of the class of viscous HJ PDEs with quadratic Hamiltonian considered here. The trajectory generation problem for certain forms of viscous Burgers equations is considered in [30], [37], [41], whereas full-state boundary feedback laws are designed in [18], [24], [28], [29], [33], [43]. Observers and output-feedback controllers are presented in [4], [5], [12], [25], [30]. Explicit boundary control and observer designs for other nonlinear parabolic PDEs also exist, see, e.g., [20], [21], [36], [47], [48]. Although it is a different problem, for completeness, it should be mentioned that the control design problem of inviscid versions of Burgers or of specific HJ PDEs is considered in, e.g., [1], [8], [14], [28]. Bilateral controllers and observers for certain classes of linear parabolic and hyperbolic PDEs are recently developed in [2], [3], [50], [51]. We should also mention here that, in comparison to [7], in the Preprint submitted to Automatica 12 March 2018 arXiv:1803.03488v1 [math.OC] 9 Mar 2018 present paper we consider, 1) a more general class of viscous HJ PDE systems, 2) the problems of trajectory generation and tracking, and 3) the problems of bilateral control and estimation. 1.3 Results Our contributions are summarized as follows. First, we solve the nonlinear trajectory generation problem for the considered viscous HJ PDE, providing explicit feedforward actions at both boundaries. The key ingredient in our approach is the employment of a feedback linearizing transformation (inspired by the Hopf-Cole transformation [15], [23]) that we introduce, which allows us to convert the original nonlinear problem to a motion planning problem for a linear heat equation. We then establish the well-posedness of the feedforward controllers for the original nonlinear PDE system, for reference outputs that belong to Gevrey class (of certain order) with sufficiently small magnitude. Second, we design full-state feedback laws in order to achieve trajectory tracking, with an arbitrary decay rate, as the system is not, in general, asymptotically stable around a given reference trajectory. Modifying, in a suitable way, the introduced feedback linearizing transformation we recast the original nonlinear control problem to a problem of fullstate feedback stabilization of a linear heat equation, with Neumann actuation at each of the two boundaries. The bilateral boundary controllers are designed using the recently introduced backstepping technique [50]. We then establish local asymptotic stability of the closed-loop system in H1 norm, employing a Lyapunov functional and we provide an estimate of the region of attraction of the controller. Our stability result is local in H1norm due to the fact that the linearizing transformation is invertible only locally and, in particularly, the size of the supremum norm of the transformed PDE state should be appropriately restricted. Third, we turn our attention to the observer-based outputfeedback trajectory tracking problem. We design a nonlinear observer, employing boundary measurements from both ends of the spatial domain. The observer design is based on the introduced linearizing transformation and on a suitable formulation of the backstepping methodology in [51] to the case of a one-dimensional spatial domain. We then show that the bilateral, observer-based output-feedback controller achieves local asymptotic stabilization of the reference trajectory in H1norm. Finally, we apply the developed methodologies to a model of highway traffic flow. We illustrate, in simulation, the effectiveness of the proposed control design technique, including also a comparison with the unilateral case (i.e., the case in which a full-state feedback controller is applied only at the one boundary). In general, less control effort is required in the bilateral case, fact that may be useful in actual implementations. 1.4 Organization We start presenting the class of viscous HJ PDEs under consideration and introducing the feedback linearizing transformation in Section 2. We then continue in a way such that a reader interested only in the designs could skip the details of the proofs. Specifically, in Section 3 we present the nonlinear feedforward control designs. In Section 4 we present the nonlinear, full-state feedback controllers and in Section 5 we prove local asymptotic stability of the closed-loop system. In Section 6 we present the nonlinear observer design and in Section 7 we prove stability of the closed-loop system under the observer-based output-feedback laws. We present an example of traffic flow control in Section 8. Concluding remarks and directions of future research are provided in Section 9. 1.5 Notation and Definitions We use the common definition of class K,K∞and KL functions from [27]. For a function u∈L2(0,1) we denote by ku(t)kL2the norm ku(t)kL2=qR1 0u(x, t)2dx. For u∈H1(0,1) we denote by ku(t)kH1the norm ku(t)kH1=qR1 0u(x, t)2dx +qR1 0ux(x, t)2dx. We denote by Cj(A)the space of functions that have continuous derivatives of order jon A. We denote an initial condition as u0(x) = u(x, t0)with some t0≥0, for all x∈[0,1]. With C[t0,∞); H2(0,1)we denote the class of continuous mappings on [t0,∞)with values into H2(0,1). We denote by C2,1 T([0,1] ×(t0, T)) the space of functions that have continuous spatial derivatives of order 2and continuous time derivatives of order 1on [0,1] ×(t0, T), and define C2,1 ∞=C2,1. Definition 1 The function f(t)belongs to GF,M,γ (S), the Gevrey class of order γin S, if f(t)∈C∞(S)and there exist positive constants F,Msuch that supt∈Sf(n)(t)≤ FMn(n!)γ, for all n= 0,1,2, . . .. 2 Problem Formulation and Feedback Linearization We consider the following viscous HJ PDE system ut(x, t) = uxx(x, t)−aux(x, t) (b+ux(x, t)) (1) ux(0, t) = U0(t)(2) ux(1, t) = U1(t),(3) where uis the PDE state, x∈[0,1] is the spatial variable, t≥t0≥0is time,  > 0is a viscosity coefficient, a6= 0 and b∈Rare constant parameters, and U0,U1are control variables. We introduce next a feedback linearizing transformation, which allows us to convert the problems of trajectory generation and tracking for the nonlinear HJ PDE (1)–(3) to the corresponding problems for a linear diffusion-advection PDE. 2 The following locally invertible transformation ¯v(x, t) = e−a u(x,t)−1,(4) and the control laws U0(t) = − aea u(0,t)¯ V0(t)(5) U1(t) = − aea u(1,t)¯ V1(t),(6) where ¯ V0,¯ V1are the new control variables yet to be chosen, transform system (1)–(3) to ¯vt(x, t) = ¯vxx(x, t)−ab¯vx(x, t)(7) ¯vx(0, t) = ¯ V0(t)(8) ¯vx(1, t) = ¯ V1(t).(9) It turns out that in the control design and analysis it is more convenient to perform an additional transformation, namely v(x, t) = ¯v(x, t)e−ab 2x,(10) in order to re-write (7)–(9) as vt(x, t) = vxx(x, t)−a2b2 4v(x, t)(11) vx(0, t) = V0(t)(12) vx(1, t) = V1(t),(13) where ¯ V0(t) = V0(t) + ab 2¯v(0, t)(14) ¯ V1(t) = eab 2V1(t) + ab 2¯v(1, t),(15) and V0,V1are the new control variables. 3 Trajectory Generation In this section we design the feedforward boundary control laws that generate the desired reference outputs. We solve the problem first for the linearized system (11)–(13) and we then provide the feedforward actions for the original system (1)–(3). We consider as outputs of the system the values u(x0, t)and ux(x0, t), where x0is some fixed point within the interval [0,1]. Theorem 1 Let yr 1(t)and yr 2(t)be in GF,M,γ ([0,+∞)) class with 1≤γ < 2. There exists a positive constant µ1 such that if F≤µ1then the functions ur(x, t) = − aln eab 2xvr(x, t)+1(16) Ur 0(t) = − a vr x(0, t) + ab 2vr(0, t) 1 + vr(0, t)(17) Ur 1(t) = −eab 2 a vr x(1, t) + ab 2vr(1, t) 1 + eab 2vr(1, t),(18) where vr(x, t) = ∞ X k=0 1 k (x−x0)2k (2k)! k X m=0 k ma2b2 4k−m ×yr 1,v (m)(t) + ∞ X k=0 1 k (x−x0)2k+1 (2k+ 1)! × k X m=0 k ma2b2 4k−m yr 2,v (m)(t)(19) yr 1,v(t) = e−ab 2x0e−a yr 1(t)−1(20) yr 2,v(t) = e−ab 2x0−a e−a yr 1(t)yr 2(t) −ab 2e−a yr 1(t)−1,(21) satisfy the boundary value problem (1)–(3) and, in particular, ur(x0, t) = yr 1(t)and ur x(x0, t) = yr 2(t). Proof Via transformations (4) and (10), in order to generate the desired trajectory ur(x, t)and to provide the feedforward laws Ur 0(t),Ur 1(t), which achieve ur(x0, t) = yr 1(t)and ur x(x0, t) = yr 2(t), it is sufficient to generate vr(x, t)that satisfies (11) with vr(x0, t) = yr 1,v(t)(22) vr x(x0, t) = yr 2,v(t),(23) where yr 1,v(t)and yr 2,v(t)are defined in (20) and (21), respectively. The feedforward laws Ur 0(t),Ur 1(t)are then given combining (12), (13) with (14), (15) and (5), (6). Moreover, vr(x, t)should be restricted appropriately such that (16) and (17) are well-posed, which holds true whenever sup x∈[0,1] |vr(x, t)|<¯ce−|ab 2|,for all t≥t0,(24) for some constant ¯c∈(0,1), in addition to vr x(x, t)being bounded for all x∈[0,1] and t≥t0. Since system (11)–(13) is in the form of a linear diffusionadvection PDE we postulate the reference trajectory vrin the form, see, e.g., [34], [37], [38] vr(x, t) = ∞ X k=0 αk(t)(x0−x)k k!,(25) 3 where the functions αk(t),k= 0,1, . . . are yet to be determined in order for (25) to satisfy (11) as well as (22) and (23). Substituting (25) into (11) we arrive at the following recursive relation for α’s αk+2(t) = 1 ˙αk(t) + a2b2 4αk(t)(26) α0(t) = yr 1,v(t)(27) α1(t) = −yr 2,v(t),(28) and thus, (25) may be written as in (19), see, e.g., [34], [37], [38]. Employing the results in, for example, [37] (Remark 4), one can conclude that the series (19) is convergent (with an infinite radius of convergence) provided that yr 1,v(t)and yr 2,v(t)belong to GF∗ 1,M∗ 1,γ ([0,+∞)) for 1≤γ < 2, for some positive constants F∗ 1and M∗ 1. We derive next explicit Gevrey-type estimates for yr 1,v(t)and yr 2,v(t)as, in order to guarantee that condition (24) holds, one has to guarantee that, in addition to the functions yr 1,v(t)and yr 2,v(t)belonging to GF∗ 1,M∗ 1,γ ([0,+∞)) for 1≤γ < 2, that the constant F∗ 1may be chosen sufficiently small when Fis sufficiently small. Toward that end, from Lemmas A.1 and A.2 in Appendix A we obtain that sup t≥0yr 1,v (n)(t)≤¯ F1¯ Mn 1(n!)γ,for all n= 0,1, . . . (29) sup t≥0yr 2,v (n)(t)≤¯ F2¯ Mn 2(n!)γ,for all n= 0,1, . . ., (30) where ¯ F1=F|a| eF|a| e−ab 2x0(31) ¯ M1=MeF|a| (32) ¯ F2=F|a| eF|a| e−ab 2x0e−F|a| +|ab| 2+F|a| (33) ¯ M2=1 + F|a| eF|a| ¯ M1,(34) and hence, one can choose F∗ 1=F|a| eF|a| e−ab 2x0 ×max n1, e−F|a| +|ab| 2+F|a| oand M∗ 1=¯ M2. Consequently, series (19) is convergent. Moreover, combining (19) and (29), (30) we get that |vr(x, t)|≤ ¯ F1 ∞ X k=0 1 k(k!)γ−2a2b2 4+¯ M1k +¯ F2 ∞ X k=0 1 k(k!)γ−2a2b2 4+¯ M2k ,(35) where we used the fact that (k!)2≤(2k)!. For all x∈[0,1] the general term, say ζk, in the first series satisfies  ζk+1 ζk =1 a2b2 4+¯ M1(k+ 1)γ−2,(36) and thus, since γ < 2, we conclude that limk→∞  ζk+1 ζk= 0<1, which in turn implies, employing D’Alembert’s criterion, that the infinite sum converges to a positive number, say l1. Similarly, the second infinite sum converges to a positive number, say l2. Therefore, from (35) we arrive at |vr(x, t)|≤max ¯ F1,¯ F2(l1+l2), for all x∈[0,1] and t≥t0,(37) and hence, by choosing µ1such that max ¯ F1,¯ F2(l1+l2) <¯ce−|ab 2|, for some constant ¯c∈(0,1), which, according to relations (31), (33) is always possible (note that l1,l2are continuous functions of Fsince the two series in (35) converge uniformly and from (32), (34) it follows that ¯ M1,¯ M2 are continuous with respect to F), condition (24) is satisfied. It follows from (16) that ur(x, t)is uniformly bounded with respect to time and spatial variable. The uniform boundedness, with respect to time and spatial variable, of vr x(x, t), vr xx(x, t), and vr t(x, t), which, from (16) and (24), imply the uniform boundedness of ur x(x, t),ur xx(x, t), and ur t(x, t), follow by differentiating (19) and employing almost identical arguments (see also, e.g., Section 3 in [34]). 2 Example 1 Consider system (1)–(3) with a=−1,b= 0 and assume that the desired reference trajectories are yr 1(t) = 0and yr 2(t) = dsin(t), where d > 0. For sufficiently small dthe conditions of Theorem 1 are satisfied. The reference trajectory as well as the reference inputs are given by ur(x, t) = ln (1 + g1(x, t)) (38) g1(x, t) = d 2√e x−x0 √2sin t+x−x0 √2−π 4 −d 2√e x0−x √2sin t+x0−x √2−π 4(39) Ur 0(t) = d 2 e−x0 √2sin t−x0 √2 1 + g1(0, t) +d 2 e x0 √2sin t+x0 √2 1 + g1(0, t)(40) Ur 1(t) = d 2 e 1−x0 √2sin t+1−x0 √2 1 + g1(1, t) +d 2 e x0−1 √2sin t+x0−1 √2 1 + g1(1, t),(41) where we also used the fact that sin (y)−cos (y) = √2sin y−π 4, for any y∈R. In Fig. 1 we show the generated trajectory uras well as its spatial derivative ur x. 4 Fig. 1. Functions (38)–(41) that solve the nonlinear trajectory generation problem for system (1)–(3) with a=−1,b= 0, and = 0.5, with reference trajectories yr 1(t) = 0 and yr 2(t) = 0.25sin(t), for x0=1 2. To see that the functions defined in (38)–(41) solve the nonlinear trajectory generation problem note first that relation (19) in the present case becomes vr(x, t) = d  ∞ X k=0 1 k (x−x0)2k+1 (2k+ 1)! sin(k)(t).(42) Using the facts that sin(t) = 1 2jejt −e−jt,j= 1 2(1 + j)2, and −j=1 2(1 −j)2we obtain vr(x, t) = d 2j ∞ X k=0 1 k (x−x0)2k+1 (2k+ 1)! 1 2k(1 + j)2kejt −1 2k(1 −j)2ke−jt,(43) and hence, employing the power series expansion for the hyperbolic sine we arrive at vr(x, t) = d √2j 1 1 + jejtsinh (x−x0) (1 + j) √2 −1 1−je−jtsinh (x−x0) (1 −j) √2.(44) Performing some tedious algebraic manipulations we get vr(x, t) = d 2√e x−x0 √2sin t+x−x0 √2−π 4 −d 2√e x0−x √2sin t+x0−x √2−π 4.(45) Equations (38)–(41) are then derived combining (45) with (16)–(18). 4 Bilateral Full-State Feedback Boundary Control Design Having available the reference trajectory for system (1)–(3), in this section, we design the boundary feedback laws that stabilize the desired reference trajectory for any initial condition. We start deriving the dynamics of the error between the actual and the reference states. We then introduce a feedback linearizing transformation for the tracking error’s dynamics, which, in turn, enables us to design full-state feedback, boundary control laws utilizing infinite-dimensional backstepping for linear systems. 4.1 Tracking error dynamics and motivation for control We define the error variables ˜u(x, t) = u(x, t)−ur(x, t)(46) ˜ U0(t) = U0(t)−Ur 0(t)(47) ˜ U1(t) = U1(t)−Ur 1(t).(48) Differentiating (46) with respect to tand x, using the fact that ur(x, t)satisfies system (1)–(3) we get that ˜usatisfies the following system ˜ut(x, t) = ˜uxx(x, t)−a˜ux(x, t) (b+ ˜ux(x, t)) −2aur x(x, t)˜ux(x, t)(49) ˜ux(0, t) = ˜ U0(t)(50) ˜ux(1, t) = ˜ U1(t).(51) A feedback control design is needed to asymptotically stabilize the origin of (49)–(51). To see this note that the zero solution of (49)–(51) is not asymptotically stable since any constant could be an equilibrium of (49)–(51). 5 4.2 Feedback linearizing transformation for the tracking error dynamics Guided from the feedback linearizing transformation (4) we define ˜ ¯v(x, t) = e−a ˜u(x,t)−1,(52) which it is readily shown that satisfies the following PDE ˜ ¯vt(x, t) = ˜ ¯vxx(x, t)−a(b+ 2ur x(x, t)) ˜ ¯vx(x, t)(53) ˜ ¯vx(0, t) = ˜ ¯ V0(t)(54) ˜ ¯vx(1, t) = ˜ ¯ V1(t),(55) where we choose ˜ U0(t) = − aea ˜u(0,t)˜ ¯ V0(t)(56) ˜ U1(t) = − aea ˜u(1,t)˜ ¯ V1(t),(57) and ˜ ¯ V0(t),˜ ¯ V1(t)are new control variables. With the additional transformation (see also Fig. 2) ˜v(x, t) = ˜ ¯v(x, t)e−ab 2x−a ur(x,t),(58) and selecting the control variables ˜ ¯ V0(t),˜ ¯ V1(t)as ˜ ¯ V0(t) = ea ur(0,t)˜ V0(t) +a b 2+ur x(0, t)e−a ˜u(0,t)−1(59) ˜ ¯ V1(t) = eab 2+a ur(1,t)˜ V1(t) +a b 2+ur x(1, t)e−a ˜u(1,t)−1,(60) we arrive at the following system ˜vt(x, t) = ˜vxx(x, t)−a2b2 4˜v(x, t)(61) ˜vx(0, t) = ˜ V0(t)(62) ˜vx(1, t) = ˜ V1(t),(63) where the control variables ˜ V0(t)and ˜ V1(t)are chosen later on (in Section 4.3) via the backstepping methodology. Note that system (61)–(63), besides being linear, does not incorporate any spatially- or time-dependent terms, which may be the case when considering trajectory tracking problems for nonlinear systems. This is possible here because the overall feedback linearizing transformation (58) may be expressed as the difference of two nonlinear functions of u and ur, which both satisfy the linear PDE (11) (or, equivalently, (61)) since both uand ursatisfy (1). Moreover, relations (62), (63) are derived differentiating (58) with respect to xand using (54), (55) as well as defining the new control inputs ˜ V0,˜ V1according to (59), (60). e−a (·)−1e−ab 2x−a ur(x,t) eab 2x+a ur(x,t) − alog (·+ 1) K{·} L{·} ˜u˜ ¯v˜v w Fig. 2. The interconnections between ˜u,˜ ¯v,˜v, and winvolved in transformations (52), (58), and (79). The operators K{·} and L{·} are defined as K{˜v}(x) = ˜v(x)−Rx −x+1 k(x, y)˜v(y)dy and L{w}(x) = w(x) + Rx −x+1 l(x, y)w(y)dy respectively. The leftmost transformation (i.e., transformation (52)) is only locally invertible. 4.3 Bilateral boundary control design Exploiting the fact that the ˜vvariable satisfies the linear diffusion-advection PDE (61)–(63) we design the boundary feedback laws as [50] ˜ V0(t) = k(0,0) ˜v(0, t)−Z1 0 kx(0, ξ) ˜v(ξ, t)dξ (64) ˜ V1(t) = k(1,1) ˜v(1, t) + Z1 0 kx(1, ξ) ˜v(ξ, t)dξ, (65) where the kernel k(x, ξ)is given explicitly, for (x, ξ)in the domain D=D1∪D2, where D1=(x, ξ) : 1 2≤x≤1, −x+ 1 ≤ξ≤x}and D2=(x, ξ) : 0 ≤x≤1 2, x≤ξ≤1−x}, by k(x, ξ) = −1 2rc1  I1rc1 x−1 22−ξ−1 22 qx−1 22−ξ−1 22 ×(x+ξ−1) ,(66) with I1denoting the modified Bessel function of the first kind of first order. Combining (56), (57) and (59), (60) with (64), (65) the boundary feedback laws in the original variables are written via (52), (58) as U0(t) = − aea ˜u(0,t)k(0,0) + ab 2e−a ˜u(0,t)−1 −ea ur(0,t)Z1 0 kx(0, ξ)e−ab 2ξ−a ur(ξ,t) ×e−a ˜u(ξ,t)−1dξ+Ur 0(t)ea ˜u(0,t)(67) U1(t) = − aea ˜u(1,t)k(1,1) + ab 2e−a ˜u(1,t)−1 +eab 2+a ur(1,t)Z1 0 kx(1, ξ)e−ab 2ξ−a ur(ξ,t) ×e−a ˜u(ξ,t)−1dξ+Ur 1(t)ea ˜u(1,t),(68) 6 where Ur 0(t)and Ur 1(t)are defined in (17) and (18), respectively, with the error variable ˜ubeing defined in (46) and the reference trajectory urbeing defined in (16). 5 Trajectory Tracking Under Full-State Feedback In order to show asymptotic stability of the closed-loop system, under the full-state feedback laws, in the original variable ˜uwe have to ensure that the linearizing transformation (52) is invertible. The inverse of transformation (52) is given by (see also Fig. 2) ˜u(x, t) = − aln (˜ ¯v(x, t) + 1) ,(69) which is well-defined when the initial conditions and solutions of the system satisfy for some c∈(0,1] sup x∈[0,1] |˜ ¯v(x, t)|< c, for all t≥t0.(70) Due to the feasibility condition (70), only a local stability result can be obtained, which is stated next. Theorem 2 Consider a closed-loop system consisting of the plant (1)–(3) and the control laws (67), (68). Under the conditions of Theorem 1 for the reference outputs, there exist a positive constant µand a class KLfunction βsuch that for all initial conditions u0∈H2(0,1) which are compatible with the feedback laws (67), (68) and which satisfy k˜u(t0)kH1< µ, (71) the following holds k˜u(t)kH1≤β(k˜u(t0)kH1, t −t0),for all t≥t0.(72) Moreover, the closed-loop system has a unique solution u∈ C[t0,∞); H2(0,1)with u∈C2,1([0,1] ×(t0,∞)). The proof of Theorem 2 is based on the following three lemmas whose proofs can be found in Appendix B. Note that the compatibility conditions in the statement of Theorem 2 are the following u0 0(0) = − aea (u0(0)−ur(0,t0)) k(0,0) + ab 2 ×e−a (u0(0)−ur(0,t0)) −1−ea ur(0,t0) ×Z1 0 kx(0, ξ)e−ab 2ξe−a u0(ξ)−e−a ur(ξ,t0) ×dξ+ur x(0, t0)ea (u0(0)−ur(0,t0)) (73) u0 0(1) = − aea (u0(1)−ur(1,t0)) k(1,1) + ab 2 ×e−a (u0(1)−ur(1,t0)) −1+eab 2+a ur(1,t0) ×Z1 0 kx(1, ξ)e−ab 2ξ−a ur(ξ,t0) ×e−a (u0(ξ)−ur(ξ,t0)) −1dξ +ur x(1, t0)ea (u0(1)−ur(1,t0)).(74) Lemma 1 There exists a class K∞function α1such that if ˜u∈H1(0,1) then ˜ ¯v∈H1(0,1) and the following holds k˜ ¯v(t)kH1≤α1(k˜u(t)kH1).(75) Lemma 2 For all solutions of the system that satisfy (70) for some 0< c < 1, if ˜ ¯v∈H1(0,1) then ˜u∈H1(0,1) and the following holds k˜u(t)kH1≤ |a|(1 −c)k˜ ¯v(t)kH1.(76) Lemma 3 Under the conditions of Theorem 1 for the reference outputs, if ˜ ¯v∈H1(0,1) then ˜v∈H1(0,1) and there exists a positive constant ξ1such that the following holds k˜v(t)kH1≤ξ1k˜ ¯v(t)kH1.(77) In reverse, if ˜v∈H1(0,1) then ˜ ¯v∈H1(0,1) and there exists a positive constant ξ2such that the following holds k˜ ¯v(t)kH1≤ξ2k˜v(t)kH1.(78) Proof of Theorem 2 We start by considering the following backstepping transformation, which is introduced in [50] (see also Fig. 2) w(x, t) = ˜v(x, t)−Zx −x+1 k(x, ξ) ˜v(ξ, t)dξ, (79) where kis defined in (66). Transformation (79), together with the control laws (64), (65), map system (61)–(63) to [50] 2 wt(x, t) = wxx(x, t)−a2b2 4+c1w(x, t)(80) wx(0, t)=0 (81) wx(1, t)=0,(82) where c1>0is arbitrary. The backstepping transformation (79) is invertible with inverse that may be expressed as ˜v(x, t) = w(x, t) + Zx −x+1 l(x, ξ)w(ξ, t)dξ, (83) 2In order to become clear how the results from [50] are employed we provide additional details in Appendix C. 7 which follows specializing the results in [51] to the present case 3(see also the discussion in [50]), where 4 l(x, ξ) = −1 2rc1  J1rc1 x−1 22−ξ−1 22 qx−1 22−ξ−1 22 ×(x+ξ−1) ,(84) with J1being the first-order Bessel function of the first kind. Having defined the backstepping transformation and its inverse it is shown, specializing the results in [51] (Section 6.3), that there exist positive constants m1and m2such that kw(t)kH1≤m1k˜v(t)kH1(85) k˜v(t)kH1≤m2kw(t)kH1.(86) Defining the Lyapunov functional S1(t) = 1 2Z1 0 w(x, t)2dx +1 2Z1 0 wx(x, t)2dx, (87) we get along the solutions of the “target” system (80)–(82) that ˙ S1(t)≤ −2c1+a2b2 4S1(t), where we took the L2- inner product of (80) with w,wxx and performed one step of integration by parts. Using (85), (86), we get for all t≥t0 k˜v(t)kH1≤m1m2√2k˜v(t0)kH1e−c1+a2b2 4(t−t0),(88) and hence, from Lemma 3, we conclude that there exists a constant ν4such that for all t≥t0 k˜ ¯v(t)kH1≤ν4k˜ ¯v(t0)kH1e−c1+a2b2 4(t−t0).(89) From Lemma 1, estimate (89) implies that one can choose µ in (71) sufficiently small, in fact, such that µ≤α−1 1c 2ν4, in order for relation k˜ ¯v(t)kH1<c 2, for some 0<c<1, to hold for all t≥t0. Hence, since supx∈[0,1] |˜ ¯v(x, t)| ≤ 2k˜ ¯v(t)kH1, for any ˜v∈H1(0,1), we conclude that condition (70), for some 0<c<1, is satisfied. Estimate (72) is then obtained, employing Lemma 2 and combining estimate (89) with estimates (75), (76), with β(s, t −t0) = ν4 |a|(1−c)α1(s)e−c1+a2b2 4(t−t0). We study next the well-posedness of the closed-loop system. We start with the target system (80)–(82). Since from 3To see this, note that a one-dimensional ball is, in fact, an interval and, its boundary, i.e., a “zero-sphere” just consists of the two endpoints of the interval. 4In terms of the z=x−1 2and y=ξ−1 2vari- ables the inverse backstepping transformation (83) can be written as ˜v1(z, t) = w1(z, t) + Rz −zL(z, y)w1(y, t)dy, where w1(z, t) = wz+1 2, t,˜v1(z, t) = ˜vz+1 2, t, and L(z, y) = lz+1 2, y +1 2. (66) we get that k∈C2(D), from transformation (83) it follows, using the fact that ˜v0∈H2(0,1) (which follows, in a similar way to the derivation of estimates (75), (77), from (52), (58) exploiting the regularity assumption on u0and the regularity properties of the reference trajectory ur(t0)in Theorem 1) and the compatibility conditions, that w0∈H2(0,1) satisfies the compatibility conditions w0 0(0) = w0 0(1) = 0. Therefore, from (80)–(82) it is shown, see, for example, [9], that there exists a unique w∈C[t0,∞); H2(0,1). The inverse transformation (83) and the fact that l∈C2(D)(which follows from expression (84)) guarantee the existence and uniqueness of ˜v∈C[t0,∞); H2(0,1). Using (52), (58) it follows that ˜u(x, t) = − alog ˜v(x, t)eab 2+a ur(x,t)+ 1, and hence, in a similar way to the derivation of estimates (76), (78), the regularity properties of urand condition (70) guarantee the existence and uniqueness of u∈C[t0,∞); H2(0,1). Employing similar arguments, with [9] (see also, e.g., [32]) it is shown that u∈C2,1([0,1] ×(t0,∞)).2 6 Nonlinear Observer and Output Feedback Law Designs In this section, we design a nonlinear observer to estimate the state ˜u(x),x∈[0,1], which may be employed in the full-state feedback laws (67), (68) giving rise to an observerbased output-feedback design or, it may be utilized independently when the goal is only state estimation. The observer utilizes measurements from both ends of the spatial domain. Furthermore, we also present static, collocated output-feedback controllers, which, however, cannot achieve an arbitrary decay rate. 6.1 Observer design Exploiting the convenient form of system (61)–(63) we introduce the following observer ˆ ˜vt(x, t) = ˆ ˜vxx(x, t)−a2b2 4ˆ ˜v(x, t) + p2(x) ×e−a ˜u(0,t)−1e−a ur(0,t)−ˆ ˜v(0, t) +p1(x)e−a ˜u(1,t)−1 ×e−ab 2−a ur(1,t)−ˆ ˜v(1, t)(90) ˆ ˜vx(0, t) = ˜ V0(t) + p00 e−a ˜u(0,t)−1e−a ur(0,t) −ˆ ˜v(0, t)(91) ˆ ˜vx(1, t) = ˜ V1(t) + p11 e−a ˜u(1,t)−1e−ab 2−a ur(1,t) −ˆ ˜v(1, t).(92) The gains p2(x),p1(x),p00, and p11 are designed via the backstepping methodology, specializing the results from 8 [51] to a one-dimensional spatial domain, as p2(x) = −Pξ(x, 0) (93) p1(x) = −Pξ(x, 1) (94) p00 =−P(0,0) (95) p11 =−P(1,1) ,(96) where the kernel Pis given explicitly, for (x, ξ)in the domain E=E1∪E2, where E1=(x, ξ) : 1 2≤ξ≤1, −ξ+ 1 ≤x≤ξ}and E2=(x, ξ):0≤ξ≤1 2, ξ≤x≤1−ξ}, by P(x, ξ) = −1 2rc2  I1rc2 ξ−1 22−x−1 22 qξ−1 22−x−1 22 ×(ξ+x−1) ,(97) where I1denotes the modified Bessel function of the first kind of first order and c2>0is arbitrary. Note that observer (90)–(92) is a copy of the (linear) system (61)–(63) plus output injection, where the output-injection terms are linear in the state ˜v, which can be seen using relations (52), (58) for x= 0 and x= 1. 6.2 Observer-based output feedback boundary control design In order to employ the full-state feedback laws (67), (68), which may achieve an arbitrary decay rate for the closedloop system, utilizing only boundary measurements, we first modify the control laws (64), (65) as ˜ V0(t) = k(0,0) ˜v(0, t)−Z1 0 kx(0, ξ)ˆ ˜v(ξ, t)dξ (98) ˜ V1(t) = k(1,1) ˜v(1, t) + Z1 0 kx(1, ξ)ˆ ˜v(ξ, t)dξ, (99) and hence, the control laws (67), (68) now become U0(t) = − aea ˜u(0,t)k(0,0) + ab 2e−a ˜u(0,t)−1 −ea ur(0,t)Z1 0 kx(0, ξ)ˆ ˜v(ξ, t)dξ +Ur 0(t)ea ˜u(0,t)(100) U1(t) = − aea ˜u(1,t)k(1,1) + ab 2e−a ˜u(1,t)−1 +eab 2+a ur(1,t)Z1 0 kx(1, ξ)ˆ ˜v(ξ, t)dξ +Ur 1(t)ea ˜u(1,t).(101) 6.3 Static collocated output-feedback controllers Provided that b6= 0, the zero solution of system (61)–(63) is asymptotically stable when ˜ V0(t) = ˜ V1(t) = 0 for all t≥t0(and hence, so is the zero solution of system (53)– (55) provided that uris uniformly bounded), i.e., when U0(t) = b 2ea ˜u(0,t)−1+Ur 0(t)ea ˜u(0,t)(102) U1(t) = b 2ea ˜u(1,t)−1+Ur 1(t)ea ˜u(1,t),(103) which may be viewed as decentralized (in the sense that each controller requires measurements of the state at the same boundary), static output-feedback control laws. However, the convergence rate to the zero equilibrium of the closed-loop solution of system (61)–(63) under the control laws (102), (103) is not arbitrary (in contrast to the achievable decay rate under the full-state feedback laws (67), (68), which is arbitrary), but depends on the parameters of the system, namely a,b, and . 7 Trajectory Tracking Under Observer-Based Output Feedback We next state and prove the following stability result for the closed-loop system, under the observer-based output feedback law. Theorem 3 Consider a closed-loop system consisting of system (1)–(3), the control laws (100), (101), and the observer (90)–(92) with (98), (99). Under the conditions of Theorem 1 for the reference outputs, there exist a positive constant µ∗and a class KL function β∗such that for all initial conditions u0,ˆ ˜v0∈H2(0,1) ×H2(0,1) which are compatible with the control laws (98)–(101) and which satisfy k˜u(t0)kH1+kˆ ˜v(t0)kH1< µ∗,(104) the following holds Ω(t)≤β∗(Ω (t0), t −t0),for all t≥t0(105) Ω(t) = k˜u(t)kH1+kˆ ˜v(t)kH1.(106) Moreover, the closed-loop system has a unique solution u, ˆ ˜v∈C[t0,∞); H2(0,1)with u, ˆ ˜v∈C2,1([0,1] ×(t0,∞)). Proof of Theorem 3 The proof is divided into three parts. Part 1:Backstepping transformation of the state estimation error We start defining the state estimation error e= ˜v−ˆ ˜v. (107) 9 with Kbeing defined for all −z≤y≤zwhen 0≤z≤1 2 and for all z≤y≤ −zwhen −1 2≤z≤0, where kis given in (66). Noting that the ˜v1variable satisfies the same exact PDE system (61)–(63) as the ˜vvariable (with the only difference that the boundary conditions for ˜v1are taken for z=−1 2and z=1 2, which correspond to x= 0 and x= 1, respectively) and that the kernel K in (C.2) is the gain kernel introduced in [50], we conclude (see [50]) that the w1variable in (C.1) satisfies the PDE w1t(z, t) = w1zz (z, t)−a2b2 4+c1w1(z, t). The boundary condition w1z−1 2, t= 0 is obtained differentiating (C.1) with respect to zand using (64). The boundary condition w1z1 2, t= 0 is obtained analogously. Appendix D Derivation of (108)–(110) from (114)–(116) via (111), (112) Using relation (97), the kernel pin (113) can be written as p(z, y) = −1 2rc2  I1pc2 (y2−z2) py2−z2(y+z),(D.1) which is defined in the domain ¯ E=¯ E1∪¯ E2, where ¯ E1=(z, y) : 0 ≤y≤1 2,−y≤z≤yand ¯ E2= (z, y) : −1 2≤y≤0, y ≤z≤ −y. Function p(z, y)satisfies the following problem in ¯ E1∪¯ E2 pyy (z, y)−pzz (z, y) = c2p(z, y)(D.2) p(z, z) = −c2 2z(D.3) p(z, −z)=0,(D.4) which can be either seen by direct computation (using the properties of Bessel functions, see, e.g., [31]) or by using relations (75)–(77) from [50] and noting that p(z, y)satisfies relation (78) from [50] with x→yand ξ→z. Using (111), we next provide few details on the derivation of (108), (110) from (114), (116) (one can then derive (108), (109) from (114), (115) via (112), employing identical arguments). In fact, the computations follow the standard, but tedious, algebraic manipulations involved in the backstepping methodology when deriving a target system for a specific case. In our case, differentiating (111) with respect to t and x, and using equations (114)–(116) as well as equations (D.2)–(D.4) we arrive at ¯et(z, t)−¯ezz(z, t) + a2b2 4¯e(z, t) +¯p2(z)¯e−1 2, t+ ¯p1(z)¯e1 2, t= ¯p2(z)¯e−1 2, t+ ¯p1(z)¯e1 2, t +pyz, 1 2¯w1 2, t+pyz, −1 2¯w−1 2, t.(D.5) Therefore, using (111) and (112) for z=1 2and z= −1 2, respectively, one can conclude that the right-hand side of (D.5) is zero when ¯p2(z) = −pyz, −1 2 and ¯p1(z) = −pyz, 1 2. In the shifted variables x=z+1 2and ξ=y+1 2, it follows from (97), (D.1) that P(x, ξ) = Pz+1 2, y +1 2=p(z, y), and hence, p2(x) = −Pξ(x, 0) and p1(x) = −Pξ(x, 1), which are the gains defined in (93) and (94), respectively. Finally, differentiating (111) with respect to z and setting z=1 2, we get employing (116), (D.4) that ¯ez1 2, t=p1 2,1 2¯w1 2, t=p1 2,1 2¯e1 2, t, and hence, relation (110) is recovered utilizing the fact that −p11 =P(1,1) = p1 2,1 2, which follows from (96). One can then derive relations (108), (109) from (114), (115) via (112), employing identical arguments. Acknowledgments Nikolaos Bekiaris-Liberis was supported by the funding from the European Commission’s Horizon 2020 research and innovation programme under the Marie Sklodowska- Curie grant agreement No. 747898, project PADECOT. Rafael Vazquez acknowledges financial support of the Spanish Ministerio de Economia y Competitividad under grant MTM2015-65608-P. References [1] Aubin, J.-P., Bayen, A., Saint-Pierre, P. (2008). Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints. SIAM Journal on Control and Optimization, 47, 2348–2380. [2] Auriol, J. and Di Meglio, F. (2016). Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time. IEEE Conference on Decision and Control, Las Vegas, NV. [3] Auriol, J. and Di Meglio, F. (2018). Two sided boundary stabilization of heterodirectional linear coupled hyperbolic PDEs. IEEE Transactions on Automatic Control, in press. [4] Balogh, A. and Krstic, M. (2000). Burgers equation with nonlinear boundary feedback: H1stability, well-posedness, and simulation. Mathematical Problems in Engineering, 6, 189–200. [5] Balogh, A. and Krstic, M. (2000). Boundary control of the Korteweg– de Vries–Burgers equation: Further results on stabilization and numerical demonstration. IEEE Transactions on Automatic Control, 45, 1739–1745. [6] Balogh, A., Aamo, O.-M., and Krstic, M. (2005). Optimal mixing enhancement in 3D pipe flow. IEEE Transactions on Control Systems Technology, 13, 27–41. [7] Bekiaris-Liberis, N. and Bayen, A. M. (2015). Nonlinear local stabilization of a viscous Hamilton-Jacobi PDE. IEEE Transactions on Automatic Control, 60, 1698–1703. [8] Blandin, S., Litrico, X., Delle Monache, M.-L., Piccoli, B., and Bayen, A. (2017). Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws. IEEE Transactions on Automatic Control, 62, 1620–1635. [9] Brezis, H. (2011). Functional analysis, Sobolev spaces and Partial Differential Equations. Springer, New York. 16 [10] Bribiesca Argomedo, F., Prieur, C., Witrant, E., Bremond, S. (2013). A strict control Lyapunov function for a diffusion equation with timevarying distributed coefficients. IEEE Transactions on Automatic Control, 58, 290–303. [11] Bribiesca Argomedo, F., Witrant, E., Prieur, C., Bremond, S., Nouailletas, R., and Artaud, J. F. (2013). Lyapunov-based distributed control of the safety factor profile in a tokamak plasma. Nuclear Fusion, 53, paper no. 033005. [12] Byrnes, C. I., Gilliam, D. S., and Shubov, V. I. (1998). On the global dynamics of a controlled viscous Burgers equation. Journal of Dynamical and Control Systems, 4, 457–519. [13] Carlson, R. C., Papamichail, I., Papageorgiou, M. (2014). Integrated feedback ramp metering and mainstream traffic flow control on motorways using variable speed limits. Transportation Research Part C, 46, 209–221. [14] Claudel, C. G. & Bayen, A. M. (2010). Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory. IEEE Transactions on Automatic Control, 55, 1142–1157. [15] Cole, J. D. (1951). On a quasilinear parabolic equation occurring in aerodynamics. Q. Appl. Math., 9, 225–236. [16] Deutscher, J. (2015). A backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica, 57, 56–64. [17] Deutscher, J. (2016). Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs. IEEE Transactions on Automatic Control, 61, 2288–2294. [18] Freudenthaler, G., Goettsch, F., Meurer, T. (2017). Backsteppingbased extended Luenberger observer design for a Burgers-type PDE for multi-agent deployment. IFAC World Congress, Toulouse, France. [19] Greenshields, B. D. (1935). A study of traffic capacity. Proceedings of the Highway Research Board, 14, 448–477. [20] Hasan, A. (2015). Backstepping boundary control for semilinear parabolic PDEs. IEEE Conf. on Decision and Control, Osaka, Japan. [21] Hasan, A., Aamo, O.-M., and Foss, B. (2013). Boundary control for a class of pseudo-parabolic differential equations. Systems & Control Letters, 62, 63–69. [22] Hasan, A. and Tang, S.-X. (2017). Local exponential stabilization of a Burgers’ PDE-ODE cascaded system. IEEE Conference on Decision & Control, Melbourne, Australia. [23] Hopf, E. (1950). The partial differential equation ut+uux=µuxx. Comm. Pure Appl. Math., 3, 201–230. [24] Iwamoto, T. and Fujimoto, K. (2016). Optimal control of the viscous Burgers equation by the Hopf-Cole transformation and its properties. IFAC Symposium NOLCOS, CA, USA. [25] Jadachowski, L., Meurer, T., Kugi, A. (2014). Backstepping observers for periodic quasi-linear parabolic PDEs. IFAC World Congress, Cape Town, South Africa. [26] Kachroo, P. and Ozbay, K. (1999). Feedback Control Theory for Dynamic Traffic Assignment, Springer, London. [27] Khalil, H. (2002). Nonlinear Systems. Prentice Hall, New Jersey. [28] Krstic, M. (1999). On global stabilization of Burgers’ equation by boundary control. Systems and Control Letters, 37, 123–142. [29] Krstic, M., Magnis, L., and Vazquez, R. (2008). Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE. IEEE Transactions on Automatic Control, 53, 1678–1683. [30] Krstic, M., Magnis, L., and Vazquez, R. (2009). Nonlinear control of the viscous Burgers equation: Trajectory generation, tracking, and observer design. Journal of Dynamic Systems, Measurement, and Control, 131, paper 021012 (8 pages). [31] Krstic, M. and Smyshlyaev, A. (2008). Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Philadelphia. [32] Ladyzenskaja, O. A., Solonnikov, V. A., and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Transl. of AMS, 23, Rhode Island. [33] Liu, W.-J. and Krstic, M. (2000). Backstepping boundary control of Burgers’ equation with actuator dynamics. Systems & Control Letters, 41, 291–303. [34] Laroche, B., Martin, P., and Rouchon, P. (2000). Motion planning for the heat equation. International Journal of Robust and Nonlinear Control, 10, 629–643. [35] Meurer, T. (2013). Control of Higher-Dimensional PDEs: Flatness and Backstepping Designs. Springer, Berlin. [36] Meurer, T. (2013). On the extended Luenberger-type observer for semilinear distributed-parameter systems. IEEE Transactions on Automatic Control, 58, 1732–1743. [37] Meurer, T. and Krstic, M. (2011). Finite-time multi-agent deployment: A nonlinear PDE motion planning approach. Automatica, 47, 2534–2542. [38] Meurer, T. and Kugi, A. (2009). Tracking control for boundary controlled parabolic PDEs with varying parameters: Combining backstepping and differential flatness. Automatica, 45, 1182–1194. [39] Newell, G. F. (1993). A simplified theory of kinematic waves in highway traffic, part I: General theory. Transportation Research Part B, 27, 281–287. [40] Roncoli, C., Papageorgiou, M., Papamichail, I. (2015). Traffic flow optimization in presence of vehicle automation and communication systems - Part II: Optimal control for multi-lane motorways. Transportation Research Part C, 57, 260–275. [41] Servais, E., d’Andrea-Novel, B., and Mounier, H. (2014). Motion planning for multi-agent systems using Gevrey trajectories based on Burgers viscous equation. IFAC WC, Cape Town, South Africa. [42] Smyshlyaev, A. and Krstic, M. (2005). Backstepping observers for a class of parabolic PDEs. Systems and Control Letters, 54, 613–625. [43] Smyshlyaev, A., Meurer, T., and Krstic, M. (2010). Further results on stabilization of shock-like equilibria of the viscous Burgers PDE. IEEE Transactions on Automatic Control, 55, 1942–1946. [44] Tang, S.-X., Camacho-Solorio, L., Wang, Y., Krstic, M. (2017). State-of-Charge estimation from a thermal-electrochemical model of lithium-ion batteries. Automatica, 83, 206–219. [45] Tang, S.-X., Wang, Y., Sahinoglu, Z., Wada, T., Hara, S., and Krstic, M. (2015). State-of-charge estimation for Lithium-ion batteries via a coupled thermal-electrochemical model. American Control Conference, Chicago, Illinois. [46] Treiber, M. and Kesting, A. (2013). Traffic Flow Dynamics: Data, Models and Simulation, Springer, Berlin. [47] Vazquez, R. and Krstic, M. (2008). Control of 1-D parabolic PDEs with Volterra nonlinearities – Part I: Design. Automatica, 44, 2778– 2790. [48] Vazquez, R. and Krstic, M. (2008). Control of 1-D parabolic PDEs with Volterra nonlinearities – Part II: Analysis. Automatica, 44, 2791–2803. [49] Vazquez, R. and Krstic, M. (2010). Boundary observer for outputfeedback stabilization of thermal convection loop. IEEE Transactions on Control Systems Technology, 18, 789–797. [50] Vazquez, R. and Krstic, M. (2016). Bilateral boundary control of onedimensional first- and second-order PDEs using infinite-dimensional backstepping. IEEE Conf. on Decision and Control, Las Vegas, NV. [51] Vazquez, R. and Krstic, M. (2016). Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls. ESAIM: Contr., Optim., & Calculus of Variat., 22, 1078–1096. 17